Guideline for Offshore Structural Reliability - DNVresearch.dnv.com/skj/OffGuide/Jacket_Examples.pdf · report: Guideline for Offshore Structural Reliability Analysis - Application

TECHNICAL REPORT

DET NORSKE VERITAS

JOINT INDUSTRY PROJECT

GUIDELINE FOR OFFSHORE STRUCTURALRELIABILITY ANALYSIS:

EXAMPLES FOR JACKET PLATFORMS

REPORT NO. 95-3204

TECHNICAL REPORT

DET NORSKE VERITAS

JOINT INDUSTRY PROJECT

GUIDELINE FOR OFFSHORE STRUCTURALRELIABILITY ANALYSIS:

EXAMPLES FOR JACKET PLATFORMS

REPORT NO. 95-3204

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:

6 September 1996Organisational 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 ProjectClient ref.:

Rolf SkjongProject No.:

22210110

Summary:

This report documents two case studies on jacket structure reliability analysis, supporting thereport: Guideline for Offshore Structural Reliability Analysis - Application to JacketPlatforms, in which experience and knowledge on application of probabilistic methods tostructural assessment are comprised and advice on probabilistic modelling and structuralreliability analysis of jacket structures is given.

The two case studies involving probabilistic response analyses of jacket structures are afatigue failure limit state (FLS) and a total collapse limit state (ULS).

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 - Application to Jacket Platforms, DNVReport no. 95-3203.

Report No.:

95-3204Subject Group:

P12 Indexing termsReport title:

Guideline for Offshore Structural ReliabilityAnalysis:Examples for Jacket Platforms

structural reliability

jacket platforms

environmental loads

capacity

Work carried out by:

Gudfinnur Sigurdsson and Espen Cramer 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:

05.09.96Rev.No.:

01Number of pages:

108 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: Examples for Jacket Plattforms Page No. 5

DNV Report No. 95-3204 Introduction

Sigurdsson,G and E. Cramer “Guideline for Offshore Structural Reliability Analysis- Examples for Jacket Platforms”, DNVReport 95-3204

Table of Contents

1 INTRODUCTION ___________________________________________________________ 7

1.1 Objective________________________________________________________________________ 7

1.2 Failure Modes ___________________________________________________________________ 7

2 DESCRIPTION OF CONSIDERED NORTH SEA JACKET ________________________ 8

2.1 General _________________________________________________________________________ 8

3 JACKET FATIGUE LIMIT STATE ___________________________________________ 10

3.1 General ________________________________________________________________________ 10

3.2 Limit State Formulation __________________________________________________________ 113.2.1 The SN-Fatigue Approach ______________________________________________________________ 11

3.2.1.1 General_________________________________________________________________________ 113.2.1.2 Uncertainty in SN-Curves __________________________________________________________ 123.2.1.3 Fatigue Damage Model ____________________________________________________________ 123.2.1.4 Limit State Formulation ____________________________________________________________ 13

3.2.2 The FM-Approach for Fatigue Assessment _________________________________________________ 133.2.2.1 General_________________________________________________________________________ 133.2.2.2 Crack Growth Rate________________________________________________________________ 143.2.2.3 Crack Size over Time______________________________________________________________ 143.2.2.4 Limit State Formulation ____________________________________________________________ 153.2.2.5 Uncertainty in FM-fatigue Approach __________________________________________________ 15

3.2.3 Inspection Updating___________________________________________________________________ 15

3.3 Load and Response Modelling _____________________________________________________ 173.3.1 General ____________________________________________________________________________ 173.3.2 Sea State Description__________________________________________________________________ 18

3.3.2.1 Main wave directions ______________________________________________________________ 183.3.2.2 Wave scatter diagram______________________________________________________________ 193.3.2.3 Wave spreading function ___________________________________________________________ 213.3.2.4 Wave spectrum model _____________________________________________________________ 233.3.2.5 Uncertainty Modelling _____________________________________________________________ 23

3.3.3 Global Structural Analysis______________________________________________________________ 243.3.3.1 General_________________________________________________________________________ 243.3.3.2 Wave Load Calculation:____________________________________________________________ 253.3.3.3 Structural Analysis: _______________________________________________________________ 253.3.3.4 Uncertainty in Global Structural Analysis ______________________________________________ 27

3.3.4 Local Stress Calculations_______________________________________________________________ 283.3.4.1 General_________________________________________________________________________ 283.3.4.2 Local Stress _____________________________________________________________________ 293.3.4.3 Uncertainties in Local Stress Calculation_______________________________________________ 31

3.3.5 Stress Range Distribution ______________________________________________________________ 32




3.4 Results_________________________________________________________________________ 343.4.1 General ____________________________________________________________________________ 343.4.2 Deterministic SN-Fatigue Analysis _______________________________________________________ 343.4.3 Probabilistic SN-Fatigue Analysis ________________________________________________________ 393.4.4 Probabilistic FM-Fatigue Analysis: _______________________________________________________ 433.4.5 Inspection Updating - Inspection Planning _________________________________________________ 44

4 TOTAL STRUCTURAL COLLAPSE LIMIT STATE______________________________ 47

4.1 General ________________________________________________________________________ 47

4.2 Limit State Formulation __________________________________________________________ 47

4.3 Load and Response Modelling _____________________________________________________ 484.3.1 Load Modelling ______________________________________________________________________ 484.3.2 Long-term Joint Environmental Model ____________________________________________________ 504.3.3 Annual Extreme Sea-state (Storm)________________________________________________________ 544.3.4 Extreme Wave Height _________________________________________________________________ 544.3.5 Hydrodynamic Loading ________________________________________________________________ 55

4.4 Capacity Model _________________________________________________________________ 58

4.5 Numerical Results _______________________________________________________________ 61

5 REFERENCES ____________________________________________________________ 68

6 APPENDIX A: WAVE ENVIRONMENT DESCRIPTION _________________________ 72

7 APPENDIX B: EIGENMODES OG STRUCTURAL RESPONSE ___________________ 75

8 APPENDIX C: PROBAN INPUT FILE : ULS APPLICATION _____________________ 79

9 APPENDIX D: FORTRAN ROUTINES : ULS APPLICATION _____________________ 83




1 INTRODUCTION

1.1 ObjectiveThe objective of the example part of the Guideline for Offshore Structural Reliability Analysisfor the structure types jacket, TLP and jack-up, is to

• exemplify the use reliability analyses to selected failure modes for the specified structure type.

Two case studies are in the following carried out for a North Sea jacket structure in order toillustrate the probabilistic approach for assessing the structural integrity. Both examples arebased on a Level-III reliability analysis procedure, where the joint probability distribution of theuncertain parameters are applied in the computation of the estimated failure probability.

For completeness, some of the text from the report, Guideline for Offshore Structural ReliabilityAnalysis - Application to Jacket Platforms (DNV 1995b) has been repeated herein.

1.2 Failure ModesThe two failure modes considered are:

• the fatigue limit state (FLS) for failure of a critical joint in the jacket structure, where crackgrowth initiating from the joint weld is considered

• the ultimate limit state (ULS) for failure of the jacket structure, where total collapse of thejacket structure is considered.


DNV Report No. 95-3204 Description of Considered North Sea Jacket


2 DESCRIPTION OF CONSIDERED NORTH SEA JACKET

2.1 GeneralThe selected North Sea jacket structure analysed in the FLS and ULS case studies is an eightlegged jacket located at 107 m water depth. The choice of this particular structure is motivatedfrom the degree of structural redundancy of this structure, believed to be typical for North Seajackets.

In the structural model only the load bearing structure is included in the analysis, i.e. the top-sideand the risers are not accounted for directly. The top-side permanent loads and live loads areincorporated through nodal and element masses at the top level of the structure.

The main characteristics of the jacket platform are given in Table 2.1. The applied FEM model isillustrated in Figure 2.1.

Table 2.1 Main characteristics of considered jacket structure

Water depth 107. m

Topside dead load 48.47 MN

Topside live load 289.80 MN

Number of elements 504

Number of nodal points 211


DNV Report No. 95-3204 Description of Considered North Sea Jacket


Figure 2.1 Applied Structural model


DNV Report No. 95-3204 Jacket Fatigue Limit State


3 JACKET FATIGUE LIMIT STATE

3.1 GeneralAs discussed in DNV (1995b), the probabilistic fatigue analysis is divided into the followingsteps:

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

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, the analyses 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 realisable fatigue evaluation of a jacket structure, it is necessary tointroduce some simplifying assumptions in the modelling:

• For a short term period (a few hours) the sea surface is considered as a realisation of a zero-mean stationary Gaussian process. The sea surface elevation is (completely) characterised bythe frequency spectrum, which for a given direction of wave propagation, is described by twoparameters, the significant wave height HS and a characteristic period like the spectral peakperiod TP or the zero-mean up-crossing period TZ .

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

• In order to apply the frequency domain approach for assessing the structural response, thewave loading on structural members must be linearised and the structural stress response isassumed to be a linear function of the loading, i.e. the structural and material models arelinear.

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

The influence and consequence of the following modelling aspects are discussed in detail in theforthcoming;

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

• The effect of the linearisation of the wave loading is of significance for some structures, andthe influence of performing the linearisation at different sea-states is investigated. The study isbased on a stochastic linearisation techniques for three different sea states.

• The influence of applying two different commonly applied parametric expressions fordefinition of the SCFs.




3.2 Limit State Formulation

3.2.1 The SN-Fatigue Approach

3.2.1.1 GeneralSN-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 SN-curves are based on a statistical analysis of experimental data. The data are presented aslinear or piecewise linear relations between log10S and log10N, where the design SN-curves areobtained applying the mean value minus two times the standard deviation of the spreading forlog10N.

log log log10 10 10

0

N K m S

N K SS s

m

= −

= ⋅

�

��

��>

−or

where

N number of cycles to failure for stress range S

K parameter in the SN-curve

m the inverse slope of the SN-curve

s0 endurance limit

The numerical values for the relevant parameters are summarised in Table 7.10 in DNV (1995a).For tubular joints, the T-curve, DNV CN 30.2 (DNV 1984b) is recommended for modelling thefatigue capacity. In air, the T-curve has m=3, which changes to m=5 for N N K S m= = ⋅ = ⋅0 0

71 10 .For cathodically protected structures in seawater the T-curve has m=3 and a cut-off value atN N= = ⋅0

82 10 .

The stress range levels below the endurance limit do not contribute to fatigue damage providedthe joint is situated in a region with sufficient cathodic protection. The endurance limit cannot,however, be relied upon if the cathodic protection is insufficient.

The fatigue strength of welded joints is dependent on the plate thickness, t, with decreasingfatigue strength with increasing thickness. For the T-curve, the reference thickness t is 32mm.For other thicknesses, a modification of the T-curve is applied,

log log log log10 10 10 10 04 32

N Km t

m S S s= − ⋅ �

��

�

�� − ⋅ >,

or

N t K Sm m= ⋅ ⋅− −( / ) /32 4 , S s> 0




where t is the thickness in mm through which the potential fatigue crack will grow. The factor( / ) /t m32 4− is denoted the thickness-effect factor.

3.2.1.2 Uncertainty in SN-CurvesThe uncertainties associated with a description in terms of empirical SN-curves are accounted forby considering a stochastic S-N relation. The parameters of the deterministic linear or bilinearrelations are treated as random variables. In the current application, the T-curve for cathodicprotection in seawater is applied where the inverse sloop m is modelled as deterministic and K isassumed Log-Normal distributed with the following characteristics:

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

= ⋅ = ⋅= = ≤= =∞ >

539 10 335 103

12 12

1 0

2 0

. .

The cut-off level N0 is modelled as Normal distributed with

[ ] [ ]E N CoV N08

02 10 010= ⋅ = .

The uncertainties associated with the fatigue capacity ∆ for random loading is modelled asunbiased Normal distributed with a CoV of 20%.

[ ] [ ]E CoV∆ ∆= =10 0 20. .

3.2.1.3 Fatigue Damage ModelThe accumulated fatigue damage is computed from the representative stress distribution and theS-N capacity model. The accumulated damage depends on the number and magnitude of thelocal 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 ≥ ∆ , where the fatigue capacity ∆ =1 , as the SN-curves are originallyderived from constant amplitude loading.

For a variable amplitude loading, the value of the Miner's sum at failure will typically be randomdue to the inherent randomness in the stress history and the potential influence of sequenceeffects.

For offshore structures, the number of stress cycles resulting in fatigue failure is typically large.The Miner's summation then contains so many load terms that this inherent uncertainty can be




neglected and the accumulated damage can be represented by the expected value of m'th momentof the local stress response process.

3.2.1.4 Limit State FormulationThe limit state function applied in the reliability analysis is defined as,

g x D( ) = −∆

where the random variable ∆ describes general uncertainty associated with the fatigue capacityand D is the accumulated fatigue damage.

The accumulated damage is defined as the expected accumulated damage per stress cycle timesthe number of stress cycles over the considered time period T,

D T Dlong term cycle= ⋅ ⋅ν 0,

where ν 0,long term is the mean number of stress cycles per time unit and Dcycle is the expecteddamage per stress cycle.

The expected damage per stress cycle is dependent on the local stress range response process andthe associated SN-curve, and will for the Weibull distributed long term stress range responseprocess be,

DK

Am

B

S

A KA

m

B

S

Acycle

mB

mB

= +�

��

�

��

�

��

�

�� + +

�

��

�

��

�

��

�

��

11

11

2

2 2 0 0γ ; ;Γ

A and B are distribution parameters in the Weibull distribution,

( ) ( )[ ]F s s AsB= − −1 exp /

and ( )γ ⋅ ⋅; and ( )Γ ⋅ ⋅; are the incomplete gamma functions, respectively.

3.2.2 The FM-Approach for Fatigue Assessment

3.2.2.1 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, there exist ameasurable quantity which reflects the degree of fatigue accumulation and that is the size of thedeveloped fatigue crack.

Applying the developed crack size as a measure for the fatigue damage, the extent of fatiguedamage on the structure between the initial condition (design) and the failure condition can berelated to a physical measurable parameter. The degree of accumulated fatigue damage in a jointcan then be assessed based on the outcome of structural inspections determining the size ofobserved fatigue cracks.




3.2.2.2 Crack Growth RateThe basis for most fracture mechanics descriptions of crack growth is a relationship between theaverage increment in crack growth during a load cycle and the range ∆K of the parameter K.The parameter K is called the stress intensity factor because its magnitude determines theintensity or magnitude of the stresses/strains in the crack tip region. The influence of externalvariables, i.e. the magnitude and type of loading and the geometry of the cracked body, ismodelled in the crack tip 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 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 all the boundaries, i.e. therelevant dimensions of the structure (width, thickness, crack size, crack front curvature etc.).

The crack growth rate in the crack depth and length direction, a and c, is defined through twocoupled differential equations,

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

∆; ( )

where the material parameters CA and CC may differ due to the general triaxial stress field. Thematerial property m depends mainly on the fatigue crack propagation, assumed to be independentof the crack size, both in the depth and surface directions.

3.2.2.3 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 of the coupled differential equations, and numerical solution proceduresmust be applied.

However, assuming a fixed aspect ratio a c/ , for illustration purposes only, the crack size(depth) over a time period with N stress cycles can be expressed as

( ) [ ]da

Y aC N t E S

m ma

aN m

π0

� = ⋅ ⋅( ) ∆

where a0 is the size of the initial crack. The m'th moment of the stress range response for theWeibull stress range distribution is,

[ ]E S AmB

m m∆ Γ= ⋅ +�

��

�

��1




3.2.2.4 Limit State FormulationThe reliability assessments for fatigue crack growth is expressed as a limit state formulation. Thefailure criteria is defined as,

a aC N− ≤ 0

where aC is the critical crack size defined as through the thickness cracking and aN is the depthof the developed crack after N stress cycles.

The safety margin M is defined as,

M a aC N= −

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

P P MF = ≤( )0

3.2.2.5 Uncertainty in FM-fatigue ApproachThere is uncertainty associated with the modelling of the FM-fatigue approach, both with respectto the initial fatigue quality and the fatigue crack growth material parameters.

The initial fatigue quality is a material and manufacturing property, thus representing materialand process defects such as inclusions, as well as damage caused during fabrication andinstallation which is not detected by quality control. In the example application, the initial fatiguequality is expressed through the depth and length of the initial flaw.

In the example application, the initial crack depth is assumed as Exponential distributed withmean value 0.11 mm and a fixed initial aspect ratio (a/c) which is varied in a parameter study.

The fatigue crack growth material parameters are dependent on the location of the consideredstructural details. In the analysis, only details exposed to sea water are considered, and the fatiguematerial parameters are modelled as,

m: Fixed with value 3.5

lnC: Normal distributed with mean value -31.01 and standard deviation 0.77.

The units for the lnC parameter is Newton and mm.

3.2.3 Inspection UpdatingIn-service inspection is performed in order to assure that existing defects in the structure do notexceed maximum tolerable sizes during the service life. The in-service inspections are commonlycarried out applying Non-Destructive Examination (NDE), where the reliability of the NDE isdescribed by its ability to detect a defect as a function of the size of the defect, and by theuncertainty associated with the sizing of an identified defect. The effect of inspection updating onthe estimated fatigue reliability of the structures is dependent on the target reliability level andthe detection ability of the particular NDE method.




The target reliability for fatigue failure depends on the consequence of failure and is brieflydiscussed in DNV (1995b). In the current example the following procedure for assessment of thetarget reliability for inspection planning is followed:

• calculate the fatigue live of the component, Tlife (deterministic assessment using coderequirements)

• define the design fatigue factor for no access for inspection and repair, λ fatigue (depending onthe classification of the component based on damage consequence) e.g. equal to 10 forsubstantial consequences

• define the time for the first inspection as TT

insplife

fatigue1− =

λ. E.g. in the example given in section

3.4.5, Tlife =80 years, the consequence of failure is evaluated to be substantial andλ fatigue =10. , resulting in T insp1 8− = years (see Figure 3.9)

• define the target reliability as the reliability at service time T insp1− . E.g. in the example given insection 3.4.5, βtarget =32.

The detection ability for the NDE method is defined as a function of a defect size, throughProbability of Detection (POD) curves.

In DNV (1995a) typical POD curves for different inspection scenarios are presented. The curvesare 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 the application example, the influence of inspection updating is accounted for in theestimation of the fatigue reliability of the structure over the service life.

In Table 3.1, typical values for x0 and b for different inspection scenarios are given. Thecorresponding POD curves are shown in Figure 3.1.




Table 3.1 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

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 3.1 POD curves for different inspection scenarios.

3.3 Load and Response Modelling

3.3.1 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.

3.3.2 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 is assigned to all of the wave-statistics defined by eachassigned scatter diagram.

3.3.2.1 Main wave directionsThe main wave direction denotes the middle direction for each of the sectors. The analysis isonly performed for waves in these discrete directions. The sector numbering and main wavedirections are shown in Figure 3.2.




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( , )θθ2

- 9 0

- 4 5

0

4 5

90

θ

Reference direction(Global X-axis)

Figure 3.2 (a) Applied 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 . The wave directions and thecorresponding discrete probabilities are shown in Appendix A.

3.3.2.2 Wave scatter diagramA bi-variate discrete form of the wave scatter diagram is applied. The scatter diagram givesthe occurrence frequency of a discrete number of combinations of ( HS ,TZ ), where 106 seastates are used to describe the sea environments, see Table 3.2 (confer Appendix A).

For the oceanographic area considered no directional-dependent wave statistics is available,and the same wave scatter diagram is consequently applied for all wave directions, see Table3.2.




Table 3.2 Applied wave scatter diagram, given as a relative occurrence frequency of100,000 observations.

Hs Tz (sec.)

(m) 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5

0.5 2 102 1962 3562 1703 522 382 233 110 25 20 10 0

1.5 0 40 3218 10066 9638 4847 2253 1067 640 244 101 41 0

2.5 0 4 165 2788 9877 7438 3642 1346 338 161 23 11 0

3.5 0 0 3 151 2059 7426 3866 1390 364 115 50 16 2

4.5 0 0 0 2 147 2372 4357 1570 507 76 57 23 4

5.5 0 0 0 0 12 165 2431 1550 500 112 44 16 8

6.5 0 0 0 0 0 8 310 1408 394 149 42 13 7

7.5 0 0 0 0 0 0 25 486 365 93 42 13 4

8.5 0 0 0 0 0 0 2 47 279 54 23 12 4

9.5 0 0 0 0 0 0 0 6 88 50 11 3 1

10.5 0 0 0 0 0 0 0 1 6 36 12 2 1

11.5 0 0 0 0 0 0 0 0 2 9 7 1 0

12.5 0 0 0 0 0 0 0 0 0 1 4 1 0

13.5 0 0 0 0 0 0 0 0 0 0 1 0 0

The discrete values of the ( HS ,TZ ) data are approximated by a joint log-normal distribution.The cumulative distribution function is

( )F h t h tH T s z

s zS Z

, log , log ,= − −�

��

�

��Φ µ

σµ

σρ1

1

2

2

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

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 3.3)

( )( )

F t ht h

T H z s

z s

Z S|

log log,=

− + −�

��

�

��

−

�

�

��

�

�

��

Φµ ρ σ

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 3.3 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

3.3.2.3 Wave spreading functionThe 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 assumed that the wave energy is spread over a set of directions in a region of π / 2 onboth sides of the main direction. The function is selected in such a way that it gives higherweights to the directions closer to the main direction. For a long crested sea the wave energyspreading is not introduced by definition. The wave energy spreading function for a givenmain 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 anon-negative number. Figure 3.4 shows the directional function for different values of N . Forlarge values 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 3.4 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, and 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.

In the current study, the influence of varying degree of wave spreading is investigated.




3.3.2.4 Wave spectrum modelA one side Gamma spectrum is applied in the analysis. The one sided Gamma spectrum isuniquely 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 πξζ

ξζ

ξ

ξ

ξ

Γ

Γ

BTZ

=�

��

�

��

−�

��

�

��

−�

��

�

��

21

3

2

2

πξζ

ξζ

ξ

ξ

ξ

Γ

Γ

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

3.3.2.5 Uncertainty ModellingUncertainties in the wave description for the following quantities are considered:

• main wave direction,

• significant wave height HS

• mean zero crossing period TZ

• wave scatter diagram

• wave energy spreading function

• one-dimensional wave spectrum

Other types of uncertainties, such as uncertainties in the still water level, the effect of currents,and the distribution of the main wave directions, are not explicitly included as these uncertaintiesare judged not to be of major importance. These uncertainties are instead implicitly accounted forby introducing modelling uncertainties.

Uncertainty in main wave directionThis uncertainty is accounted for explicitly by defining probability density functions for themain wave directions. By conditioning on the main wave direction in the computation offatigue damage, the overall damage is obtained by a weighted integration of the conditionaldamage over all possible directions, weighted w.r.t. the probability density for the wavedirection.

The probability distribution of the main wave direction is given as a discrete distributionwhere eight different directions are considered, see Appendix A.




Uncertainty in wave scatter diagramA continuous joint log-normal distribution is applied to represent the long-term wave scatterdiagram for ( HS , TZ ). The uncertainty in the diagram is included by considering uncertaintyin the distribution parameters for the joint distribution. The underlying distribution function is

defined by the estimated distribution parameters $

,$

,$

,$

,$µ σ µ σ ρ1 1 2 2

�

��

�

�� . These best estimates are

next multiplied by random variables X1 - X5 , yielding the distribution parameters:

µ µ σ σ µ µ σ σ ρ ρ1 1 1 1 1 2 2 2 3 2 2 4 5= = = = =$

;$

;$

;$

;$

X X X X X

X1- X5 are defined as mutually independent unbiased Normal distributed random variables,having Coefficients of Variations according to Table 3.3.

Table 3.3 Uncertainty measures (CoV) in the modelling of the bi-variate Log-normallong-term distribution of the wave environment.

VX1V X2

V X3V X4

V X5

0.10 0.06 0.10 0.06 0.02

Uncertainty in wave energy spreading functionThe sensitivity of the fatigue life to the wave spreading is investigated through a parameter study.

Uncertainty in wave spectrum model

A one-dimensional wave spectrum ( )Sη ω , defined by the one sided Gamma spectrum, isapplied. The uncertainty in the wave spectrum is accounted for by modelling the spectralparameters ( )ξ ζ, as Normal distributed random variables. The mean values and coefficient ofvariations are given by

[ ] [ ]

[ ] [ ]

E CoV

E CoV

ξ ξ

ζ ζ

= =

= =

5 0 05

4 0 05

.

.

It should be noted that the mean values of the spectral parameters, ( )ξ ζ, , i.e.ζ = 4 and ξ = 5,correspond to the PM spectrum.

3.3.3 Global Structural Analysis

3.3.3.1 GeneralThe structural response to wave induced loading is determined by the use of finite elementmethods (FEM). This includes modelling of the structural stiffness, the damping (only for




dynamic analysis), the influence of marine growth, the stiffness from the foundation and thewave induced loading.

The structure is modelled using PREFRAME (DNV (1984a)). The finite element model is anidealised representation of the real structure, and the following simplifications are made:

• Smaller eccentricities are not modelled.

• Eccentricities in the joints are not modelled

• The soil/structure interaction is simplified by assuming fixed support in the soil

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

• The jacket is analysed as a frame with members connected at idealised rigid joints. In realitythe joints are flexible, and on the global level, the joint flexibility is known to have someinfluence on the response, (Bouwkamp et al. (1980), Fessler and Spooner (1981), UEG(1984)). The joint flexibility affects the bending moments in braces, the axial forcedistribution and the natural frequencies.

3.3.3.2 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 the wave amplitude. The linear wave theoryis based on the assumption that the wave height is much smaller than both the wave lengthand the 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= +C D C Dd n n m nρ ρπ

2 4

2 &

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

The drag and inertia coefficients are difficult to measure under realistic flow conditions andlarge uncertainties are related to their magnitude, (Sarpkaya and Isaacson (1981). However, tosimplify the analysis these coefficients are assumed to be constant for all the structural forcesegments, and the following values are applied,

Cd = 0 7. Cm = 2 0.

The wave load calculation is performed using WAJAC (DNV (1992)).

3.3.3.3 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 eight different wave directions, analysing the structure subjected to waves of differentangular frequencies. For the computation of the transfer function, 49 different wave periods(frequencies) are selected, consisting of (seconds)

(1.00, 1.40, 1.60, 1.70, 1.80, 1.90, 1.95, 2.00, 2.10, 2.20, 2.35, 2.50, 2.65, 2.70, 2.90, 3.15,3.50, 3.70, 3.90, 4.10, 4.50, 4.70, 4.80, 5.10, 5.25, 5.40, 5.55, 5.80, 6.00, 6.30, 6.45, 6.60,6.80, 6.90, 7.20, 7.60, 7.90, 8.30, 8.50, 8.80, 9.25, 0.50, 13.00, 15.00, 18.00, 21.00, 25.00,30.00, 35.00)

The wave periods have been selected in order to adequately define the transfer function overthe expected range of wave energy. Special care has been given in the modelling of thetransfer function for wave periods close to the eigenperiods of the structure (the three largesteigen-periods are calculated to be in the area 2.0 - 1.90 sec., see Appendix B).

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, astochastic linearisation is applied, where the response is computed using the non-linear forceand then linearised in one sea state, (Borgman (1967)).

In the following, the influence of applying the following three different sea states in thestochastic linearisation has been considered:

H T

H T

H T

S Z

S Z

S Z

= =

= =

= =

35 6 5

5 5 7 5

8 5 9 5

. . sec.

. . sec

. . sec

m

m

m

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 the load response isapproximately linear 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 transfer function are calculated using the DNV-SESAM program modules, i.e. WAJAC(DNV (1992)) and SESTRA (DNV (1991)).

In order to study the effect of dynamics in the fatigue analysis, both the quasi-static and thedynamic transfer functions have been calculated. In Figure 3.5, the quasi-static and dynamictransfer functions for the axial force in one of the most fatigue critical braces in the jacketstructure are shown.




0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0Frequency (rad/sec.)

0.0E+0

5.0E+5

1.0E+6

1.5E+6

Mem

ber f

orce

tran

sfer

func

tion

Quasi-static analysis

Dynamic analysis

Figure 3.5 Quasi-static and dynamic transfer functions for an axial force in one of themost critical braces in the structure with respect to fatigue failure.

3.3.3.4 Uncertainty in Global Structural AnalysisUncertainty in wave load models:

The uncertainty/bias introduced using Airy's wave theory in the fatigue analysis is believed tobe insignificant for most structures, water depths and wave climates of interest.

A full probabilistic model of the loading is very complicated due to the complex interrelationsbetween the parameters in Morrison's equation. Although the prediction of the drag and inertiacoefficients at a given time is rather uncertain, the prediction of the average value over alonger period is associated with less uncertainty and supports the selection of a relativelysimple probabilistic model.

Uncertainties in the load calculations due to the effect of currents, the relative particlevelocities, marine growth, free surface effects and tidal effects are included throughuncertainty modelling of the transfer function.

Uncertainty in structural analysis:The uncertainty/bias introduced in the derivation of the transfer function could be related tothe significant wave height HS , e.g. by multiplying the calculated transfer functions ( )Hcalc ωobtained in the structural analysis by a 2nd order polynomial function of 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 the calculated transfer function, Xb

and Xc should be set equal to zero (0.0) and the mean value of Xa should be set equal to one(1.0). Herein, no uncertainties are assigned to Xa , Xb and Xc .

Uncertainty in structural behaviour:The uncertainties in the structural behaviour are due to the uncertainties in both the structuraland soil-structure stiffness properties, the damping properties and the model uncertaintiescoming from the mathematical idealisation of the structure. The latter model uncertainty isbelieved to be rather small and is included in the uncertainty model in connection with thecomputation of local stresses.

The uncertainty related to the stiffness properties results in uncertainty associated with theestimated modal eigenfrequencies and the corresponding mode shapes. The uncertainty in thedamping mainly influences the dynamic amplification. These uncertainties are included in themodelling of the transfer functions as

( ) ( )H X HH applω ω= ⋅

where ( )Happl ω is the transfer function given above and X H represents the modellinguncertainties in the structural behaviour. X H is defined as a normally distributed randomvariable with,

[ ] [ ]E X CoV XH H= =10 01. .

3.3.4 Local Stress Calculations

3.3.4.1 GeneralThe 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. API (1991, 1993), AWS (1984), DoE (1984)), use differentdefinitions of the SCFs. The hot-spot stress is the present application defined as: the greatestvalue around the brace/chord intersection of the extrapolation to the weld toe of the geometricstress distribution near the weld. This hot-spot stress definition incorporates the effects of theoverall geometry but omits the stress concentrating influence of the weld itself which results in alocal stress concentration.

Parametric formulas exist only for simple joints with members in one plane. In real structuresone finds very few of these simple joints. No reference is made to sign, location, or orientation ofthe stress values representative of the SCFs. Little information is available on SCFs in




overlapping and/or multiplanar and/or grouted and/or ring stiffened joints. An inherentshortcoming of the available SCF equations for K-joints is that they were derived under balancedaxial forces or self-equilibrated bending moments. Experimental work performed by Dijkstra andde Back (1980), shows that the SCFs are highly dependent on the type of loading on theindividual member.

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

In the current analysis, the SCFs are calculated using FRAMEWORK ( DNV (1993b)) where theparametric formulas proposed by Efthymiou (Efthymiou (1985, 1988) have been applied. Theresults obtained using these formulas are compared to results obtained using the parametricformulas proposed by Kuang (Kuang et al. (1977)).

3.3.4.2 Local StressIn 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 is then calculated as:

σ hot ax ipbipb

local opbipb

localSCFN

ASCF

M

Iz SCF

M

Iy= ⋅ − ⋅ ⋅ − ⋅ ⋅' '

whereN the axial force in the braceMipb the In-Plane Bending moment in the braceMopb the Out-of-Plane Bending moment in the braceA the cross section areaI the moment of inertia for the pipe sectiony zlocal local' , ' the co-ordinates of the stress point relative to the section centre of gravity, in

the in-plane/out-of-plan axis systemSCFax SCF for axial stressSCFipb SCF for in-plan bending stressSCFopb SCF for out-of-plan bending stress

The SCFs are calculated for eight locations around each brace/chord intersection. See DNV(1993b) for a more detailed description of the hot-spot stress calculations.

Based on the transfer functions H Fiη ω( ) for all section forces (i.e. i=1: axial force, i=2: IPBmoment and i=3: OPB moment), the cross section properties and the SCFs, the spectral densityof 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 = '

In Figure 3.6, the stress response spectral density function for one of the most critical hot-spots inthe considered jacket structure with respect to fatigue failure is shown. In the figure, the spectraldensity functions are obtained using both the quasi-static and the dynamic transfer functions. Thestress spectral densities are presented using both the PM and JONSWAP unidirectional wavespectra for wave direction θ=0deg. , HS =55. m and TZ =75. sec .

It is seen that the PM and the JONSWAP wave spectra give approximately identical results, butthat the dynamic amplification for this selected sea state is of significance for the stress response.




0.0 1.0 2.0 3.0 4.0Frequency (rad/sec.)

0.0

10.0

20.0

30.0

40.0

50.0

60.0

Stre

ss s

pect

ral d

ensi

ty

0.00 1.00 2.00 3.00 4.00Frequency (rad/sec.)

0.00

10.00

20.00

30.00

40.00

50.00

60.00

Stre

ss s

pect

ral d

ensi

ty

Applying PM wave spectru m

Quasi-static response

Dynamic response

Applying JOSWAP wave spectrum

Quasi-static response

Dynamic response

Figure 3.6 Stress spectral densities, obtained using both quasi-static and dynamictransfer functions, and two different wave spectra, the Pierson-Moskowitzspectrum (PM) and the JONSWAP spectrum.

3.3.4.3 Uncertainties in Local Stress CalculationIt is a common practice to check the fatigue life at 8 points along the brace/chord intersection.However, the parametric formulas for the SCFs do not provide information about the variationof SCFs along the intersection brace/chord, which leads to uncertainties in the estimation ofthe maximum resulting hot-spot stress over the intersection due to axial force, in-plane andout-of-plane bending moments.




Because the position of the hot-spot is not known, a common procedure is to simply add themaximum stresses derived separately from the axial and bending loads in order to estimate thehot-spot stress, which will usually result in conservative estimates. The degree ofconservatism depends on the actual geometry and the contribution of bending stresses to thetotal hot-spot stress.

Uncertainties associated with the modelling of the SCFs are defined at two levels:

• One single common uncertainty factor is assigned to all the stress concentration factors.This uncertainty measure accounts for fabrication inaccuracies and approximations made inthe stress calculation or joint classification, and is modelled through X SCF Common− .

• Individual uncertainty factors are in addition assigned to the SCFs for each degree offreedom, i.e. for axial load X SCF ax−

, in-plan bending moment X SCF ipb−and out-of-plan

bending moment X SCF opb−.

All uncertainties in the SCFs are modelled as independent unbiased Normal distributedrandom variables with coefficient of variation as presented in Table 3.4.

Table 3.4 Uncertainty measures (CoV) on the modelling of the SCFs.

Variable CoV

XSCF Common− 0.05

X SCF ax− 0.20X SCF ipb− 0.20

X SCF opb− 0.20

3.3.5 Stress Range DistributionCalculation of the fatigue life involves estimation of the total number of stress cycles and the"crack driving force", i.e. the m'th moment of the stress range distribution, [ ]E S m . In the

example, these measures are established by the peak counting method. implying that the numberof stress cycles is equal to the number of up-crossings of the mean level, and that the stress rangeis defined as two times the peak value.

The fatigue damage can be derived directly through a weighted summation of the accumulatedfatigue damage within each sea-state the structure is exposed to over the lifetime. Alternatively,it can be derived from an estimated long-term stress range distribution, where the long-termstress range distribution is calibrated to a weighted sum of the stress range distribution withineach short term sea-state. The latter approach is desirable due to computational efficiency whenthe probabilistic fatigue evaluation of the structure involves inspection updating.




The direct approach and the stress-range calibration approach results in comparable fatiguedamage estimates, as can be seen from Figure 3.7, where the calibration of the long-term stressrange distribution is based on the approach presented in DNV (1995b).

0 10 20 30 40 50 60 70 80 90Service time (years)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

Rel

iabi

lity

inde

x

Calibtation of Weibull Load Model

Sum of short term damages

Weibull load model

Weibull parameters: E[Ln(A)] = 1.7 ; CoV[Ln(a)]=0.12 B=0.83

Figure 3.7 Calibration results: The fitted Weibull stress range distribution is calibratedto the original long term stress range distribution in order to obtain the samefatigue reliability as the original load model.




3.4 Results

3.4.1 GeneralThe most fatigue sensitive structural elements are identified through a frequency domain SN-fatigue analysis (stochastic fatigue analysis). The load-response model from Section 3.3 isutilised.

A stochastic linearisation is applied where three different sea states are considered for thelinearisation,:

H TH TH T

S Z

S Z

S Z

= == == =

35 6 555 7 58 5 9 5

. . sec.

. . sec

. . sec

mmm

Insignificant differences in the calculated fatigue lives are obtained for the three sea states, whichindicate that the loading on the considered structure is dominated by linear inertia forces.

The base case for the fatigue analysis is using transfer functions obtained from a dynamicanalysis, the parametric equations proposed by Efthyminu for deriving the SCFs, the PM seaspectrum, and the assumption of long crested (uni-directional) sea.

The results for the base case are compared with results from equivalent fatigue analyses wheredifferent common modelling alternatives are compared. The following variations are considered:using a quasi-static approach for deriving the transfer functions, Kuang's model for deriving theSCFs, JONSWAP sea spectrum and the influence of different levels of short crested sea.

In the following, the fatigue lives for the different members in one of the most critical joints inthe jacket structure, joint 589, is considered in the comparison analysis.

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

3.4.2 Deterministic SN-Fatigue AnalysisIn Table 3.5, the geometrical characteristics for the selected joint with associated joint membersare shown.

In Table 3.6-9, the derived fatigue lives for the defined base case and the respective comparisonanalyses are presented. The base case is presented first in each table. It is seen that the differentmembers of the considered joint have quite comparable fatigue lives, except for member 152where a fatigue life more than seven times the other members is obtained.

In table 3.6, the base case results are compared with the corresponding fatigue results applying aquasi-static, instead of a dynamic, approach for deriving the transfer functions. It is seen that thequasi-static approach, the derived fatigue lives are, as expected, longer than for the dynamicapproach (by a factor 1.3-4.0). However, the unconservative estimates obtained by the quasi-static approach are not valid for member number 152, indicating that the dynamically derivedfatigue life for member 152 can have been influenced by numerical inaccuracies in the analysis.Joint 152 is therefore not considered further in the comparison analysis, but the results obtainedare shown.

In table 3.7, the fatigue lives obtained for the base case using the Efthyminu empirical model forderiving SCFs are compared with the equivalent model using the Kuang model for the SCFs. The




obtained fatigue lives are longer with a factor of 2-7 for the Kuang model (not considering joint152)

In table 3.8, the fatigue lives obtained for the base case using the PM wave spectrum arecompared with the equivalent results using the JONSWAP wave spectrum. Only a minorincrease in the fatigue lives is observed using the JONSWAP spectrum.

In table 3.9, the fatigue lives are derived for different degrees of wave spreading. It is seen that,as expected, the estimated fatigue lives are increasing with the level of wave spreading, but thatthis increase in only minor.

Table 3.5 Selected joint with associated members considered in the fatigue life comparisonanalysis 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 thickness (m) Chord

Brace

0.065

0.025

0.065

0.045

0.065

0.040

0.065

0.045

0.065

0.030

0.065

0.060

Joint type KTT YT KTK KTK KTK KTK




Table 3.6 Comparison of dynamic and quasi-static fatigue analysis results.

Joint Number 589

Member Number 123 152 372 373 401 402

Dynamic Analysis: Fatigue life (years)

Cycles per year

Hot spot Position

Hot spot Number

246

1.22 ⋅107

Brace-side

10

1780

1.17 ⋅107

Chord-side

13

80

9.85 ⋅106

Chord-side

4

87

8.90 ⋅106

Chord-side

10

158

9.88 ⋅106

Chord-side

4

63

1.20 ⋅107

Chord-side

16*

Quasi-Static Analysis: Fatigue life (years)

Cycles per year

Hot spot Position

Hot spot Number

323

5.16 ⋅106

Brace-side

10

228

4.58 ⋅106

Chord-side

13

114

4.99 ⋅106

Chord-side

4

119

5.27 ⋅106

Chord-side

10

273

5.22 ⋅106

Chord-side

4

247

6.14 ⋅106

Chord-side

10*

*Please note that depending on the applied model, different hot-spot positions or hot spotnumbers can be found to be critical.




Table 3.7 Comparison of fatigue analysis results using the Efthyminu and the Kuangempirical models for deriving the SCFs.

Joint Number 589

Member Number 123 152 372 373 401 402

Efthyminu: Fatigue life (years)

Cycles per year

Hot spot Position

Hot spot Number

246

1.22 ⋅107

Brace-side*

10

1780

1.17 ⋅107

Chord-side

13*

80

9.85 ⋅106

Chord-side

4

87

8.90 ⋅106

Chord-side

10

158

9.88 ⋅106

Chord-side

4

63

1.20 ⋅107

Chord-side

16

Kuang: Fatigue life (years)

Cycles per year

Hot spot Position

Hot spot Number

1910

1.22 ⋅107

Chord-side*

10

951

1.11 ⋅107

Chord-side

16*

215

1.01 ⋅107

Chord-side

4

266

9.24 ⋅106

Chord-side

10

478

1.02 ⋅107

Chord-side

4

135

1.22 ⋅107

Chord-side

16

Efthyminu SCF’s: SCFax

SCFipb

SCFopb

4.943

3.152

7.445

16.88

4.287

8.843

4.958

3.162

7.782

4.220

2.899

6.925

3.264

2.500

4.900

6.607

4.253

7.883

Kuang SCF’s: SCFax

SCFipb

SCFopb

3.320

2.500

4.238

20.40

3.866

7.699

3.830

2.634

4.796

3.325

2.500

3.514

2.500

2.500

2.541

5.543

3.656

5.961





Table 3.8 Comparison of fatigue results using the PM and the JONSWAP wave spectrum.

Joint Number 589

Member Number 123 152 372 373 401 402

PM Spectrum: Fatigue life (years)

Cycles per year

Hot spot Position

Hot spot Number

246

1.22 ⋅107

Brace-side

10

1780

1.17 ⋅107

Chord-side

13

80

9.85 ⋅106

Brace-side

4

87

8.90 ⋅106

Brace-side

10

158

9.88 ⋅106

Brace-side

4

63

1.20 ⋅107

Brace-side

16*

JONSWAP Spectrum Fatigue life (years)

Cycles per year

Hot spot Position

Hot spot Number

304

1.22 ⋅107

Brace-side

10

1890

1.15 ⋅107

Brace-side

13

88

9.84 ⋅106

Brace-side

4

91

8.80 ⋅106

Brace-side

10

190

1.00 ⋅107

Brace-side

4

79.6

1.18 ⋅107

Brace-side

10*





Table 3.9 Comparison of fatigue results using different levels of wave spreading.

Joint Number 589

Member Number 123 152 372 373 401 402

One Dimensional: Fatigue life (years)

Cycles per year

246

1.22 ⋅107

1780

1.17 ⋅107

80

9.85 ⋅106

87

8.90 ⋅106

158

9.88 ⋅106

63

1.20 ⋅107

Spread cos8(θθθθ): Fatigue life (years)

Cycles per year

301

1.23 ⋅107

2170

1.14 ⋅107

91

9.79 ⋅106

95

9.18 ⋅106

178

1.03 ⋅107

70

1.22 ⋅107


Cycles per year

324

1.23 ⋅107

2440

1.12 ⋅107

98

9.81 ⋅106

99

9.31 ⋅106

191

1.05 ⋅107

72

1.23 ⋅107


Cycles per year

328

1.23 ⋅107

2690

1.11 ⋅107

105

9.84 ⋅106

103

9.41 ⋅106

204

1.06 ⋅107

72

1.23 ⋅107

3.4.3 Probabilistic SN-Fatigue AnalysisFor the probabilistic SN-fatigue analysis, the load and response model described in section 3.3 isapplied. The fatigue damage is calculated using the Miners sum and SN T-curve, and the limitstate formulation defined in section 3.2.1 is applied. The stress range distribution within each seastate is assumed Rayleigh distributed.

In table 3.10, the applied modelling parameters and uncertainty measures discussed in section 3.2and section 3.3 are summarised.

The estimated fatigue reliability over the service life of the structure is shown in Figure 3.8applying transfer functions derived from a dynamic and quasi-static analysis. It is observed thatthe quasi-static approach gives approximatly 50% longer estimated fatigue live at the samefatigue reliability level than the dynamic approach.

The results are given both using the advanced load model description within each sea state andfor the calibrated long term Weibull stress distribution. It is seen that the calibrated long termstress distribution is able to represent the more advanced stress response model very accurately.The calibrated long term Weibull stress response model is applied in the FM-fatigue analysis forthe inspection planning.

The significant importance factors obtained from the fatigue reliability analysis are shown inTable 3.11 for the advanced load model and in Table 3.12 for the calibrated long term loadmodel. It should be noted that these estimated importance factors vary only slightly over theservice live of the structure. It is seen, as expected, that the uncertainties associated with the




modelling of the fatigue capacity (the SN-curve) and the estimation of the local stress response(the SCFs) are the most important uncertainty contributions to the fatigue reliability assessment.




Table 3.10 Model parameters and uncertainty measures applied in the probabilistic SN-fatigue analysis. Units in [N, mm]

Uncertainty Source Distribution Mean CoV

SN-curve: log10K m1

m2

N0

NormalFixedFixedNormal

12.6613.0∞

2·108

1.96·10-2

----

0.10

Miner’s Sum: ∆ Normal 1.0 0.20Wave scatter diagram: X1 ( [ ]E HS )

X2 ( [ ]E TZ )

X3 ( [ ]Std HS )

X4 ( [ ]Std TZ )

X5 ( [ ]ρ H TS Z, )

NormalNormalNormalNormalNormalFixed

1.01.01.01.01.01.0

0.100.060.100.060.02

--

Wave spectrum (PM) ξ ζ

NormalNormal

5.04.0

0.050.05

Structural response: X H Normal 1.0 0.10Local stresses (SCFs) X SCF Common−

X SCF ax−

X SCF ipb−

X SCF opb−

NormalNormalNormalNormal

1.01.01.01.0

0.050.200.200.20




0 10 20 30 40 50 60 70 80 90Service time (years)

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

Rel

iabi

lity

inde

xWeibull parameters: E[Ln(A)] = 1.7 ; CoV[Ln(a)]=0.12 B=0.83

Dynamic analysis

Advanced load model

Weibull load model

Quasi-Static analysis

Advanced load model

Weibull load model

Weibull parameters: E[Ln(A)] = 1.61 ; CoV[Ln(a)]=0.12 B=0.75

Figure 3.8 The estimated SN-fatigue reliability of the joint as function of the service life

Table 3.11 The importance factors in the reliability analysis, using the advanced stressmodel

Variable log10K ∆ X1 X2 XSCF Com− X SCF ax− X H

Importance fact. (%) 40-45 5-10 1-5 4-8 ≈3 20-25 10-15

Table 3.12 The importance factors in the reliability analysis, using the Weibull stress model

Variable log10K ∆ ln( )A

Importance fact. (%) 45-50 5-10 40-45




3.4.4 Probabilistic FM-Fatigue Analysis:The main advantage for using a Fracture Mechanics (FM) fatigue model compared to a SNfatigue model, is that the outcome of structural inspections can be incorporated in the FM fatiguemodel. For the SN fatigue model, the inspection results can not be used directly for updating ofthe degree of fatigue damage accumulation.

In order to utilise the inspection results also for the SN fatigue approach, the FM fatigue model iscalibrated directly to the original SN fatigue model. Applying a probabilistic FM fatigue analysis,the initially estimated uncertainty model can be modified based on the additional informationgained about the system over the service life, e.g. the inspection results of no crack detection orcrack detection (with possible crack size assessment). The additional information from thefatigue crack inspections is then applied directly for updating of the estimated fatigue reliabilitylevel over the remaining service life after the inspection.

The applied FM fatigue approach is based on the 2-dimensional crack propagation model, wherethe crack growth rate is calculated using Paris equation. The geometry function is calculatedusing the parametric equations proposed by Raju and Newman (Raju and Newman (1981,1986)). In order to account for the local stress concentrations due to the weld toe, the stressmagnification factor M k given in PD6493 (BSI 1991) is applied.

The parameters describing the crack propagation in the fracture mechanics approach are fittedsuch that the probability of having a through thickness crack as function of time applying thefracture mechanics approach is comparable with the equivalent result applying the SN-approach.The fitting is based on a least-squares analysis.

Three fitting parameters are proposed for this purposes:

1. the expected value of the initial aspect ratio crack-depth/half-crack-length ratio [ ]E a c0 0/

2. the expected value of the model uncertainty for the geometry function [ ]E Ymodel

3. the standard deviation of the material parameter ln( )Ca , i.e. Std Ca(ln( ))

Using all three parameters in the calibration will not provide a unique fitting solution. Theexpected value of the aspect ratio [ ]E a c0 0/ is therefore pre-selected and the other twoparameters are calibrated in order to match the probabilistic FM fatigue results with theequivalent probabilistic SN fatigue results.

Three different values are selected for the expected value of the initial aspect ratio, i.e.[ ]E a c0 0 01 0 3 0 5/ . , . , .= . With basis in brace number 372, the calibrated parameters in the FM

fatigue model for the different selections of the expected value of the initial aspect ratio areshown in Table 3.13.

The results from the probabilistic fatigue analysis are shown in Table 3.14, both for the originalSN-fatigue analysis and the three different FM-fatigue analyses. It is seen that all three cases givea good approximation to the SN reliability results. In the subsequent updating analysis, the fitteddistribution parameters obtained from Case-1 are applied. Similar analysis using the parametersfrom Case-2 and Case-3 have been performed, where empirically the same results as for Case-1are obtained. This indicates that the reliability updating is not sensitive to the choice ofparameter set as long as the FM-results give a good fit to the SN-results.




The applied uncertainty modelling for the probabilistic FM analysis is shown in Table 3.15.

Table 3.13 Three different selection of parameters to be applied in the FM-fatigueanalysis.

FM-fatigue [ ]E a c0 0/ [ ]E YmodelStd Ca(ln( ))

Case-1 0.10 0.938 0.443

Case-2 0.30 0.958 0.446

Case-3 0.50 0.990 0.471

Table 3.14 Comparison of the fatigue reliability results from the SN-fatigue analysis andthe calibrated FM-fatigue analysis.

Reliability Index: ββββ = - ΦΦΦΦ-1 [ PF ]Time SN-fatigue FM-fatigue

Case-1FM-fatigue

Case-2FM-fatigue

Case-33 (years) 4.30 4.30 4.28 4.28

8 (years) 3.21 3.21 3.23 3.23

20 (years) 2.23 2.23 2.26 2.26

50 (years) 1.30 1.30 1.29 1.29

3.4.5 Inspection Updating - Inspection PlanningApplying the FM fatigue model, it is possible to derive the influence of an inspection on theestimated fatigue failure probability for specified inspection quality. The inspection quality isexpressed through the likelihood of detecting an existing crack as a function of the size (length)of the crack.

In the estimation of the inspection plan for the jacket in the case study, the target inspectionreliability level is defined as the reliability level of the structure at one tenth of the fatigue designlife, being after 8 years of service with reliability index β = 32. . In Table 3.15, the applieduncertainty model for the probabilistic FM fatigue analysis is shown together with the appliedinspection model.

In figure 3.9, it is shown how the estimated fatigue reliability of the inspected joint increase afterthe inspection when no fatigue cracks were detected. Based on the specified target reliabilitylevel, a new structural inspections will have to take place after 15 years of service.

In Table 3.16, the importance factors for the modelled stochastic variables are presented prior tofirst inspection, and after one and two inspections. It is observed that the modelling detectionability of the last inspection is having a large influence of the estimated reliability level.




Table 3.15 Model parameters and uncertainty measures applied in the probabilistic FM-fatigue analysis. Units in [N, mm, s]

Uncertainty Source Distribution Mean CoVFM model a0

a c0 0/ acritic

ln CA*

m T0

ExponentialNormalFixedNormalFixedFixed

0.110.1065.0

-31.013.50.0

1.00.10

--0.014

----

Inspection Model 2 1cpod−

2 2cpod−

POD curve x0 2 95= .x0 2 95= .

b = 0 905.b = 0 905.

Stress Response Model ln A B rmemb stress.

**

ν 0

NormalFixedNormalFixed

1.70.840.200.312

0.12--

0.10--

Geometry Function Ymodel

YMK***

t plate

lattachm****

NormalNormalFixedFixed

0.9381.065.080.0

0.100.10

----

* C CCm

A= ⋅−11.** Ratio of membrane stress to total stress.*** Model uncertainty on the M K factor given in PD-6493 (BSI (1991))**** Length of weld attachment considered PD-6493 (BSI (1991))




0 5 10 15 20 25 30Service time (years)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0R

elia

bilit

y in

dex 1.

Insp

ectio

n at

T=8

yea

rs

2. In

spec

tion

at T

=15

year

s

Target reliability

3. In

spec

tion

at T

=24

year

s

Reliability over Time

No inspection performed

No crack found at 1st inspection

No crack found at 2nd inspection

Figure 3.9 Estimated updated reliability against fatigue failure based on the inspectionoutcome of no crack detection.

Table 3.16 The importance factors for the fatigue reliability analysis over the servicelife, accounting for the number of structural inspections.

Importance Factors (%)

Variable: ln( )A ln( )CA YmodelYMK a 0 2 1cpod− 2 2cpod−

Prior to Insp., t = 8 years 58-60 22-24 10-12 4-5 2-3 not-used not-used

After 1.insp., t =15 years 30-36 10-15 5-8 1-2 < 1 40-50 not-used

After 2.insp., t = 24 years 40-45 14-16 8-10 < 1 2-4 6-7 20-25


DNV Report No. 95-3204 Total Structural Collapse Limit State


4 TOTAL STRUCTURAL COLLAPSE LIMIT STATE

4.1 GeneralIn reliability analyses of jacket structures for structural collapse (ULS), the uncertaintiesassociated with the determination of the hydrodynamic loading are generally much greater thanfor the estimation of the collapse capacity. As the problem formulation is load driven, a propermodelling of the hydrodynamic loading is therefore important.

In DNV (1995b), a procedure for probabilistic modelling of the structural collapse is discussed.In the following, an example of a reliability analysis of total collapse of a jacket structure isconsidered.

The selected North Sea jacket structure is the same as for the FLS study presented in the previoussection, being an existing North Sea structure located at 107 m water depth. The choice of thisparticular structure is motivated from the degree of structural redundancy, believed to be typicalfor North Sea jackets.

In the structural model only the load bearing part of the jacket is included directly in themodelling. The top-side permanent loads and live loads are incorporated through nodal andelement masses at the top level of the structure.

The main characteristics of the jacket platform are presented in section 2.

4.2 Limit State FormulationThe limit state function for a given load direction θ , is expressed as,

( )g cc l cc lθ θ θ θ, = −

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

In the current example, the jacket collapse capacity is obtained by performing a non-linear push-over analysis with the computer program USFOS (USFOS (1996)). The hydrodynamic loads onthe jacket are calculated by WAJAC (DNV (1992)).




4.3 Load and Response Modelling

4.3.1 Load ModellingThe annual extreme base-shear loading acting of the structure, Lθ , is expressed as,

L X L LL jack jack windθ θ θ= ⋅ +−

Ljack θ is the calculated hydrodynamic loading with associated model uncertainty X L jack− θ , andLwind θ is the wind loading. Possible hydrodynamic loading on the deck structure is not includedin the analysis.

Ljack θ is generally stochastic due to randomness (aleatory uncertainties) in the sea state, e.g.uncertainties associated with the description of the wave height, the wave period and the currentspeed.

For a given environmental situation, the maximum base shear loading over the wave cycle,Ljack θ , is obtained by time stepping a Stokian 5’th wave through the structure, where thehydrodynamic loads on the members are calculated using Morison’s equation. The calculatedcurrent speed is added vectorially to the wave induced particle speed. In the example analysis, theMorison’s coefficients are assumed deterministic, with the following values

C CD M= =0 70 16. .

The model uncertainty X L jack− θ is introduced to account for the overall uncertainties associatedwith the applied load models for a given environmental condition (e.g. the uncertainties in themodelling of the hydrodynamic parameters and the marine growth). The model uncertainty isassumed equal and fully correlated for all load directions Θ .

As discussed in DNV (1995b), numerous studies have been performed on the quantification ofmodel uncertainties associated with the load prediction. However, no general recommendationcan at present be made based on these studies.

In the current example, the global model uncertainty, X L jack− , is modelled as unbiased normaldistributed with CoV=20%.

The load directions Θ are discretised into eight directions with occurrence frequency asspecified in Table 4.1 (see also Appendix A and Figure 2.1), where for a given environmentalload direction, the wave, wind and current directions are assumed to be identical.

The wind loading is for all load directions defined by

L C Uwind wind wθ= ⋅ 2

where Uw is the 1-hour mean wind speed and Cwind is the wind coefficient, dependent on thestructural configuration. In Table 4.2, the applied wind coefficients are specified for differentwind directions.




Table 4.1 Load direction occurrence frequency

Load direction θ : NW W SW S SE E NE N

( )PΘ θ 0.11 0.13 0.20 0.22 0.10 0.07 0.05 0.12

Table 4.2 Wind coefficient Cwind for the different wind directions

Wind direction: NW and SE SW and NE N, S, W and E

Cwind [MN/(m/s)2] 0.0020 0.0030 0.0025

The sea water level is assumed deterministic and equal to the mean water level.

The current speed profile is given by

V zV

ZZ

Z Z

V Z Z

cC MWL

C MWL

( )

/

=⋅�

��

�

��

<

≥

�

�

−

−

0

1 70

0

where VC MWL− is the current speed at the mean water level (MWL), Z0 is the MWL depth(107m) and Z is the distance from the sea-bed. The current profile is shown in Figure 4.1.

Current Speed (m/sec)0.00

20.00

40.00

60.00

80.00

100.00

120.00

Dist

ance

from

sea-

bed

MWL

Figure 4.1 Current speed profile




4.3.2 Long-term Joint Environmental ModelIn order to establish the probability distribution of the annual maximum base-shear load actingon the structure for a given direction, a joint distribution of the corresponding wave height(Stokes 5th), wave period and the current speed is needed. A proper description of theenvironmental condition is therefore required, i.e. the long term variations of the sea statecharacteristics and the short term description of the environmental condition (wave, current,wind) for a given sea state. In the following the model is outlined:

The long term description of the environmental conditions for each load direction θ is definedby the joint probability density function,

( ) ( ) ( ) ( ) ( )f h t v u f t h f v h f u h f hHs Tz Vc U w s z c w Tz Hs z s Vc H s c s U w Hs w s H s s, , , = ⋅ ⋅ ⋅

I.e., given Hs (and Θ ), the random variables Tz , Vc , and Uw are assumed to be mutuallyindependent. For the oceanographic area in which the applied jacket is located, no directionalinformation for the wave statistic is available, and the same wave scatter diagram is applied forall directions. The discrete values of the ( HS ,TZ ) data are shown in Figure 3.1.

The conditional distribution of the current speed at MWL, given the significant wave height, isassumed to be Normal distributed with mean and standard deviation (unit: m/sec.),

[ ]E V h hc s s= + ⋅0 2 0 04. . [ ]Std V hc s =01.

The conditional distribution of the 1-hour mean wind speed, given the significant wave height isassumed to be Normal distributed with mean and standard deviation (unit: m/sec.) (Dalane andHaver (1995));

[ ]E U h hw s s= + ⋅55 18. . [ ]Std U h hw s s= − ⋅49 0 26. .

The conditional distribution of the mean crossing period Tz , given the significant wave height isassumed to be Lognormal distributed (unit: sec.) (Bitner-Gregersen and Haver (1989));

( ) ( )f t ht

tT H z s

z

zz s

= −−�

�

��

�

�

��

12 2

2

2π σµ

σexp

ln

where

[ ] [ ]µ σ= =E T h Var T hz s z sln | ln |2

and the following functions define the estimates for µ and σ2

( )µ= + ⋅a a hsa

1 23 ( )σ2

1 2 3= + ⋅ ⋅b b b hsexp

The parameters a a a b b1 2 3 1 2, , , , and b3 are determined by calibration to observed data, i.e. thescatter diagram for the structural location considered, using least squares technique. Applyingthe scatter diagram in Table 3.1, the following results are obtained

a1 122= . a2 0382= . a3 0 456= .




b1 0 0623= . b2 0 255= . b3 0 297=− .

The fitted and observed results are shown in Figure 4.2 - 4.4. Figure 4.2 and figure 4.3 show theobserved and fitted mean and standard deviation of ( )ln Tz . In figure 4.4 the observed and fittedmarginal distribution of Tz are shown. It should be noted that the marginal distribution of Tz

does not follow any known distribution type and is in figure 4.4 plotted on Weibull paper only tocompare the observed data and the fit. As can be seen, the obtained fits are in good accordancewith the observed data.

The same marginal distribution of the significant wave height is assumed for all directions, and a3-parameter Weibull distribution is applied;

( ) ( )f hh h

H ss s

s=

−⋅ − −�

��

��

��

��

−β γα

γα

β

β

β1

exp

The distribution parameters α (scale parameter), β (shape parameter) and γ (locationparameter) are established by non-linear least square fitting to the observed data, where thefollowing results are obtained for the considered location,

α =2 23. β=137. γ=0 281.

In figure 4.5 the observed and fitted marginal distributions of Hs are shown. As can be seen, theobtained fit is in good accordance with the observed data.




Figure 4.2 Observed and fitted mean value of ( )ln Tz .

Figure 4.3 Observed and fitted standard deviation of ( )ln Tz .




Figure 4.4 Observed and fitted marginal distribution of Tz (plotted on Weibull paper)

Figure 4.5 Observed and fitted marginal distribution of Hs




4.3.3 Annual Extreme Sea-state (Storm)The annual largest loading is assumed to occur when the largest wave, in the largest annual storm(i.e. annual largest Hs ) passes the structure. The annual largest significant wave height,Hs max, θ , for a given load direction θ , is derived from the long term distribution of the arbitrarysignificant wave heights as

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

N

s s

storm

, max , max ,maxΘ Θθ θθ

=

where Nstorm θ is the number of annual storms in direction θ , obtained by

( )N N Pstorm stormθ θ= ⋅ Θ

where N storm =1460 is the total number of 6-hours storms in one year and ( )PΘ θ is the directionoccurrence probability previously specified.

4.3.4 Extreme Wave HeightThe distribution of the largest wave height for a given stationary sea state or storm (i.e. specifiedHs and Tz ) out of N wave wave cycles may be obtained by

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

N

s s

wave

max , max , max, ,Θ Θθ θ=

where the number of waves N wave is estimated as

NT hwave

z s

= τ

, max

and τ = 6 hours (21,600 sec.) is the specified duration of a storm.

The Weibull distribution is applied for the wave height distribution in a stationary sea state(Forristall (1978) ),

( )F h h hhH H s

ss , , , expΘ θ δ

α

= − − ⋅�

��

�

��

��

��

1

where the parameters are defined as α = 213. and δ=2 26. .

The distribution of the largest wave height in a given storm with H Hs s= ,max is then obtained as




( )F h h N hhH H s wave

ssmax , max

max ,maxmax

,max

, exp expΘ

θ δα

= − ⋅ − ⋅�

��

�

��

�

�

�

��

��

��

�

��

��

The wave period corresponding to the largest wave height is defined as,

T TH zmax= ⋅1.2

Applying the model described above, the wave characteristics shown in Table 4.3 are obtained,where the subscript “ x% ” denotes the annual probability of exceedance, i.e. 50% is the medianvalue, 1% is 100 years condition and 0.01% is the 10,000 years condition.

Table 4.3 Obtained wave characteristics

Annual maximum significantwave height Hs,max (m)

Annual maximum wave heightHmax (m)

Direction Nstorm θ Hs,max,50% Hs,max,1% Hs,max, .0 01% Hmax,50% Hmax,1% Hmax, .0 01%

All 1460 10.1 13.8 17.4 18.3 25.8 33.4

NW 161 7.9 11.9 15.7 14.5 22.2 29.9

W 190 8.1 12.1 15.9 14.8 22.5 30.1

SW 292 8.5 12.4 16.2 15.6 23.2 30.8

S 321 8.6 12.5 16.3 15.7 23.4 31.0

SE 146 8.0 11.8 15.7 14.3 22.1 29.7

E 102 7.5 11.5 15.4 13.7 21.5 29.1

NE 73 7.1 11.2 15.1 13.0 20.9 28.6

N 175 8.0 12.0 15.8 14.6 22.4 30.0

4.3.5 Hydrodynamic LoadingKnowing the wave height and period, and the current speed profile, the hydrodynamic loading iscalculated using WAJAC. In order to make the reliability analysis more efficient, a responsesurface has been established for all eight directions. The response surface provides the total baseshear on the jacket as function of the wave height, wave period (5th order Stokes wave) and thecurrent speed at MWL.

Figure 4.6 shows the total hydrodynamic base shear loading for the eight considered directions asfunction of the wave height, for specified wave period and current speed. Figure 4.7 shows thebase shear loading for the NW direction as function of the wave height for different wave periodand current speed combinations.




20 25 30 35 40Wave height (m) (5th order Stokes)

50

100

150

200

250

300

Bas

e Sh

ear (

MN

)

Current speed Vc(MWL) = 0.8 m/secWave period T= 15 sec

Load direction

NW

SE

SW

E

20 25 30 35 40Wave height (m) (5th order Stokes)

50

100

150

200

250

300

Bas

e Sh

ear (

MN

)

Load direction

W

E

S

N

Current speed Vc(MWL) = 0.8 m/secWave period T= 15 sec

Figure 4.6 The total base shear due to hydrodynamic loading as function of wave height

It is observed in Figure 4.7 that the base shear increases exponential with the wave height, butthat the current speed and the wave period also have some significance.

For a given wave period and current speed, the base shear L jack can be approximated with a

simple empirical relation as L a Hjackb= ⋅ . As an example, for the NW direction with Vc =0 8.

m/sec and T = 15 s, the parameters are a = 0.20 and b = 1.93. The obtained fit is shown togetherwith the original derived wave loading results in Figure 4.8.




20 24 28 32 36 40Wave height (m) (5th order Stokes)

0

50

100

150

200

250

300B

ase

shea

r (N

M)

Vc : Current speed at MWL (m/sec) ; T : Wave period (sec.)

Vc = 0.0 T = 13

Vc = 0.0 T = 15

Vc = 0.0 T = 17

Vc = 0.5 T = 13

Vc = 0.5 T = 15

Vc = 0.5 T = 17

Vc = 0.8 T = 13

Vc = 0.8 T = 15

Vc = 0.8 T = 17

Figure 4.7 The total base shear hydrodynamic loading as function of the wave height.

20.00 25.00 30.00 35.00 40.00

Wave height (m)

50

100

150

200

250

300

Bas

e sh

ear (

MN

)

Calculated by WAJAC

Power-fit : log(L)=B*log(H)+A

Fitted Equation:log(L) = 1.92731 * log(H) + -1.6047Alternate equation:L = (X**1.92731) * 0.20095

L : Base shear in MNH : wave height in m

Load direction : NW, Vc(MWL) = 0.8 m/sec., T = 15 sec.

Figure 4.8 Analytical fit of the base shear as function of the wave height




4.4 Capacity ModelThe collapse capacity CCθ , for a given load direction θ , is expressed as

CC X CCCC model calcθ θ= ⋅−

CCcalc θ is the calculated capacity and XCC model− is the overall modelling uncertaintyassociated with the estimated collapse capacity. The model uncertainty XCC analysis− , accountingfor possible inconsistencies in the analysis, is here assumed 1.0.

The model uncertainty XCC model− accounts for all the uncertainties in the modelling of thestructural non-linear FEM analysis (geometry and material properties) and is assumed identicaland fully correlated for all load directions. XCC model− is taken as unbiased Normal distributedwith CoV=0.15.

The calculated base shear capacities CCcalc θ and the load level λ for the eight load directionsare shown in the table below, where the load level λ defines the scaling factor on thehydrodynamic loading resulting in total collapse of the structure. The load condition consideredis a 5th order Stokian wave profile with H = 26 m and T = 14.0 s, vC MWL− = 1.0 m/s and uw = 0.0.

Table 4.4 The calculated structural collapse capacity for the eight load directions (MN).

Direction θ NW W SW S SE E NE N

Load-level λ 2.52 2.98 3.21 3.49 2.71 3.52 3.22 3.01

Capacity CCcalc 194.9 246.8 301.5 297.1 222.1 289.3 299.0 247.6

Since, the same long term joint environmental distribution is applied for all load direction, theload-level λ in Table 4.4 indicates that the NW and SE directions are the most criticaldirections. It should, however, be noted that the load-level parameter in the table can not berelated directly to the probability of failure, as the occurrence probabilities for the different loaddirections are not equal. E.g. P SW P NWΘ Θ( ) ( )θ θ≡ ≈ ⋅ ≡2 , i.e. the probability of an extremeload condition is higher for the SW than the NW direction (see Table 4.1) and the non-linearrelationship between the base shear and the wave height and period varies for the differentdirections.

The calculated capacity CCcalc θ is in general a function of the hydrodynamic load profileapplied in the push-over analysis. In order to evaluate the sensitivity of the capacity to theapplied load profile, 15 different profiles have been studied. The results are shown in Figure 4.9.It is seen that for larger waves, the push-over capacity is almost independent of the applied waveheight but that a dependence exist for the wave period. The obtained relationship between thecapacity and the wave heights and wave periods are included in the reliability analysis.




20.0 22.0 24.0 26.0 28.0 30.0Wave heights (m)

190.0

200.0

210.0

220.0

230.0

240.0

Col

laps

e C

apac

ity (M

N)

Wave comming from North-WestCollapse Capacity for different wave periods

(solid line show the applied fit)

Wave periode = 19.0 sec.



Collapse Capacity as function of different wave profiles(wave-height and period - 5th order Stoke wave)

Figure 4.9 Collapse capacity as function of wave height and wave period




0.0 0.1 0.2 0.3 0.4 0.5Displacement (m)

0.0E+0

5.0E+7

1.0E+8

1.5E+8

2.0E+8

2.5E+8

Bas

e sh

ear (

N)

H = 20.0 mT = 19.0 secT = 16.5 secT = 14.0 sec

0.0 0.1 0.2 0.3 0.4 0.5Displacement (m)

0.0E+0

5.0E+7

1.0E+8

1.5E+8

2.0E+8

2.5E+8

Bas

e sh

ear (

N)

T = 19.0 secT = 16.5 secT = 14.0 sec

H = 24.0 m

0.0 0.1 0.2 0.3 0.4 0.5Displacement (m)

0.0E+0

5.0E+7

1.0E+8

1.5E+8

2.0E+8

2.5E+8

Bas

e sh

ear (

N)

T = 19.0 secT = 16.5 secT = 14.0 sec

0.0 0.1 0.2 0.3 0.4 0.5Displacement (m)

0.0E+0

5.0E+7

1.0E+8

1.5E+8

2.0E+8

2.5E+8

Bas

e sh

ear (

N) T = 19.0 sec

T = 16.5 secT = 14.0 sec

H = 26.0 m H = 28.0 m

0.0 0.1 0.2 0.3 0.4 0.5Displacement (m)

0.0E+0

5.0E+7

1.0E+8

1.5E+8

2.0E+8

2.5E+8

Bas

e sh

ear (

N) T = 19.0 sec

T = 16.5 secT = 14.0 sec

H = 30.0 m

Load-Displacement curves for different wave patternsWave direction : North-West

Figure 4.10 Load-displacement curve for different wave profiles




4.5 Numerical ResultsThe statistical properties of the stochastic variables included in the reliability model aresummarised in Table 4.5. The hydrodynamic parameters Cd and Cm in Morison’s equation andthe marine growth Mg are modelled as deterministic, with the following values:

Cd =0 70. Cm =16.

MgZ Z

Z Z=

> +

≤ +

�

��

��

0 0 2 0

0 01 2 0

0

0

. .

. .

m

m m

where Z0 is the MWL depth (107 m) and Z is the distance from the sea-bed.

PROBAN (DNV (1993a)) is applied for the reliability calculations, and the implementation isshown schematically in Figure 4.11. The derived FORTRAN routines are enclosed in AppendixD.

A response surface for the hydrodynamic loading has been applied, with the current speed atMWL, the wave height and the wave period (5th Stokian wave) as response co-ordinates. AFORTRAN code has been developed in order to define the response surfaces for all the eightload directions automatically, where WAJAC is used for the load calculation. The program codeis shown in Appendix D. The PROBAN input-file for one load direction is given in Appendix C.

Table 4.5 Statistical properties for the stochastic parameters (summary).

Parameter Description Distribution

Hs Significant wave height Weibull distributed :

( )F h hH s

ss

= − − −��

��

��

��

1 exp γ

α

β

α =2 23. β=137. γ=0 281.

Hs max, Annual largest Hs ( ) ( )[ ]F h F hH s H s

N

s s

storm

, max ,max , max=

FH HsWave height distribution ina stationary sea state (storm)

Conditional Weibull distribution

( )F h h hhH H s

ss

= − − ⋅�

��

�

��

��

��

1 exp δ

α

α = 213. δ=2 26. .

FH HsmaxLargest wave height in astationary sea-state ( ) ( )[ ]F h h F h hH H s H H s

N

s s

wave

max max max=

≈ − ⋅ − ⋅�

��

�

��

�

�

�

��

��

��

��

��exp exp maxN h

hwaves

δα




Tz Mean zero-crossing period Conditional Log-Normal distribution

[ ] ( )µ = = + ⋅E T h a a hz s s

aln | 1 2

3

[ ] ( )Var T h b b b hz s sln | exp= + ⋅ ⋅1 2 3

a1 122= . a2 0 38= . a3 0 46= .

b1 0 062= . b2 0 26= . b3 0 30= − .

U Hw s 1-hour mean wind speed Conditional Normal distribution

µ = + ⋅55 18. . hs Std hs= − ⋅4 9 0 26. .

V Hc s Current speed at MWL Conditional Normal distribution

µ = + ⋅0 2 0 04. . hs Std =01.

X L jack− Global model uncertainty inthe hydrodynamic loading

Normal distributed

µ =10. COV = 0 2.

XCC model− Global model uncertainty inthe calculated collapsecapacity

Normal distributed

µ =10. COV = 015.

In Table 4.6, the annual probability of failure (SORM) and the importance factors are shown.

In Table 4.7, the design point values for the stochastic variables are shown. As can be seen, thewind loading is less than 2% of the total loading and can in this example be omitted in theanalysis. For the hydrodynamic loading, the wave loading contributes with about 75% and thecurrent loading with about 25% of the total hydrodynamic loading.




PROBAN

ULSfuncProban function library

Purpose : Calculate hydrodynamic baseshear loading, using response surface.Input : H, T, Vc, Cd, Mg, BS_modelOutput : Base shear loading

BaseShear

Response surface modul, calculatingthe response for given function coordinates.Input : H, T, Vc, Cd, MgOutput : Base shear loading

Response

Data file contaning response surface coordinates and responses.Input : H, T, Vc, Cd, MgOutput : Base shear loading

Data base (file)

Purpose: Modify the calculated collapsecapacity of the jacket as function of H and TInput : H, T, calculated base shear capacity Output : Modified base shear capacity

CollCap

Response surface modul, calculatingthe response for given function coordinatesInput : H, TOutput : Change in the calc. capacity

Resp5D

Data file contaning response surface coordinates and responses.Input : H, TOutput : Change in the calc. capacity

Data base (file)

Proban input file

Figure 4.11 Reliability calculation - implementation overview




Table 4.6 Annual probability of failure and importance factors

Annual failure Importance factors (%)

Load direction probability Env. loading Model uncertainties

Θ Pf annualθ Ljack & Lwind X L jack− XCC model−

NW 4.8·10-5 79 9 12

W 1.5·10-5 79 9 13

SW 1.3·10-5 75 9 16

S 0.6·10-5 79 8 13

SE 2.6·10-5 77 9 14

E 0.2·10-5 80 8 12

NE 0.3·10-5 78 9 14

N 1.3·10-5 79 9 13

All directions 13 10 5⋅ − *

*(series system, no correlation between the different directions)

Table 4.7 Design point values for the stochastic variables

Load Hs Tz H T Vc Uw Lwind L jack X L jack− XCC model−

direction (m) (sec) (m) (sec) (m/sec) (m/sec) (MN) (MN)

NW 14.4 12.3 28.5 14.9 0.80 31.4 2 126 1.23 0.79

W 15.0 12.5 30.3 15.0 0.82 32.5 3 153 1.25 0.77

SW 14.8 12.2 30.8 14.7 0.82 32.2 3 177 1.26 0.74

S 15.7 12.9 32.3 15.4 0.85 33.7 3 179 1.26 0.76

SE 14.5 12.3 28.9 14.8 0.80 31.5 2 137 1.25 0.77

E 15.6 12.8 32.1 15.4 0.85 33.5 3 176 1.25 0.76

NE 14.8 12.2 30.8 14.7 0.82 32.2 3 174 1.27 0.74

N 15.0 12.6 30.5 15.1 0.83 32.6 3 153 1.25 0.77

As shown in Table 4.6, the annual failure probability for the different directions is in the order of10 5− , and the design point for the wave height H shown in Table 4.7 is close to the 10,000 yearscondition for the wave height, confer Table 4.3. This indicates that the inherent uncertainty in thewave height is the dominating parameter in the reliability analysis. The contribution from themodel uncertainties are, however not negligible.




In order to study the importance of the model uncertainties, the uncertainty in the mean zero-upcrossing period Tz and the uncertainty in the current speed at MWL Vc , the following casesare studied:

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

The NW load direction is considered. For given outcome of Hs , the Tz and Vc aremodelled as deterministic, equal to the median value defined in Table 4.5, i.e.

( )( )T a a hz sa= + ⋅exp 1 2

3

V hc s= + ⋅0 2 0 04. .

Results : Pfannualθ = ⋅ −4 6 10 5. i.e. for given outcome of Hs , it follows that Tz and

Vc can be modelled as deterministic (see Table 4.6)

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

Only the NW load direction is considered in the studies. Mean values in the rangeof 0.9-1.1 and CoV in the range of 0.0-0.4 are studied, considering both the Normaldistribution and Log-Normal distribution. The results are shown in Table 4.8 and4.9

Table 4.8 Annual failure probability Pfannualθ as function mean and CoV,

X L jack− Normal distributed

CoV

[ ]E X L jack−0.0 0.1 0.2 0.3 0.4

0.9 0.90·10-5 1.2·10-5 2.2·10-5 4.5·10-5 8.5·10-5

1.0 2.1·10-5 2.7·10-5 4.8·10-5 9.2·10-5 17.1·10-5

1.1 4.8·10-5 5.6·10-5 9.5·10-5 18.1·10-5 28.7·10-5


X L jack− Log-Normal distributed

CoV

[ ]E X L jack−0.0 0.1 0.2 0.3 0.4

0.9 0.90·10-5 1.2·10-5 2.5·10-5 6.5·10-5 16.8·10-5

1.0 2.1·10-5 2.7·10-5 5.8·10-5 12.9·10-5 32.0·10-5

1.1 4.8·10-5 5.7·10-5 10.6·10-5 21.2·10-5 55.0·10-5




As can be seen, the choice of distribution for X L jack− is not significant for theresults, only for unrealistic height CoV some differences are obtained. It is alsoshown that a 10% variation in the mean value (bias), change the failure probabilityby approximately a factor two. In the base case model CoV=0.2 was proposed. Byreducing this value to 0.1, the failure probability will reduce with a factor two.Ignoring the model uncertainty in the calculated hydrodynamic loading will lead tounderestimation of the failure probability by a factor two.

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

Only the NW load direction is considered in the studies. Mean values in the rangeof 0.9-1.1 and CoV in the range of 0.0-0.4 are studied, where XCC model− ismodelled as Normal distributed. The results are shown in Table 4.10.

For CoV < 0.2, the effect of model uncertainty in the calculated collapse capacityfor the failure probability is similar as for X L jack− , but for large CoV (> 0.2) theestimated failure probability increases dramatically.


XCC model− Normal distributed

CoV

[ ]E XCC model− 0.0 0.1 0.15 0.2 0.3 0.4

0.9 4.3·10-5 6.0·10-5 10.3·10-5 26.1·10-5 385·10-5 1950·10-5

1.0 1.8·10-5 2.6·10-5 4.8·10-5 11.2·10-5 301·10-5 1730·10-5

1.1 0.81·10-5 1.2·10-5 2.3·10-5 6.1·10-5 246·10-5 1570·10-5

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

The calculated collapse capacity is obtained by applying the 100 years wavecondition shown in table 4.3, and the median value of the wave period. Only theNW load direction is included in the studies, where following wave condition isobtained;

H = 22.2 m T = 13.3 sec.

Results : Pfannualθ = ⋅ −53 10 5. i.e. the 100 years wave condition can be applied in

the push-over analysis in order to obtain the calculated collapse capacityand the relationship between the capacity and the wave height and periodcan be omitted.

• CASE-5: Effect of ignoring the model uncertainties




X L jack− and XCC model− are set equal to unity in the reliability model.

Results : Pfannualθ = ⋅ −0 7 10 5. i.e. ignoring the model uncertainties in the reliability

model leads to underestimation of the failure probability by a factor 7 (seven).

It should be noted that the results presented in the cases above are in principle only valid for thecurrent example, but it is expected that the results will be similar for other North Sea jackets.


DNV Report No. 95-3204 References


5 REFERENCES

API (1991); API Recommended Practice for Planning, Designing and Constructing FixedOffshore Platforms, American Petroleum Institute RP 2A, Nineteenth Edition, August 1991.

API (1993); API Recommended Practice for Planning, Design and Constructing Fixed OffshorePlatforms - Load and Resistance Factor Design, American Petroleum Institute RecommendedPractice (RP 2A-LRFD), First Edition, July 1993.

American Welding Society (AWS) (1984). Structural welding code, AWS D1.1-84.

Bitner-Gregersen, E.M., and Haver, s: (1989); Joint Long Term Description of EnvironmentalParameters for Structural Response Calculation, Proceedings of the 2nd International Workshopon Wave Hindcasting and Forecasting, Vancouver, B.C., April 25-28, 1989.

Borgman, L.E. (1967); Spectral analysis of ocean wave forces on piling, Journal of Waterwaysand Harbour Division, ASCE, Vol. 93, pp.129-156.

Bouwkamp, J.G. et al. (1980); Effects of joint flexibility on the response of offshore towers,Proc. OTC, Paper 3901, Houston, Texas.

BSI PD6493 (1991); Guidance on methods for assessing the acceptability of flaws in fusionwelded structures, British Standard Institute, 1991.

Chakrabarti, S.K. (1971); Discussion on dynamics of single point mooring in deep water, Journalof Waterways, Harbours and Coastal Eng. Div., ASCE, Vol.97, No. WW3.

Chakrabarti, S.K. (1976); Total forces on submerged randomly oriented tube due to waves, Proc.OTC\, Paper 2495, Houston, Texas.

Dalane, J.I. and Haver, S. (1995); Requalification of an Unmanned Jacket Structure UsingReliability Methods, OTC 7756, Houston, May 1995.

Department of Energy (DoE) (1984a); Background to new fatigue design guidance for steelwelded joints in offshore structures, United Kingdom Department of Energy 1984, London.




Department of Energy (DoE) (1984b); Offshore Installations: Guidance on Design andConstruction, United Kingdom Department of Energy, HSMO, London, April 1984.

Dijkstra, O.D. and de Back, J. (1980); Fatigue strength of tubular T- and X-joints, Proc. OTC\,Paper 3696, Houston, Texas.

DNV (1995a), Guideline for Offshore Structural Reliability Analysis - General, DNV TechnicalReport 95-2018, Det Norske Veritas, 1995.

DNV (1995b), Guideline for Offshore Structural Reliability Analysis - Application for JacketPlatforms, DNV Technical Report 95-3203, Det Norske Veritas, 1995.

DNV (1993a) PROBAN; PROBAN Version 4 - Theory Manual, Det Norske Veritas ResearchReport no. 93-2056, Høvik, Norway 1993.

DNV (1993b) FRAMEWORK, Steel Frame Design - Theoretical Manual, DNV SesamTechnical Report 93-7076, 01-AUG-1993.

DNV (1992) WAJAC, Wave and Current Loads on Fixed Rigid Frame Structures - User’sManual, Det Norske Veritas Sesam AS, technical report 92-7052, May 25, 1992.

DNV (1991) SESTRA; Super Element Structural Analysis - User’s Manual, Veritas SesamSystem A.S, technical report 91-7033, September 1, 1991.

DNV (1984a), PREFRAME, Pre-processor for Frame Structures - User’s Manual, Veritec,technical report 82-6003, Revision 5, October 1, 1984.

DNV (1984b); Fatigue Strength Analysis for Mobile Offshore Units, Classification Notes no.30.2., Det Norske Veritas, August, 1984.

Efthymiou, M. et al. (1985); Stress concentration in T/Y and gap/overlap K joints, Proc.BOSS'85, Delft, The Netherlands.

Efthymiou, M. (1988); Development of SCF formulae and generalised influence functions foruse in fatigue analysis, Shell International Petroleum Mij. B.V., OTJ’88 Resent Developments inTubular Joints Technology, Surrey, UK, 5 October 1988.




Fessler, H. and Spooner, H. (1981); Experimental determination of stiffness of tubular joints,Proc. IOS'81, Glasgow, Scotland.

Forristall, G.Z. (1978); On the Statistical Distribution of Wave Height in a Storm, Journal ofGeophysical Research, Vol. 83, No. C5, 1978.

Hogben, N. et al. (1977); Estimation of fluid loading on offshore structures, Proc. Institution ofCivil Engineers, Vol.63, Part 2, Sept., London, UK.

Kuang, J.G., Potvin, A.B. and Leick, R.D. (1977); Stress concentration in tubular joints, Societyof Petroleum Engineers Journal 5472, August, 1977.

Lalani, M., Tebbett, I.E. and Choo, B.S. (1986); Improved fatigue life estimation of tubularjoints, Proc. OTC, Paper 5306, Houston, Texas.

Morison, J.R., O'Brian, M.P., Johnson, J.W. and Schaaf, S.A. (1950); The force exerted bysurface waves on piles, Petroleum Transactions, AIME, Vol.189, pp.149-154.

Raju I.S and Newman J.C (1981): An empirical stress-intensity factor equation for surfacecrack, Engineering Fracture Mechanics, Vol 15, pp.185-192.

Raju I.S. and Newman J.C. (1986): Stress-intensity factors for circumferential surface cracks inpipes and rods under bending and tension loads, Fracture Mechanics, Vol.17, AST, STP 905, pp.709-805.

Pierson, W.J. and Moskowitz, L. (1964); A proposed spectral form for fully developed wind seasbased on similarity theory of S.A. Kitaigorodskii, Journal of Geophysical Research, Vol.69,Nr.24, December.

Sarpkaya, T. and Isaacson, M. (1981); Mechanics of wave forces on offshore structures, VanNostrand Reinhold Company, New York.

Underwater Engineering Group (UEG) (1984); Node flexibility and its effects on jacketstructure, UEG publication Report UR22, London.

USFOS (1996); USFOS - A Computer Program for Progressive Collapse Analysis of SteelOffshore Structures, SINTEF Report no. STF71 F88039, Dated 1996-01-01.




Wheeler, J.D. (1970); Methods for calculation forces produced by irregular waves, Journal ofPetroleum Technology.


DNV Report No. 95-3204 APPENDIX A: Wave Environment Description


6 APPENDIX A: WAVE ENVIRONMENT DESCRIPTION

****** ****** ****** ****** ** *** ************ ******** ******** ******** *************

** ** ** ** ** ** ** ** ** ** **** ** ** ** ** ** ** ********* ********** ******* ********* ** ** ********* ********* ******* ********** ** ** **

** ** ** ** ** ** ** **** ** ** ** ** ** ** ** ** ** ********** ******** ******** ********* ** ** ******** ****** ****** ****** ** ** ** **

********************************************** ** F R A M E W O R K ** ** Postprocessing of Frame Structures ** **********************************************

Marketing and Support by DNV Sesam

Program id : 2.1-03 Computer : VAX 4000-105ARelease date : 22-JUL-1994 Impl. update : NoneAccess time : 11-AUG-1995 15:32:05 Operating system : VMS V5.5-2H4User id : V24GUIDE CPU id : 1059811820Account : 22210110 Installation : Veritec, V2VAX

Copyright DET NORSKE VERITAS SESAM AS,P.O.Box 300, N-1322 Hovik, Norway

Dir WavSta Prob--------------------------------

0.000 SCAT1 1.100E-0145.000 SCAT1 1.300E-0190.000 SCAT1 2.000E-01

135.000 SCAT1 2.200E-01180.000 SCAT1 1.000E-01225.000 SCAT1 7.000E-02270.000 SCAT1 5.000E-02315.000 SCAT1 1.200E-01

Total probability: 1.00

Hs Tz Prob Typ L-param N-param Gamma SigA SigB WaveSpr----------------------------------------------------------------------5.000E-01 1.500 2.000E-05 P-M NONE5.000E-01 2.500 1.020E-035.000E-01 3.500 1.962E-025.000E-01 4.500 3.562E-025.000E-01 5.500 1.703E-02en000E-01 6.500 5.220E-035.000E-01 7.500 3.820E-03




5.000E-01 8.500 2.330E-035.000E-01 9.500 1.100E-035.000E-01 10.500 2.500E-045.000E-01 11.500 2.000E-045.000E-01 12.500 1.000E-041.500E+00 2.500 4.000E-041.500E+00 3.500 3.218E-021.500E+00 4.500 1.007E-011.500E+00 5.500 9.638E-021.500E+00 6.500 4.847E-021.500E+00 7.500 2.253E-021.500E+00 8.500 1.067E-021.500E+00 9.500 6.400E-031.500E+00 10.500 2.440E-031.500E+00 11.500 1.010E-031.500E+00 12.500 4.100E-042.500E+00 2.500 4.000E-052.500E+00 3.500 1.650E-032.500E+00 4.500 2.788E-022.500E+00 5.500 9.877E-022.500E+00 6.500 7.438E-022.500E+00 7.500 3.642E-022.500E+00 8.500 1.346E-022.500E+00 9.500 3.380E-032.500E+00 10.500 1.610E-032.500E+00 11.500 2.300E-042.500E+00 12.500 1.100E-043.500E+00 3.500 3.000E-053.500E+00 4.500 1.510E-033.500E+00 5.500 2.059E-023.500E+00 6.500 7.426E-023.500E+00 7.500 3.866E-023.500E+00 8.500 1.390E-023.500E+00 9.500 3.640E-033.500E+00 10.500 1.150E-033.500E+00 11.500 5.000E-043.500E+00 12.500 1.600E-043.500E+00 13.500 2.000E-054.500E+00 4.500 2.000E-054.500E+00 5.500 1.470E-034.500E+00 6.500 2.372E-024.500E+00 7.500 4.357E-024.500E+00 8.500 1.571E-024.500E+00 9.500 5.070E-034.500E+00 10.500 7.600E-044.500E+00 11.500 5.700E-044.500E+00 12.500 2.300E-044.500E+00 13.500 4.000E-055.500E+00 5.500 1.200E-045.500E+00 6.500 1.650E-035.500E+00 7.500 2.431E-025.500E+00 8.500 1.550E-025.500E+00 9.500 5.000E-035.500E+00 10.500 1.120E-035.500E+00 11.500 4.400E-045.500E+00 12.500 1.600E-045.500E+00 13.500 8.000E-056.500E+00 6.500 8.000E-056.500E+00 7.500 3.100E-036.500E+00 8.500 1.408E-026.500E+00 9.500 3.940E-036.500E+00 10.500 1.490E-036.500E+00 11.500 4.200E-046.500E+00 12.500 1.300E-046.500E+00 13.500 7.000E-057.500E+00 7.500 2.500E-04




7.500E+00 8.500 4.860E-037.500E+00 9.500 3.650E-037.500E+00 10.500 9.300E-047.500E+00 11.500 4.200E-047.500E+00 12.500 1.300E-047.500E+00 13.500 4.000E-058.500E+00 7.500 2.000E-058.500E+00 8.500 4.700E-048.500E+00 9.500 2.790E-038.500E+00 10.500 5.400E-048.500E+00 11.500 2.300E-048.500E+00 12.500 1.200E-048.500E+00 13.500 4.000E-059.500E+00 8.500 6.000E-059.500E+00 9.500 8.800E-049.500E+00 10.500 5.000E-049.500E+00 11.500 1.100E-049.500E+00 12.500 3.000E-059.500E+00 13.500 1.000E-051.050E+01 8.500 1.000E-051.050E+01 9.500 6.000E-051.050E+01 10.500 3.600E-041.050E+01 11.500 1.200E-041.050E+01 12.500 2.000E-051.050E+01 13.500 1.000E-051.150E+01 9.500 2.000E-051.150E+01 10.500 9.000E-051.150E+01 11.500 7.000E-051.150E+01 12.500 1.000E-051.250E+01 10.500 1.000E-051.250E+01 11.500 4.000E-051.250E+01 12.500 1.000E-051.350E+01 11.500 1.000E-05

Total probability: 1.0


DNV Report No. 95-3204 APPENDIX B: Eigenmodes og Structural Response


7 APPENDIX B: EIGENMODES OG STRUCTURAL RESPONSE











DNV Report No. 95-3204 APPENDIX C: Proban Input File : ULS Application


8 APPENDIX C: PROBAN INPUT FILE : ULS APPLICATION

Proban input file for load direction North-West:%----------------------------------------------------------------------

% Proban input file

% file name : NW.inp => Load direction : North-West

% Reliability analysis of Jacket due to total collapse of the jacket

% Project : Guideline

% Gudfinnur Sigurdsson June 18. 1996

%----------------------------------------------------------------------

%

%-------------------------------------

% marginal distribution of Hs :

%-------------------------------------

CREATE VARIABLE alfa 'alfa ' FIXED 2.230

CREATE VARIABLE beta 'beta ' FIXED 1.374

CREATE VARIABLE gamma 'gamma ' FIXED 0.281

CREATE VARIABLE Hs 'Hs ' DISTRIBUTION Weibull Alp-Beta-Low alfa beta gamma

%

%-------------------------------------------------------

% Annual number of storms for the different direction

%-------------------------------------------------------

% Nstorm = total number of 6 hours storms = 1460

% NW_Nstorm = Annual number of storms in NW-dir = 0.11*1460 = 161

% W_Nstorm = Annual number of storms in W-dir = 0.13*1460 = 190

% SW_Nstorm = Annual number of storms in SW-dir = 0.20*1460 = 292

% S_Nstorm = Annual number of storms in S-dir = 0.22*1460 = 321

% SE_Nstorm = Annual number of storms in SE-dir = 0.10*1460 = 146

% E_Nstorm = Annual number of storms in E-dir = 0.07*1460 = 102

% NE_Nstorm = Annual number of storms in NE-dir = 0.05*1460 = 73

% N_Nstorm = Annual number of storms in N-dir = 0.12*1460 = 175

%---------------------------------------------------------------------

%

% Annual maximum Hs:

%----------------------

COPY VARIABLE Hs NW_Hs

ASSIGN EXTREME-VALUE NW_Hs MAX-OF-N 161

%

%-------------------------------------

% conditional distribution of Tz

%-------------------------------------

CREATE VARIABLE a1 'a1 ' FIXED 1.22






CREATE VARIABLE b1 'b1 ' FIXED 0.0623

CREATE VARIABLE b2 'b2 ' FIXED 0.255

CREATE VARIABLE b3 'b3 ' FIXED -0.297

%--------------------------------------

% NW-direction

%----------------

CREATE VARIABLE NW_Hs_a3 ' Hs**a3' FUNCTION Power NW_Hs a3

CREATE VARIABLE NW_Hs_b3 'Hs * b3' FUNCTION Product ( ONLY NW_Hs b3 )

CREATE VARIABLE NW_ElnTz 'E[Ln Tz) = a1+a2*Hs**a3' FUNCTION Linear-Comb ( ONLY a1

1 a2 NW_Hs_a3 )

CREATE VARIABLE NW_exp_Hs_b3 'exp(Hs*b3)' FUNCTION Exp NW_Hs_b3

CREATE VARIABLE NW_Std_lnTz 'Std(Ln(Tz))' FUNCTION Linear-Comb ( ONLY b1 1 b2

NW_exp_Hs_b3 )

CREATE VARIABLE NW_LnTz 'Ln(Tz) ' DISTRIBUTION Normal Mean-StD NW_ElnTz NW_Std_lnTz

CREATE VARIABLE NW_Tz 'Tz NW-dir' FUNCTION Exp NW_LnTz

%

%----------------------------------------------------------------------------------

% Modelling of wave height appying Forristall approach

% The modelling is done applying dummy (0,1) uniform dist. i.e. U(0;1)

% i.e. distribution transformation

%

% F_Hmax = exp(-Nwave * exp( -delta*(hmax/hs)**alfa ) )

%

% where ; alfa = 2.13 delta = 2.26

%

% => hmax = hs * ( -1/delta * ln( - ln(U)/Nwave ) )**(1/alfa)

%

% where Nwave is number of wave in the storm

% i.e. Nwave=tau/Tz = 21600/Tz (tau is the storm duration = 6 hours=21600 sec.)

%-------------------------------------------------------------------------------------

CREATE VARIABLE U1 'Dummy var: uniform(0;1)' DISTRIBUTION Uniform Limits 0 1.0

CREATE VARIABLE lnU1 'Dummy var: ln(U1)' FUNCTION Log U1

CREATE VARIABLE D1_213 'Dummy var: 1/2.13' FUNCTION Division 1 2.13

%-------------

%NW-direction

%-------------

CREATE VARIABLE NW_Nwave 'Number of waves in a storm' FUNCTION Division 21600 NW_Tz

CREATE VARIABLE NW_lnU1_N 'Dummy var: ln(U1)/Nwave' FUNCTION Division lnU1 NW_Nwave

CREATE VARIABLE NW_ln_U1_N 'Dummy var: - ln(U1)/Nwave' FUNCTION Division NW_lnU1_N -1

CREATE VARIABLE NW_lnlnU1N 'Dummy var: ln(- ln(U1)/Nwave)' FUNCTION Log NW_ln_U1_N

CREATE VARIABLE NW_lnlnU1N_1 'Dummy var: ln(-ln(U1)/Nwave)/-2.26'

FUNCTION Division NW_lnlnU1N -2.26

CREATE VARIABLE NW_lnlnU1N_2 'Dummy var: (ln(-ln(U1)/Nwave)/-2.26)**(1/2.13)'




FUNCTION power NW_lnlnU1N_1 D1_213

%

%---------------------------------------------------------------------------------

% nn_Hmax : max wave for direction nn, using Forristall model for wave heights

%---------------------------------------------------------------------------------

CREATE VARIABLE NW_Hmax 'Hmax : NW-dir' FUNCTION Product ( ONLY NW_Hs NW_lnlnU1N_2 )

%

%--------------------------------------------------

% Wave period for extreme wave (5th order Stoke):

%--------------------------------------------------

CREATE VARIABLE NW_Tmax 'Wave period : 1.2*Tz' FUNCTION Product ( ONLY NW_Tz 1.2 )

%

%-------------------------------------------------------------------

% Current speed : Normal dist with E[] = Ecurr and Std= STDcurr

%-------------------------------------------------------------------

CREATE VARIABLE NW_Ecurr 'mean Current speed in NW-dir'

FUNCTION Linear-Comb ( ONLY 1 0.2 0.04 NW_Hs )

CREATE VARIABLE STDcurr 'Std of the Current speed' FIXED 0.1

CREATE VARIABLE NW_Curr 'Current speed in NW-dir' DISTRIBUTION Normal Mean-Std NW_Ecurr STDcurr

%

%-------------------------------------------------------------------

% Wind speed : Normal dist with E[] = Ewind and Std= STDwind

%-------------------------------------------------------------------

CREATE VARIABLE NW_Ewind 'mean 1-hour wind speed in NW-dir'

FUNCTION Linear-Comb ( ONLY 1 5.5 1.8 NW_Hs )

CREATE VARIABLE NW_STDwind 'Std for 1-hour wind speed in NW-dir'

FUNCTION Linear-Comb ( ONLY 1 4.9 -0.26 NW_Hs )

CREATE VARIABLE NW_Wind 'Wind speed in NW-dir' DISTRIBUTION Normal Mean-Std NW_Ewind NW_STDwind

CREATE VARIABLE NW_Wind2 '(NW_Wind speed)**2' FUNCTION Power NW_Wind 2

%

%-------------------------------------------------------------------------

% Wind load : = Cwind * Wind**2

% Cwind = 0.002 for directions : NW and SE

% Cwind = 0.003 for directions : SW and NE

% Cwind = 0.0025 for directions : N, S, W and E

% the wind speed in given in m/sec and the loading is in MN

%--------------------------------------------------------------------------

CREATE VARIABLE NW_Cwind 'Wind load parameter in NW-dir' FIXED 0.002

CREATE VARIABLE NW_WindLoad 'Wind loading' FUNCTION product ( ONLY NW_Cwind NW_Wind2 )

%

CREATE VARIABLE Cd 'Cd' Fixed 0.7

CREATE VARIABLE Mg 'Marine growth' Fixed 0.01

%

%--------------------------------------------------------------------------------

% NB !! the response surface for the hydrodynamic loading is given in Newtons




% but the capacity and the wind loading is in MN =>

% the L_jack is scaled by factor 1.e-6

%--------------------------------------------------------------------------------

% Total loading :

%--------------------

CREATE VARIABLE X_L_jack 'Model unc. in hydrodynamic load calc' DISTR Normal Mean-CoV 1. 0.2

CREATE VARIABLE NW_L_jack 'Calculated Base Shear in NW-dir'

FUNCTION BaseShear NW_Hmax NW_Tmax NW_Curr Cd Mg X_L_jack

CREATE VARIABLE NW_BS 'Total Calculated Base Shear in NW-dir'

FUNCTION Linear-Comb ( ONLY 1.e-6 NW_L_jack 1 NW_WindLoad )

%-------------------------

% Collapse Capacity :

%-------------------------

CREATE VARIABLE X_CC_model 'Model unc. in the capacity' DISTRIBUTION Normal Mean-CoV 1 .15

CREATE VARIABLE NW_CC_calc 'Calc. Collapse capacity' FIXED 194.9

CREATE VARIABLE NW_CC_mod 'Modification due to H and T'

FUNCTION CC_modify NW_Hmax NW_Tmax NW_CC_calc

CREATE VARIABLE NW_CC 'Collapse capacity NW-Dir' FUNCTION Product ( ONLY X_CC_model NW_CC_mod )

%

%---------------------------

% Limit State : CC - BS

%---------------------------

CREATE VARIABLE NW_CC_BS 'CollCap - Base Shear' FUNCTION Difference NW_CC NW_BS

%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% End of input file for NW-direction%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


DNV Report No. 95-3204 APPENDIX D: FORTRAN Routines : ULS Application


9 APPENDIX D: FORTRAN ROUTINES : ULS APPLICATIONIn the following, the soubroutines applied in the ULS application are shown. Both the routinesapplied together with PROBAN in the reliability analysis, and the program applied fordevelopment of the response surface for base-shear loading on the structure are shown. Theresponse surface routines are not included, as these are have not been developed within theproject.

PROGRAM make_rspprogram make_rsp

C

C------------------------------------------------------------------------

C program for establishing respons surface for base-shear loading

C on a jacket for eight different direction automatically

C the following cooridinates are applied :

C GROW : marine growth

C CURR : current speed at MWL

C CD : Cd parameter in Morisons equation

C H : wave height (5th order Stokes wave)

C T : wave period (5th order Stokes wave)

C

C Coded by : Gudfinnur Sigurdsson

C------------------------------------------------------------------------

implicit real*8 (a-h,o-z)

character*12 resfile(8)

character*4 dir(8)

character*4 Char(5)

real*8 mmg(100),hh(100)

dimension z(10),force(100),ccd(100),ccm(100),waveper(100),

1 curr(100)

C

resfile(1) = 'Dir-NW.res'

resfile(2) = 'Dir-W.res'

resfile(3) = 'Dir-SW.res'

resfile(4) = 'Dir-S.res'

resfile(5) = 'Dir-SE.res'

resfile(6) = 'Dir-E.res'

resfile(7) = 'Dir-NE.res'

resfile(8) = 'Dir-N.res'

dir(1) = '0. '

dir(2) = '45. '

dir(3) = '90. '




dir(4) = '135.'

dir(5) = '180.'

dir(6) = '225.'

dir(7) = '270.'

dir(8) = '315.'

open(70,file='test1',status='unknown')

D0 = 107.3

THFAC = 0.9

A0 = -2.6

A1 = 6.59

A2 = 0.382

c

idim=5

Char(1)='GROW'

Char(2)='CURR'

Char(3)='CD '

Char(4)='H '

Char(5)='T '

C number of coordenates for Marine growth = 1

nummg=1

startmg=0.01

deltamg=0.03

C number of coordenates for Cm = 1 (not included in the response surface

numcm=1

startcm=1.6

deltacm=0

C number of coordenates for current speed = 5

numcurr=5

startcurr=0.0

deltacurr=0.2

C number of coordenates for Cd = 1

numcd=1

startcd=0.7

deltacd=0.3

C number of coordenates for wave height = 6

numh=6

starth= 15.0

deltah= 3.0

C number of coordenates for wave period = 4




numt=4

startt=13.0

deltat=2.0

do 5 j=1,nummg

mmg(j)=startmg+(j-1)*deltamg

5 continue

do 6 j=1,numcurr

curr(j)=startcurr+(j-1)*deltacurr

6 continue

do 7 j=1,numcd

ccd(j)=startcd+(j-1)*deltacd

7 continue

do 8 j=1,numh

hh(j)=starth+(j-1)*deltah

if(hh(j).gt.34.) hh(j) = 34.

8 continue

c-- max wave height is set to 34.0 m

C because we get breaking wave at T=12.5 sec for h=35 m

do 11 j=1,numt

waveper(j)=startt+(j-1)*deltat

11 continue

do 9 j=1,numcm

ccm(j)=startcm+(j-1)*deltacm

9 continue

do 999 idir = 1,8

open(60,file=resfile(idir),status='unknown')

write(60,*) idim

write(60,530) nummg,Char(1)

write(60,530) numcurr,Char(2)

write(60,530) numcd,Char(3)

write(60,530) numt,Char(5)

write(60,530) numh,Char(4)

do 80 i=1,100

idiff=nummg-(i-1)*6

ndiff=6

if(idiff.gt.0) then

if(idiff.lt.6) ndiff=idiff

ist=(i-1)*6+1

isl=(i-1)*6+ndiff




write(60,520)(mmg(j),j=ist,isl)

else

goto 81

endif

80 continue

81 continue

do 82 i=1,100

idiff=numcurr-(i-1)*6

ndiff=6

if(idiff.gt.0) then


ist=(i-1)*6+1

isl=(i-1)*6+ndiff

write(60,520)(curr(j),j=ist,isl)

else

goto 83

endif

82 continue

83 continue

do 84 i=1,100

idiff=numcd-(i-1)*6

ndiff=6

if(idiff.gt.0) then


ist=(i-1)*6+1

isl=(i-1)*6+ndiff

write(60,520)(ccd(j),j=ist,isl)

else

goto 85

endif

84 continue

85 continue

do 86 i=1,100

idiff=numt-(i-1)*6

ndiff=6

if(idiff.gt.0) then


ist=(i-1)*6+1

isl=(i-1)*6+ndiff




write(60,520)(waveper(j),j=ist,isl)

else

goto 87

endif

86 continue

87 continue

do 88 i=1,100

idiff=numh-(i-1)*6

ndiff=6

if(idiff.gt.0) then


ist=(i-1)*6+1

isl=(i-1)*6+ndiff

write(60,520)(hh(j),j=ist,isl)

else

goto 89

endif

88 continue

89 continue

cm = ccm(1)

do 120 img=1,nummg

grow=mmg(img)

do 110 icu=1,numcurr

U=curr(icu)

do 100 icd=1,numcd

cd=ccd(icd)

do 95 it=1,numt

t=waveper(it)

write(60,500) grow,U,cd,t

do 90 ih=1,numh

h=hh(ih)

force(ih)=0.

c

c--- det tages ikke hensyn til overflade haevning her :

D = D0

c

c

c data to subroutine waveload

c

Z(1) = CD

Z(2) = CM

Z(3) = GROW

Z(4) = U




Z(5) = D

Z(6) = H

Z(7) = T

write(70,*)z

CALL WAVELOAD(dir(idir),Z,FH)

force(ih)=fh

write(70,*)'fh =',fh

90 continue

do 91 i=1,100

idiff=numh-(i-1)*6

ndiff=6

if(idiff.gt.0) then


ist=(i-1)*6+1

isl=(i-1)*6+ndiff

write(60,510)(force(k),k=ist,isl)

else

goto 92

endif

91 continue

92 continue

95 continue

100 continue

110 continue

120 continue

close(60)

999 continue

c

500 format('Mg Uc Cd T: ',4(1pe10.3,3x))

510 format(10x,6(1pe12.5,1x))

520 format(6(1pe12.5,1x))

530 format(i10,5x,a4)

stop

END

c

c-------------------------------------------------------------------------

subroutine waveload (dir,xvar,force)

c-------------------------------------------------------------------------

c


integer i,iunix,system

double precision xvar(10),force,h




character*4 dir

external system

c control of input

c control for negative input values

do 100 i=1,10

if(xvar(i).lt.0.)then

xvar(i)=0.

endif

100 continue

c control for maximum values

c drag coeff. CD

if(xvar(1).gt.9.9)then

xvar(1)=9.9

endif

c initia coeff. CM


xvar(2)=9.9

endif

c marine grow GROW

if(xvar(3).gt.1.)then

xvar(3)=1.

endif

c current U


xvar(4)=9.9

endif

c waterdepth D

c if(xvar(5).gt.175.)then

c xvar(5)=175.0

c endif

c wavehight


xvar(6)=40.0

endif

c perriod T


xvar(7)=25.0

endif

c increase the significanse of wavehight




h=xvar(6)

c calculate the forces by wajac

c make inputfile to wajac

call inwaj(dir,xvar)

c run wajac

iunix=system('wajac < wajac.inp >! wajac.res')

C--remove tmp files

iunix=system('rmtmp')

c find maxs base shear from wajac results

call reswaj(force)

return

end

c***************************************************************

subroutine inwaj(dir,xvar)

c---------------------------------------------------------------

c

c This subroutine makes the inputfile 'wajac.inp'. The original

c inputfile is given in the file 'wajac'. The program do:

c 1) reads the file 'wajac.old'

c 2) makes the changes defined by the parameters in xvar

c 3) write the new file on 'wajac.inp'

c

c input:

c cd = xvar(1) : drag coefficient

c cm = xvar(2) : inertia coefficient

c grow= xvar(3) : marine growth

c u = xvar(4) : current

c d = xvar(5) : waterdept

c h = xvar(6) : wavehight

c t = xvar(7) : wave period

c dir = wave direction

c

c---------------------------------------------------------------


double precision xvar(10),cd,cm,grow,u,d,h,t

double precision crntprof(10),crntdepth(10)

integer numcrnt




character string*79,inpfile*20,oldfile*20,dir*4

c initiating

cd = xvar(1)

cm = xvar(2)

grow= xvar(3)

u = xvar(4)

d = xvar(5)

h = xvar(6)

t = xvar(7)

write(70,*)'xvar',xvar

c--

c the current profile is calculated as

c curr(z) = curr(MWL) * (z/Depth)**(1/7) z < Depth

c curr(z) = curr(MWL) z > Depth

c where curr(MWL) = u

c Depth = d

c--

c the current speed is calculated in seven depths i.e.

c

numcrnt = 7

crntdepth(1) = 0.0

crntdepth(2) = 2.0

crntdepth(3) = 10.0

crntdepth(4) = 50.0

crntdepth(5) = 80.0

crntdepth(6) = 107.3

crntdepth(7) = 150.0

do 10 icrnt=1,numcrnt

if (crntdepth(icrnt).LE.d) then

crntprof(icrnt) = u * (crntdepth(icrnt)/d)**(1./7.)

else

crntprof(icrnt) = u

endif

10 continue

inpfile='wajac.inp'

oldfile='wajac.old'

open(1,file=inpfile,status='unknown')

open(2,file=oldfile,status='old')

100 read(2,'(a)',end=300)string

if(string(1:4).eq.'COEF ') then

write(1,'(a30,f10.8,a10,f10.8)')

+ string(1:30),cd,string(41:50),cm

elseif(string(1:4).eq.'MGRW') then

write(1,'(a30,f10.8,a20)')




+ string(1:30),grow,string(41:60)

elseif(string(1:4).eq.'CRNT') then

write(1,'(a40,f10.2,2x,e10.2,a20)')

+ string(1:40),crntdepth(1),crntprof(1),

+ string(61:79)

do 110 i=2,numcrnt

read(2,'(a)',end=300)string

string=' '

write(1,'(a40,f10.2,2x,e10.2,a20)')

+ string(1:40),crntdepth(i),crntprof(i),

+ string(61:79)

110 continue

elseif(string(1:4).eq.'DPTH') then

write(1,'(a10,f10.6)')string(1:10),d

elseif(string(1:3).eq.'SEA') then

write(1,'(a15,f4.1,1x,f4.1,6x,a40,a4)')

+ string(1:15),h,t,string(31:70),dir

else

write(1,'(a79)')string

endif

goto 100

300 continue

close (1)

close (2)

return

end

c***************************************************************

subroutine reswaj(force)

c---------------------------------------------------------------

c

c This subroutine reads the result from 'wajac.res'. The

c program find the maxs base shear (load)

c

c output:

c force = max base shear force

c---------------------------------------------------------------


character string*79,resfile*20

integer iload,i,imax

double precision force,f(100),f1,f2,f3,fmax,a,b,c,xmax

resfile='wajac.res'

open(1,file=resfile,status='old')




iload = 0

fmax = 0.

100 read(1,'(a)',end=300)string

if(string(8:11).eq.'STEP ') then

iload = iload + 1

read(string(41:52),*)X_force

read(string(57:68),*)Y_force

f(iload) = (X_force**2+Y_force**2)**0.5

endif

goto 100

300 continue

close (1)

do 400 i=1,iload

if(f(i).ge.fmax) then

imax = i

fmax = f(imax)

endif

400 continue

if(imax.eq.1 .or. imax.eq.iload) then

write(70,*)' worning !! ; max load is outside the area'

write(70,*)' imax = ',imax

if(imax.eq.1) then

f1 = f(imax)

f2 = f(imax+1)

f3 = f(imax+2)

endif

if(imax.eq.iload) then

f1 = f(imax-2)

f2 = f(imax-1)

f3 = f(imax)

endif

write(70,*)'f1,f1,f3 ',f1,f2,f3

else

f1 = f(imax-1)

f2 = f(imax)

f3 = f(imax+1)

endif

c fit a second order parabel to f1,f2 and f3

c f(x)= a+bx+cx**2, vhere f(0)=f1,f(1)=f2,f(2)=f3

a = f1

b = 0.5*(-3*f1+4*f2-f3)

c = f2 - a - b




c maximum force

xmax = -b/(2*c)

fmax = a+b*xmax+c*xmax*xmax

force = fmax

write(70,*)'f1,f1,f3,fmax ',f1,f2,f3,fmax

return

end




SUBROUTINE ULSfuncSUBROUTINE ULSfunc( OPTION, FUNCNO, MODE, NF, FC,

CIO I I I I I

+ NX, X, NDX, XC, XINC,

CIO I I I I I

+ NDX1, XC1, XINC1, NDX2, XC2, XINC2,

CIO I I I I I I

+ IVAL, CVAL, F, DF, D2F, ISTAT )

CIO O O IO IO O O

C

INTEGER FUNCNO, NF, FC(NF), NX, NDX, XC(NDX), NDX1,

+ XC1(NDX1), NDX2, XC2(NDX2), IVAL, ISTAT(NF)

CHARACTER*(*) OPTION, MODE, CVAL

DOUBLE PRECISION X(NX), XINC(NDX), XINC1(NDX1), XINC2(NDX2),

+ F(NF), DF(NF,NDX), D2F(NF,NDX1,NDX2)

C-----------------------------------------------------------------------

C PART OF: PROBAN

C

C PURPOSE: Library of ULS analysis: Probability assessment of Collapse of jackets

C Project : Guideline

C

C RESTRICTIONS:

C

C INPUT:

C OPTION Determines what must be passed back to the calling routine:

C = 'SUBLNAME' : Return CVAL = Name of a sublibrary

C = 'SUBLDESC' : Return CVAL = Description of a sublibrary

C = 'SUBLNFUNC' : Return IVAL = Number of functions in a

C sublibrary

C = other value: passed on to function

C SUBLNO Sublibrary number

C FUNCNO Function number inside sublibrary

C MODE Passed on to function

C NF Passed on to function

C FC Passed on to function

C NX Passed on to function

C X Passed on to function

C NDX Passed on to function

C XC Passed on to function

C XINC Passed on to function

C NDX1 Passed on to function

C XC1 Passed on to function

C XINC1 Passed on to function

C NDX2 Passed on to function




C XC2 Passed on to function

C XINC2 Passed on to function

C F Passed on to function

C DF Passed on to function

C

C OUTPUT:

C IVAL Varies, depending on OPTION

C CVAL Varies, depending on OPTION

C CVAL may also be used to pass a one line error message

C back to PROBAN. If any ISTAT value is not = 0, the

C contents of CVAL will be presented as an error message if

C CVAL is not empty (CVAL will always be empty on input).

C F Function value(s)

C DF Gradient(s)

C D2F Second order derivative(s)

C ISTAT Return status:

C = 0 : OK.

C = 1 : Numerical or other error, cancel further calculations

C =-1 : Numerical error, X yields F < 0

C =-2 : Numerical error, X yields F = 0

C =-3 : Numerical error, X yields F > 0

C

C CODED BY:

C Robert Olesen, VSS, November 1991 (template layout)

C Gudfinnur Sigurdsson DNVI-240 April 1996

C

C REVISIONS:

C

C COPYRIGHT (C) 1991, VERITAS SESAM SYSTEMS

C

C-----------------------------------------------------------------------

C METHOD:

C

C INTERNAL VARIABLES:

C

C COMMON VARIABLES:

C Var: IO Type Common/File Description

C

C-----------------------------------------------------------------------

C

ISTAT(1) = 0

C

IF (OPTION(1:4).NE.'SUBL') THEN

C

IF (FUNCNO.EQ.1) THEN




C

C---- Calculation of total base-shear loading

CALL BaseShear( OPTION, MODE, NF, FC, NX, X, NDX, XC, XINC,



C

ELSEIF (FUNCNO.EQ.2) THEN

C

C---- Calculation of Collapse capacity

C

CALL CollCap( OPTION, MODE, NF, FC, NX, X, NDX, XC, XINC,



C

C

ELSE

C

C---- Called with illegal function number.

C Pass an error message back to PROBAN in CVAL.

C

WRITE(CVAL,9100) FUNCNO

ISTAT(1) = 1

C

END IF

C

ELSE IF (OPTION.EQ.'SUBLNAME') THEN

C

C---- Return name of this sublibrary. The name must contain

C alphanumeric characters or hyphens ('-') and can be at most 12

C characters long. eg: 'Sublibrary' or: 'MySubl-11'

C

CVAL = 'ULS-FUNC'

C

ELSE IF (OPTION.EQ.'SUBLDESC') THEN

C

C---- Return description of this sublibrary. The description should

C not be longer than 50 characters

C

CVAL = 'Collapse Analysis of jackets'

C

ELSE IF (OPTION.EQ.'SUBLNFUNC') THEN

C

C---- Return number of functions in this sublibrary

C

IVAL = 2




C

END IF

RETURN

C

9100 FORMAT('ULSfunc called with illegal function number: ',I12)

C End of ULSfunc

END




SUBROUTINE BaseShear

SUBROUTINE BaseShear( OPTION, MODE, NF, FC, NX, X, NDX, XC, XINC,

CIO I I I I I I I I I


CIO I I I I I I


CIO O O IO IO O O

C

INTEGER NF, FC(NF), NX, NDX, XC(NDX), NDX1, XC1(NDX1),

+ NDX2, XC2(NDX2), IVAL, ISTAT(NF)




C-----------------------------------------------------------------------

C PART OF: PROBAN

C

C PURPOSE: Calculate Base Shear loadin on jacket using

C response surface

C

C INPUT:

C H : wave height (5th Stokes)

C T : wave period (5th Stokes)

C Curr : Current speed at MWL

C Cd : Drag coefficient in Morison equation

C Mg : Marine growth

C BS_model : model uncertainty in the Base Shear Calculation


C = 'FUNCVAL' : Return F(1..NF) = value of the function

C coordinate(s) specified in FC(1..NF)

C = 'FUNCNAME' : Return CVAL = Name of the function

C = 'FUNCDESC' : Return CVAL = Description of the function

C = 'FUNCNCOORD' : Return IVAL = Dimension of the function

C value. Return IVAL = 0 if this number is

C given as input by the user

C = 'COORDNAME' : Return CVAL = Name of the coordinate of

C the function specified in FC(1). This

C will not be requested if the function has

C only one coordinate.

C = 'COORDDESC' : Return CVAL = Description of the coordinate




C = 'FUNCNARG' : Return IVAL = Number of arguments in the




C function. Return IVAL = 0 if this number

C is given as input by the user

C = 'ARGNAME' : Return CVAL = Name of the argument in the

C function specified by XC(1).

C = 'ARGDESC' : Return CVAL = Description of the argument

C in the function specified by XC(1).

C = 'FUNCTYPE' : Return CVAL = Type of the function

C CVAL = 'INDEX': Index type function

C Everything else means continuous.

C = 'FUNCGRAD' : Return IVAL = gradient calculation

C capability of the function

C = 0: The function cannot return gradients

C = 1: The function can return first order

C gradients

C = 2: The function can return first and

C second order gradients

C MODE Mode of calculation. Only used when function calculation

C is required:

C = 'FIRST CALL' : This is the first entry during an analysis

C into the function for a given variable.

C The variable name is passed in CVAL.

C = 'DESIGN POINT' : Function is called with design point

C value as input (FORM/SORM only)

C NF Number of active function coordinates

C FC Indices of the active function coordinates

C NX Number of arguments in function

C X Function arguments if calculation is required

C NDX Not used.

C XC XC(1) is used to identify an argument to get name

C and description.

C XINC Not used.

C NDX1 Not used.

C XC1 Not used.

C XINC1 Not used.

C NDX2 Not used.

C XC2 Not used.

C XINC2 Not used.

C

C OUTPUT:










C F Function value(s) (See OPTION = 'FUNCVAL' above)

C ISTAT Return status for calculation for each function coordinate:

C = 0 : OK.





C

C CODED BY:


C Gudfinnur Sigurdsson, DNVI, June 1996

C

C REVISIONS:

C


C

C-----------------------------------------------------------------------

C

DOUBLE PRECISION H,T,Curr,Mg,Cd,BS_model,BS_load

DOUBLE PRECISION Z(5),FNCVAL

INTEGER I, L, IX, LF

C

C-----------------------------------------------------------------------

C METHOD:

C

C

C INTERNAL VARIABLES:

C

C COMMON VARIABLES:

C Var: IO Type Common/File Description

C

C

C-----------------------------------------------------------------------

C---- Initialize status values

C

DO 800 I=1,NF

ISTAT(I) = 0

800 CONTINUE

C

C-------------------------- Function value ------------------------------

C

IF( OPTION.EQ.'FUNCVAL') THEN

C

C---- Introduce Memnonics

C




H = X(1)

T = X(2)

Curr = X(3)

Cd = X(4)

Mg = X(5)

BS_model = X(6)

C---



C Curr : Current speed at MWL

C Cd : Drag coefficient in Morison equation

C Mg : Marine growth

C BS_model : model uncertainty in the Base Shear calculation

C

C---- Return function value(s).

C

C Loop over the requested function coordinates

C

DO 1100 L=1,NF

C

C---- Extract the requested function coordinate

C

LF = FC(L)

C

C LF=1, Base Shear calculated applying response surface

C LF=2, New model-2 (not available)

C LF=3, New model-3 (not available)

C

IF (LF.EQ.1) THEN

C

Z(1) = Mg

Z(2) = Curr

Z(3) = Cd

Z(4) = T

Z(5) = H

C

C Calling the response surface modul ‘response’ for calculation the total

C Base-Shera loading.

C Parameter list : (idim,Z,BS_load)

C idim : dimension of the response surface = 5

C Z(idim) : the coordinate for which the response is calculated for

C BS_laod : the calculated response (output from the modul)

C

Call response(5,Z,BS_load)

FNCVAL = BS_model*BS_load




C

C ELSEIF (LF.EQ.2) THEN

C

C

C---- Calculate new model for Baseshear loading

C

C ELSEIF (LF.EQ.3) THEN

C

C---- Calculate new model for Baseshear loading

C

C

ENDIF

C

C---- Calculate the value and set the proper status

C

F(L) = FNCVAL

ISTAT(L) = 0

C

1100 CONTINUE

C

C----------------------- Function properties ---------------------------

C

ELSE IF (OPTION.EQ.'FUNCNAME') THEN

C

C---- Return name of function

C The name must contain alphanumeric characters or hyphens ('-')

C and can be at most 12 characters long.

C eg: 'X22yyy' or: 'Func-Name'

C

CVAL = 'BaseShear'

C

ELSE IF (OPTION.EQ.'FUNCDESC') THEN

C

C---- Return description of function

C The description should not be longer than 50 characters

C

CVAL = 'Base Shear loading on Jacket'

C

ELSE IF (OPTION.EQ.'FUNCNCOORD') THEN

C

C---- Return dimension of the function

C Return IVAL = 0 if the dimension is to be input by the user

C If more Limit-load models are added, IVAL must be changed

C to be equal to the number of models included




C----

IVAL = 1

C

ELSE IF( OPTION.EQ.'FUNCTYPE' )THEN

C

C---- Return the type of a function ('INDEX' or otherwise)

C Presumed not to be an index function. Change to 'INDEX' if it is

C

CVAL = 'DUMMY'

C

ELSE IF( OPTION.EQ.'FUNCGRAD') THEN

C

C---- Return gradient calculation status:

C 0: No gradients are provided

C 1: First order gradients are provided

C 2: First and second order gradients are provided

C

IVAL = 0

C

ELSE IF (OPTION.EQ.'COORDNAME') THEN

C

C---- Return name of a coordinate of the function.




C

C---- Extract the coordinate which name is requested

C

LF = FC(1)

C

C---- Branch on the coordinates and return the proper name

C

C IF (LF.EQ.1) THEN

C CVAL = 'MODEL-1'

C ELSE IF (LF.EQ.2) THEN

C CVAL = 'MODEL-2'


C CVAL = 'MODEL-3'

C ENDIF

C

IF (LF.EQ.1) THEN

CVAL = 'Base-Shear'

ENDIF

C

ELSE IF (OPTION.EQ.'COORDDESC') THEN




C

C---- Return description of a coordinate of the function.

C This need not be defined if the function has only one coord.


C

C---- Extract the coordinate which description is requested

C

LF = FC(1)

C

C---- Branch on the coordinates and return the proper description

C----

IF (LF.EQ.1) THEN

CVAL = 'Base Shear load on Jacket using Response surface'


C CVAL = 'Model-2'


C CVAL = 'Model-3'

ENDIF

C

ELSE IF (OPTION.EQ.'FUNCNARG') THEN

C

C---- Return number of arguments in function

C Set this number to 0 if it is to be input by the user

C

IVAL = 6

C

ELSE IF (OPTION.EQ.'ARGNAME') THEN

C

C---- Return name of argument XC(1) in function. The name must

C obey the same restrictions as the function name

C

C---- Extract the argument which name is requested

C

IX = XC(1)

C

C---- Branch on the arguments and return the proper name

C

IF (IX.EQ.1) THEN

CVAL = 'Wave Heigth'

ELSE IF (IX.EQ.2) THEN

CVAL = 'Wave Period'


CVAL = 'Current'


CVAL = 'Cd'





CVAL = 'Mg'


CVAL = 'Model uncer'

END IF

C

ELSE IF (OPTION.EQ.'ARGDESC') THEN

C

C---- Return description of argument XC(1) in function. The

C description should not use more than 50 characters

C

C---- Extract the argument which description is requested

C

IX = XC(1)

C

C---- Branch on the arguments and return the proper description

C

IF (IX.EQ.1) THEN

CVAL = 'Wave Heigth - 5th order Stokes wave'


CVAL = 'Wave period - 5th order Stokes wave'


CVAL = 'Current speed'


CVAL = 'Drag Coefficient - Cd'


CVAL = 'Marine Growth - Mg'


CVAL = 'Model uncertainty'

END IF

C

END IF

C

RETURN

END




SUBROUTINE CollCap

SUBROUTINE CollCap( OPTION, MODE, NF, FC, NX, X, NDX, XC, XINC,

CIO I I I I I I I I I


CIO I I I I I I


CIO O O IO IO O O

C

INTEGER NF, FC(NF), NX, NDX, XC(NDX), NDX1, XC1(NDX1),

+ NDX2, XC2(NDX2), IVAL, ISTAT(NF)




C-----------------------------------------------------------------------

C PART OF: PROBAN

C

C PURPOSE: Modify the calculated Collapse Capacity of a Jacket

C as function of H and T

C

C INPUT:



C Cap_calc : calculated collapse capacity (define factor=1.0 in respone

C surface

C


C = 'FUNCVAL' : Return F(1..NF) = value of the function

C coordinate(s) specified in FC(1..NF)

C = 'FUNCNAME' : Return CVAL = Name of the function

C = 'FUNCDESC' : Return CVAL = Description of the function

C = 'FUNCNCOORD' : Return IVAL = Dimension of the function

C value. Return IVAL = 0 if this number is

C given as input by the user

C = 'COORDNAME' : Return CVAL = Name of the coordinate of




C = 'COORDDESC' : Return CVAL = Description of the coordinate




C = 'FUNCNARG' : Return IVAL = Number of arguments in the

C function. Return IVAL = 0 if this number




C is given as input by the user

C = 'ARGNAME' : Return CVAL = Name of the argument in the

C function specified by XC(1).

C = 'ARGDESC' : Return CVAL = Description of the argument

C in the function specified by XC(1).

C = 'FUNCTYPE' : Return CVAL = Type of the function

C CVAL = 'INDEX': Index type function

C Everything else means continuous.

C = 'FUNCGRAD' : Return IVAL = gradient calculation

C capability of the function

C = 0: The function cannot return gradients

C = 1: The function can return first order

C gradients

C = 2: The function can return first and

C second order gradients

C MODE Mode of calculation. Only used when function calculation

C is required:

C = 'FIRST CALL' : This is the first entry during an analysis

C into the function for a given variable.

C The variable name is passed in CVAL.

C = 'DESIGN POINT' : Function is called with design point

C value as input (FORM/SORM only)

C NF Number of active function coordinates

C FC Indices of the active function coordinates

C NX Number of arguments in function

C X Function arguments if calculation is required

C NDX Not used.

C XC XC(1) is used to identify an argument to get name

C and description.

C XINC Not used.

C NDX1 Not used.

C XC1 Not used.

C XINC1 Not used.

C NDX2 Not used.

C XC2 Not used.

C XINC2 Not used.

C

C OUTPUT:







C F Function value(s) (See OPTION = 'FUNCVAL' above)




C ISTAT Return status for calculation for each function coordinate:

C = 0 : OK.





C

C CODED BY:


C Gudfinnur Sigurdsson, DNVI, June 1996

C

C REVISIONS:

C


C

C-----------------------------------------------------------------------

C

DOUBLE PRECISION H,T,Cap_calc,CC_capacity

DOUBLE PRECISION FNCVAL

INTEGER I, L, IX, LF

C

C-----------------------------------------------------------------------

C---- Initialize status values

C

DO 800 I=1,NF

ISTAT(I) = 0

800 CONTINUE

C

C-------------------------- Function value ------------------------------

C

IF( OPTION.EQ.'FUNCVAL') THEN

C

C---- Introduce Memnonics

C

H = X(1)

T = X(2)

Cap_calc = X(3)

C---- Return function value(s).

C

C Loop over the requested function coordinates

C

DO 1100 L=1,NF

C

C---- Extract the requested function coordinate




C

LF = FC(L)

C

C LF=1, calculate collapse capacity

C LF=2, (not available)

C LF=3, (not available)

C

IF (LF.EQ.1) THEN

FNCVAL = Cap_calc*CC_capacity(H,T)

C

ENDIF

C

C---- Calculate the value and set the proper status

C

F(L) = FNCVAL

ISTAT(L) = 0

C

1100 CONTINUE

C

C----------------------- Function properties ---------------------------

C

ELSE IF (OPTION.EQ.'FUNCNAME') THEN

C

C---- Return name of function




C

CVAL = 'CC_modify'

C

ELSE IF (OPTION.EQ.'FUNCDESC') THEN

C

C---- Return description of function


C

CVAL = 'Modification of Collapse capacity for Jacket'

C

ELSE IF (OPTION.EQ.'FUNCNCOORD') THEN

C

C---- Return dimension of the function

C Return IVAL = 0 if the dimension is to be input by the user

C If more Limit-load models are added, IVAL must be changed

C to be equal to the number of models included

C----




IVAL = 1

C

ELSE IF( OPTION.EQ.'FUNCTYPE' )THEN

C

C---- Return the type of a function ('INDEX' or otherwise)

C Presumed not to be an index function. Change to 'INDEX' if it is

C

CVAL = 'DUMMY'

C

ELSE IF( OPTION.EQ.'FUNCGRAD') THEN

C

C---- Return gradient calculation status:

C 0: No gradients are provided

C 1: First order gradients are provided

C 2: First and second order gradients are provided

C

IVAL = 0

C

ELSE IF (OPTION.EQ.'COORDNAME') THEN

C

C---- Return name of a coordinate of the function.




C

C---- Extract the coordinate which name is requested

C

LF = FC(1)

C

C---- Branch on the coordinates and return the proper name

C

IF (LF.EQ.1) THEN

CVAL = ' '

ENDIF

C

ELSE IF (OPTION.EQ.'COORDDESC') THEN

C

C---- Return description of a coordinate of the function.

C This need not be defined if the function has only one coord.


C

C---- Extract the coordinate which description is requested

C

LF = FC(1)

C




C---- Branch on the coordinates and return the proper description

IF (LF.EQ.1) THEN

CVAL = 'Modify Collapse Capacity using Response surface'


C CVAL = ' Model-2'


C CVAL = ' Model-3'

ENDIF

C

ELSE IF (OPTION.EQ.'FUNCNARG') THEN

C

C---- Return number of arguments in function

C Set this number to 0 if it is to be input by the user

C

IVAL = 3

C

ELSE IF (OPTION.EQ.'ARGNAME') THEN

C

C---- Return name of argument XC(1) in function. The name must

C obey the same restrictions as the function name

C

C---- Extract the argument which name is requested

C

IX = XC(1)

C

C---- Branch on the arguments and return the proper name

C

IF (IX.EQ.1) THEN

CVAL = 'Wave Heigth'


CVAL = 'Wave Period'


CVAL = 'CC_calc'

END IF

ELSE IF (OPTION.EQ.'ARGDESC') THEN

C

C---- Return description of argument XC(1) in function. The

C description should not use more than 50 characters

C---- Extract the argument which description is requested

C

IX = XC(1)

C

C---- Branch on the arguments and return the proper description

IF (IX.EQ.1) THEN

CVAL = 'Wave Heigth - 5th order Stokes wave'





CVAL = 'Wave period - 5th order Stokes wave'


CVAL = 'Calculated collapse capacity'

END IF

END IF

RETURN

END




FUNCTION CC_capacityC**********************************************************************

C* Function : Calculates the modification of the collapse capasity for known

C* wave height and period

C**********************************************************************

DOUBLE PRECISION FUNCTION CC_capacity(H,T)

INTEGER Nmax,Npoint,Ndim,NumInt

INTEGER idim1,idim2,idim3,idim4,idim5,Maxdim

PARAMETER (Maxdim=5,Nmax=20,Ndim=2,NumInt=2)

DOUBLE PRECISION wigtpotens,capacity

DOUBLE PRECISION point(Ndim),H,T

DOUBLE PRECISION H_T,Delta,Ccap

COMMON/Ccap1/ Npoint(Maxdim)

COMMON/Ccap2/ H_T(Maxdim,Nmax),Delta(Maxdim)

COMMON/Ccap3/ Ccap(1,1,1,Nmax,Nmax)

CHARACTER FIRSTCALL*1

DATA FIRSTCALL /'Y'/

SAVE FIRSTCALL

C**********************************************************************

C* INPUT: *

C* H DP Wave height

C* T DP Wave period

C**********************************************************************

C* CODED BY : Gudfinnur Sigurdsson DATE: June 1996 *

C* REVISED BY: DATE: *

C**********************************************************************

C

IF(FIRSTCALL.EQ.'Y')THEN

call Inp_HT(Maxdim,Nmax,Ndim,Npoint,Delta,H_T,Ccap)

FIRSTCALL='N'

ENDIF

point(1) = H

point(2) = T

C---

C calculate the collapse capasity by using

C interpolation i.e. response surface, where the

C modul resp5D is applied

C---

idim1=1

idim2=1

idim3=1

idim4=Nmax




idim5=Nmax

wigtpotens=2.d0

call resp5D(idim1,idim2,idim3,idim4,idim5,Nmax,Maxdim,Ndim,

1 NumInt,Npoint,wigtpotens,H_T,Ccap,point,

2 Delta,capacity)

CC_capacity = capacity

C

RETURN

END




SUBROUTINE Inp_HTSUBROUTINE Inp_HT(Maxdim,Nmax,Ndim,Npoint,Delta,H_T,

1 Ccap)

INTEGER NUM_H,NUM_T,Maxdim,Nmax,Ndim

INTEGER i,j,ISTAT,iH,iT

INTEGER Npoint(Maxdim)

DOUBLE PRECISION Delta(Maxdim),Maxval,Minval

DOUBLE PRECISION H_T(Maxdim,Nmax),Ccap(1,1,1,Nmax,Nmax)

CHARACTER streng*80,HTfile*25

do 1 i=1,Maxdim

Delta(i) = 0.d0

Npoint(i) = 1

do 1 j=1,Nmax

H_T(i,j) = 0.d0

1 continue

ISTAT=0

HTfile=' '

WRITE(*,110)

40 WRITE(*,*)'DATA FILE FOR Capacity vs. H & T DATA'

WRITE(*,*)' (DEFAULT : H_T.dat) '

READ(*,'(a)') HTfile

IF(HTfile.EQ.' ') HTfile='H_T.dat'

OPEN (77, FILE=HTfile, STATUS='OLD',IOSTAT=ISTAT)

IF (ISTAT .NE. 0) THEN

WRITE(*, *) ' Wrong filename, PLEASE TRY AGAIN!'

WRITE(*, *) ' '

GOTO 40

ENDIF

30 READ(77,'(a80)',END=999) streng

if(streng(1:1).eq.' ') then

READ(streng,*) NUM_H

else

goto 30

endif



READ(streng,*) NUM_T

else

goto 31




endif



READ(streng,*) (H_T(4,j),j=1,NUM_H)

else

goto 32

endif



READ(streng,*) (H_T(5,j),j=1,NUM_T)

else

goto 33

endif

do 2000 iH=1,NUM_H



READ(streng,*) (Ccap(1,1,1,iH,iT),iT=1,NUM_T)

else

goto 34

endif

2000 continue

999 CONTINUE

CLOSE(77)

Npoint(4)=NUM_H

Npoint(5)=NUM_T

do 4 i=Maxdim-Ndim+1,Maxdim

maxval=H_T(i,Npoint(i))

minval=H_T(i,1)

Delta(i)=dabs(maxval-minval)

4 continue

110 FORMAT(////)

RETURN

END

Documents

Guideline for Offshore Structural Reliability - DNVresearch.dnv.com/skj/OffGuide/Jacket_Examples.pdf · report: Guideline for Offshore Structural Reliability Analysis - Application