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Guideline for Offshore Structural Reliability Analysis - General Page No._____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

DNV Report No. 95-2018 Chapter 7

Skjong,R, E.B.Gregersen, E.Cramer, A.Croker, Ø.Hagen, G.Korneliussen, S.Lacasse, I.Lotsberg, F.Nadim,K.O.Ronold (1995)“Guideline for Offshore Structural Reliability Analysis-General”, DNV:95-2018

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JANUARY 11, 1995

7. CAPACITY 163

7.1 Resistance Model and Validation of Resistance Model 163

7.2 Material Data 1647.2.1 Yield Strength for Different Materials 1647.2.2 Young's Modulus and Poisson's Ratio. 1647.2.3 Fracture Toughness 1657.2.4 Capacity of Columns 1657.2.5 Capacity of Stiffened Plates 1677.2.6 Capacity of Cylindrical Shells 1677.2.7 Capacity of Tubular Joints – Static Strength 1677.2.8 Strength of Grouted Connections 1687.2.9 Capacity of Concrete 168

7.3 Geotechnical Data 1687.3.1 Types of Uncertainty 1687.3.2 Uncertainty in Soil Properties 1697.3.3 Cyclic Loading Effects 1737.3.4 Uncertainty in Calculation Model 1757.3.5 Uncertainty in Load Effects 176

7.4 Geometry Data 177

7.5 Fatigue Data 1787.5.1 Welded Connections and Base Material Steel Structures 1787.5.2 High Strength Steel 1797.5.3 Cast and Forged Steels 1797.5.4 Welded Connections and Base Material Aluminium Structures 1797.5.5 Chains 1797.5.6 Wire Ropes 180

7.5.6.1 General 1807.5.6.2 Scatter in S-N Data 1817.5.6.3 Length Effect 1817.5.6.4 Effect of Mean Load Level 1817.5.6.5 Threshold Value 1817.5.6.6 Load Spectrum 182

7.5.7 Stress Concentration Factors 1827.5.8 Miner-Palmgren Hypothesis of Cumulative Damage 183

7.6 Non-Destructive Examination 1837.6.1 Background 1837.6.2 POD Data 184

7.7 Example Application 1867.7.1 Physical Problem 1867.7.2 Deterministic Analysis 1867.7.3 Inspection Updating 188

7.8 Geotechnical Example 1927.8.1 Physical Problem 1927.8.2 Analysis Methods 1997.8.3 Model Description 1997.8.4 Results 202




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7.8.5 Discussion and Conclusions 203

REFERENCES 205




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7. Capacity

7.1 Resistance Model and Validation of Resistance Model

The resistance model is physically based or empirically based, whereby an equation defines theresistance of a component for the particular loading situation under investigation. The resistanceequation can be written:

r r r X X XJ= =( ) ( , ,... )X 1 2 (7. 1)

where J is the number of basic variables in the resistance function.

The resistance equation must include all the relevant basic variables which govern the resistancein the limit state. It is important that the basic variables in the resistance equation arerepresentative. Any basic variables that are not included in the calculation model must have aninsignificant influence on the model.

Theoretical resistance values rt using the measured basic variables will enable a database oftheoretical results to be obtained which can be compared with the experimental results re . Beforereliability analyses are performed, it is important to validate the resistance model. To compareexperimental and theoretical values the following plots are made:• a plot of the observed resistance values re against the calculated resistance values rt using

the measured properties,• a plot of the observed resistance against each of the observed basic variables.

These plots enable:• corresponding relations to be established,• a check to determine whether the calculation model adequately accounts for the respective

variables,• correlation between experimental and theoretical values to be established.

It is very important that a plot of the observed resistance against each of the observed basicvariables does not show any deviating trends. If such trends are found in the plots, then themodel should be reconsidered.

The uncertainty in the calculation model should then be evaluated. This uncertainty may becaused by:• insufficient composition of the resistance equation,• incompleteness of the resistance equation, where not all variables, parameters or load paths

have been included.




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7.2 Material Data

7.2.1 Yield Strength for Different Materials

The characteristic yield strength is defined as the 5 percent quantile of the test data, see NPD(1994) and DNV (1977). For steel materials with a characteristic value about 350 MPa, thecoefficient of variation is approximately 6 percent, Hart et al. (1985). For steel materials with acharacteristic yield strength larger than 400 MPa the coefficient of variation is less; and 5 percentmay be used. It has been generally supposed that the material yield strength possesses alognormal distribution.

It should be noted that the measured yield strength is a function of strain rate. Studies haveshown that the static yield strength is typically 28 MPa less than the yield strength obtained froma standard test such as conducted by producing mills, Galambos (1978). Reference is also madeto Galambos (1988) and Brockenbrough (1992).

Table 7. 1 Yield strength

Material Distribution Yield strength (MPa) Coefficient of variation (%)Steels Lognormal < 350 8.0Steels Lognormal 350-400 6.0Steels Lognormal > 400 5.0

7.2.2 Young's Modulus and Poisson's Ratio.

The Young's modulus for steel may be characterized by a normal distribution with mean value21 105. ⋅ MPa and a coefficient of variation of 5%. This value is used by Smith et al. (1987). Avalue of 6% is the largest value found in the literature, Galambos and Ravindra (1978). Aconstant value of 0.3 may be used for the Poisson's ratio.

Table 7. 2 Young's modulus and Poisson's ratioNormal distributions

Young's modulus Poisson's ratioMaterial Mean value

(MPa)Coefficient ofvariation (%)

Mean value Coefficient ofvariation

Steels 21 105. ⋅ 5.0 0.30 ?Aluminiumalloys 7 0 104. ⋅

?0.30

?

Concrete 3 0 104. ⋅ ? 0.2 ? ?




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7.2.3 Fracture Toughness

Fracture toughness of steels in terms of KIC and Crack Tip Opening Displacement (CTOD) maybe assumed to be lognormally distributed. However, a two-parameter Weibull distribution gavebest fit to fracture toughness test data (CTOD) reported by Tronskar et al. (1992). The CTODvalues were taken from a total test population of 89 for normalised steels and 106 for QT steels(Quenched and Tempered steels). The test specimens were metallographically examined and splitin two groups according to the test results, namely those with a crack tip in the grain-coarsenedHAZ (Heat Affected Zone), and those with the crack tip in the material outside the HAZ. Thedistributions shown in Table 7. 3 were arrived at and used for reliability evaluations. Thematerial data used to establish these statistical distributions were obtained from a number ofsources. The data were considered to be representative for the two classes of steels, for thecorrect section thicknesses, the arc energy range (2.8-4.5 KJ/mm), and the submerged weldingprocess. 14 different 50 mm thick BS 4360 Grade 50D/E steels with yield strength 350 MPa,representative for offshore steels used for fabrication of offshore structures in the 1980'es, wereincluded in the analysis. These steels are denoted normalised steels in Table 7. 3. The materialyield strength for the QT steels is 500 MPa. The QT test steels comprised plate thicknesses in therange 25 to 80 mm with the majority in the range 30 to 50 mm. A total of 14 different QT steelswere included in the data base.

Table 7. 3 Weibull distributions for CTOD from Tronskar et al. (1992)

W d( , , )β γ1 : F x d xCTOD( ) exp( ( ) )= − − −1 1γ β

Type of steel Crack tip d β γ1Normalised in HAZ 3.857 1.265 0.0Normalised in base material 0.1924 3.082 0.0QT in HAZ 4.078 1.555 0.0QT in base material 0.846 4.158 0.0

7.2.4 Capacity of Columns

The background for the buckling curves used in design of steel structures in European designstandards are based on work carried out within the European Convention for ConstructionalSteelwork which is presented in Manual on Stability of Steel Structures, see ECCS (1976). Thedesign curves are presented by their characteristic values which are defined as mean valuesminus two standard deviations along the slenderness axis. The test results are assumed normallydistributed.

For the reliability of the buckling curves in the AISC Specification, reference is made toGalambos (1988). For background of buckling curves see also Bjørhovde (1992) and Sherman(1992).

Traditional design is based on load effects determined by elastic frame analyses.Non-linear structural analysis may also be used for documentation of the integrity of offshorestructures in the Ultimate Limit State provided that it is sufficiently documented. However, it isforeseen that use of non-linear structural analysis for a general design situation will require asignificant amount of documentation relative to the amount needed when using traditional design




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codes based on linear elastic analysis. Items to be considered for documentation of a non-linearstructural analysis as basis for acceptance of a design are:

• Considerations that the non-linear analysis program used contains an appropriate code for theactual physical behaviour of the considered structure and structural details such that the actualfailure modes are captured.

• Effect of local details for end restraints or force-deformation relationships for the joints. • Effect of fabrication tolerances (member straightness and joint eccentricities) and residual

stresses on buckling capacity. • Failure criteria in terms of maximum strain at failure for components containing relevant

imperfections from, e.g., welding and at regions containing notches. • Repeated yielding in case of reversed loading owing to, e.g., wave action. • Sensitivity of input parameters and analysis assumption for evaluation of acceptance criteria

(reliability analysis may be used).

This implies that an evaluation of the structural integrity based on use of non-linear computerprograms will require more effort by skilled engineers than that of traditional analysis. Inaddition an accepted design procedure based on use of non-linear analysis of offshore structurescannot at present be seen exist.




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7.2.5 Capacity of Stiffened Plates

Different formulations for the capacity of stiffened plates have been presented in the variousdesign codes such as DNV (1992a), BS 5400 (1982), ECCS (1990) and API RV 2V (1987a). Themodel uncertainty for the axial capacity in these codes has been presented by Bonello et al.(1993). See Table 7. 4. The NPD (1994) formulation is the same as the one used by DNV.

Table 7. 4 Comparison of Model Uncertainty based on 23 Tests of AxiallyCompressed Panels (Bonello et al., 1993)

Code Minimum value Maximum value Mean value COV (%)BS 5400 (1982) 0.89 1.46 1.10 13DNV, 30.1(1992a)

0.94 1.52 1.10 16

ECCS (ColumnAppr.) (1990)

0.86 1.44 1.08 14

ECCS (Orth.Appr.) (1990)

0.94 1.68 1.14 15

API RV 2V(1987)

0.70 2.51 1.34 34

Method proposedby ImperialCollege

0.94 1.34 1.10 8

7.2.6 Capacity of Cylindrical Shells

Several codes and standards have been developed for design of shell stuctures such as DNV(1992a), BS5500 (1985), NPD (1994) and API BULLETIN 2U (1987b). Studies on modeluncertainties have been performed by Ellinas et al. (1984) and Faulkner (1992). It should bechecked that the conclusions from these studies still are relevant because some of the designcodes referred to such as the DNV rules have been updated after these studies were performed.Faulkner (1992) refers to the 1982 and 1987 editions of DNV (1992a). Also an evaluation ofNPD (1994) is missing. The design equations for ring-stiffened cylinders in NPD (1994) isimproved relative to those in the older versions of the DNV classification notes and may berecommended for use in reliability analysis. The American design rules for design of shellstructures have lately been investigated by Miller and Saliklis (1993). This document is alsoreferred to as a database for a reliability evaluation of other codes and regulations such as NPD(1994) and DNV (1992).

7.2.7 Capacity of Tubular Joints – Static Strength

The design equations for static strength of grouted joints in the DNV Rules (1993) are mainlyderived from test data reported by Gibstein (1973, 1976). The scatter in the test data as comparedwith the capacity equations are small. The equations are presented as lower bound values, but




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due to small scatter the difference from that of the mean values are negligible. As these datarepresent only one test laboratory one should also consider scatter in other test data to getrepresentative COVs to be used for a reliability analysis.

Test data for static capacity of tubular joints from a number of test laboratories have beencollected by Department of Energy (1990, Background to New Static Strength Guidance forTubular Joints in Steel Offshore Structures). A standard for design of steel offshore structures isunder development and is planned to be out for voting in 1996. Work on the capacity equationsfor tubular joints has been performed for this purpose as reported by Ellinas et al. (1993). Seealso report from MSL to Health & Safety Executive (1992). For reliability analysis of thecapacity of tubular joints, it is recommended to look into the background for the database, as it isimportant to know which failure criterion is used for the test. For example: How is failure intension defined? Is it by first discovered crack as used by DNV (1993), NPD (1990), and APIRP2A - LRFD (1993a), or is it by the ultimate capacity as used by Department of Energy (1992)?It should also be mentioned that the definitions of the characteristic design equations in thevarious design codes are different such that they will be associated with different bias. For biasand COVs for the design equations in API RP2A - LRFD (1993a) reference is made to Yura et al.(1980). A background for the design equations in NPD (1990) is given by Lotsberg (1990a).

7.2.8 Strength of Grouted Connections

For capacity of grouted pile/sleeve connections reference is made to Sele and Skjolde (1993).The form of the design equations in the different codes is different, and this leads to significantdifferences in requirements to pile/sleeve lengths. It is indicated in this paper that the design canbe efficiently improved by reducing the modelling uncertainty in the design equations by usingparameters in the equations that better represent the physical behavior. This should also be keptin mind if a reliability analysis is performed on the basis of such equations.

7.2.9 Capacity of Concrete

The true strength of a reinforced concrete member differs from the nominal strength calculatedby the design engineers. This is owing to variations in material strengths and geometry of themember, as well as to the variabilities inherent in the equations used to compute the memberstrength. Properties of strength distributions used in reliability analysis have been presented inthe literature, see Mirza et al. (1979, 1982). The uncertainties associated with an actual designequation should be evaluated for the relevant failure mode. From tests, the COV for thecompressive strength of concrete has been shown to be approximately 0.18. (The same COV isalso indicated for the tensile strength of concrete.) The distributions referred to were assumed tobe normal. The COV for the yield strength of the reinforcement is given as 0.12.

7.3 Geotechnical Data

7.3.1 Types of Uncertainty

Geotechnical data may suffer from aleatory as well as epistemic uncertainties. Geotechnicalparameters should be represented by their probability distribution functions, preferably specifiedin terms of point estimates of their respective mean values and standard deviations. In certain




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problems involving a large volume of soil, the spatial structure of the variation of a soilparameter may also be needed. The spatial structure is described by the autocorrelation functionor the semi-variogram of the parameter.

To obtain the mean value and variance of a soil property, the basic theory of statistics is used. Asoil property may be obtained from an "involved" calculation and depends on a number ofrandom parameters. An example of such a soil property is the cyclic shear strength for a designstorm which depends on the history of the wave loading. For estimation of mean value andvariance of such a property, a Monte-Carlo simulation scheme may be worthwhile.

The choice of the statistical tool to be used for quantification of the uncertainty in geotechnicaldata depends on the nature of the problem and the number of data points available. Table 7. 5summarizes the statistical approaches that have been applied to geotechnical problems. The tabledoes not give an exhaustive list of all existing approaches.

7.3.2 Uncertainty in Soil Properties

The selection of the distribution function for a soil property is usually a simple matter. Based onTable 7. 6, one should use a normal or a lognormal distribution for most soil properties. Onemay want to use a bounded uniform distribution, if one expects the values within a range to beequally probable. When enough data are available, it is good practice to plot the data and rungoodness-of-fit tests.

Table 7. 5 Statistical and Reliability Tools and their Application to Geotechnical Analysis

Method Application Recommendation

Statistical approachesShort-cut estimates(Baecher, 1985)

× When little data are available.× Obtains mean and variance, and gives

bound for standard deviation [multipliesrange of values by factor < 1 to accountfor lack of data points].

× Useful for "symmetrical" data.

× Use to check variance when indoubt whether value used istoo low.

× Quick method.

Mean and variance,empirical histogramsor fitted probabilitydistribution functions

× Best method when enough data .× Distribution function obtained from

plotting on probability paper orgoodness-of-fit tests.

× Use whenever possible.

Stochastic interpola-tion (geostatistics)(Nadim, 1988;Keaveny et al., 1989)

× To do statistical site description.× Software adapted to geotechnical para-

meters exists.× Important advantage in data presen-

tation.

× Apply if enough data areavailable.

× Use for any soil property ormeasured parameter (e.g.depth, layer thickness).




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Table 7. 6 Probability density functions for different soil characteristics

Soilcharacteristic

Soil typeProbabilitydistribution

function

Mean value(North Sea

soils)

Coefficient ofvariation

(North Seasoils)

Cone resistance SandClay

LognormalNormal/Lognormal

* *

Undrained shear strength,su

***Marine clay(triaxial tests)Clay (from sim-ple index tests)Clayey silt

Lognormal

Lognormal

Normal

*

5 - 20%

10 - 35%

10 - 30%

su normalised w.r.t.vertical effective stress Clay Normal/Lognormal ** 5 - 15%

Plastic limit Clay Normal 0.13 - 0.23 3 - 20%

Liquid limit Clay Normal 0.30 - 0.80 3 - 20%

Submerged unit weight All soils Normal 4.5 - 11( kN / m3 )

0 - 10%

Friction angle Sand Normal * 2 - 5%

Void ratio and porosity,including initial voidratio

All soils Normal* 7 - 30%

Overconsolidation ratio Clay Normal/Lognormal 1.2 - 40 10 - 35%

* Values very much site- and soil type-dependent

** Function of overconsolidation ratio

*** Undrained shear strength is anisotropic and depends on the type of stresses imposed.The coefficient of variation for tests of good quality (consolidated triaxialcompression/extension, direct simple shear, true triaxial, plane strain) is expected to bethe same. For extension tests, because of the fewer data available and at times moredifficult testing conditions, the coefficient of variation may be higher.

A summary of means and coefficients of variation for various soil characteristics is provided inNGI Report 51411-3.

In geotechnical problems that involve large soil volumes, the "averaged" property governs thefoundation analysis, because local fluctuations average out over large volumes. A normaldistribution is then an asymptotically correct distribution. For the soil variable that cannot




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possibly become negative, a lognormal distribution should be preferred. For example, alognormal distribution is preferable when the undrained shear strength is low, or when thesubmerged unit weight is low (e.g. less than 6 kN/m3 ).

Before any statistical analysis is performed, one should determine whether the parameter enteringthe formulation is meant to represent the characteristic of a large volume (or area) of soil or torepresent a punctual characteristic.

It is recommended that site description strategy be included whenever a large volume of soil isinvolved. The approach involves the following two steps: (1) identification of the correlationstructure of the soil data, and (2) use of the kriging stochastic interpolation technique to estimatethe soil property at the location of interest.

Such procedures and the relevant computer codes were described in Nadim (1988) and Keavenyet al. (1989). The procedures are straightforward, and give unbiased, illustrative and easy-to-useresults in terms of mean and variance. The methods compensate for (1) the inherent spatial vari-ation due to the natural heterogeneity of soil and (2) the limited availability of information, andgives the value of the measurement noise automatically. The key to quantifying the uncertaintyis the selection of a correlation function for the soil property. A minimum quantity of "goodquality" data is, however, necessary to apply these methods.

The uncertainties discussed above represent values for the parameters that are used for theanalysis of bearing capacity of gravity structures, jack-up platforms and suction anchors, where"shallow" type of foundation failure is modelled. However, for piled structures, the failure ismodelled differently. For this reason, uncertainties specific to piles are described below.

Piled Foundations

Current axial pile capacity calculation methods use simplified models that have been derivedpredominantly from onshore load tests on small piles. These empirical methods have lead to anumber of parameters specific to the design of piles.

For clays, the design variables considered are the skin friction factor along the pile (α, β, or λ),the undrained shear strength (su) and at times correction factors to account for specific effects, forexample pile length or cyclic effects. In addition, the end bearing factor (Nc) is used to calculateend bearing.

For sands, the design variables considered include coefficient of lateral soil stress (K), soil-pilefriction angle (δ), limiting skin friction (flim), bearing capacity factor (Nq) and limiting endbearing (qlim).

The uncertainty in the in situ effective overburden stress (s'vo) which enters most of thecalculations is the same as, or less than (due to integration over the depth), the uncertainty for thesubmerged unit weight Table 7. 6.

In clay, the differences between existing design methods lie in the manner of estimating the unitskin friction. There is a general agreement in the profession that the unit end bearing can becalculated with a bearing capacity factor close to 9 times the undrained shear strength of the clay.In most cases, the end bearing component represents only a small part of the total bearingcapacity of a pile in clay subjected to gravity and environmental loading.




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The following uncertainties for pile design parameters in clay are suggested. Note that the APIRP2A, 20th edition (1993) design procedure is used as reference and mean value for the limitskin friction factor, α:

Parameter Clay type Mean COV

α NC clay API (1993) 10%OC clay API (1993) 15%

β, λ All clays ? ?

Nc All clays 9.0 10%

In sand, the differences among the many design methods are found with respect to mainly threefactors:

1) The value of the earth pressure coefficient, and thus the value of the effective stress,allowed in the calculations of axial pile capacity for compression and tension loading.

2) The limiting side friction value, and the extent to which it depends on relative density.

3) The limiting end bearing value and its dependence on relative density.

Elf Aquitaine Production and NGI did a survey of the opinion of experts in the prediction of axialpile capacity in sands (Lacasse and Goulois, 1989). The results of the survey indicated that thebias and coefficient of variation depended on the relative density of sand. The bias in thedifferent soil parameter estimates entering into the calculation of axial pile capacity variedappreciably, but suggested important conservatism. The bias can be expressed in terms of thebias factor, i.e., the expected ratio between the true value of a soil parameter and the predictedvalue of that parameter. Based on this work, the following uncertainties are suggested for pileanalysis in sand:

Parameter Relative density of sand, % Bias factor * COV

K < 35 1.00 15%35 - 65 1.00 15%65 - 90 1.10 25%> 90 1.10 25%

δ < 35 1.00 15%35 - 65 1.00 15%65 - 90 1.00 15%> 90 1.00 15%

flim < 35 1.00 20%35 - 65 1.10 20%65 - 90 1.15 25%> 90 1.25 25%

Nq < 35 1.00 10%35 - 65 1.20 20%65 - 90 1.25 40%




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> 90 1.25 40%

qlim < 35 1.00 10%35 - 65 1.20 20%65 - 90 1.25 30%> 90 1.25 30%

* A bias factor greater than 1.0 indicates conservative calculation, i.e., the calculation method underestimates the true capacity quantity

The bias factors listed above are with respect to the values recommended in API RP2A (1993c).

7.3.3 Cyclic Loading Effects

The loads induced by sea waves, wind and current, acting on an offshore structure, expose thefoundation soil to repetitive loading-unloading-reloading. The period of each load cycle isbetween 5 and 20 s. The pore pressure generated by cyclic loading may accumulate, leading to areduction in the shear strength. Depending on the soil type, foundation dimensions, and drainageboundary conditions, the accumulation of pore pressure may have to be accounted for during afew load cycles, during an entire storm, or during the platform lifetime. On the other hand, therapid rate of loading tends to increase the shear strength of clayey soils. The cyclic soil strengthand deformation characteristics are problem- and definition-dependent, and should be consideredon a case-by-case basis. The main factors governing the cyclic soil characteristics are:

• Soil type• Design storm composition• Foundation dimensions and drainage boundary conditions• Overconsolidation ratio• Ratio of cyclic load amplitude to permanent load

If the permanent load on the foundation is significantly larger than the cyclic load amplitude(one-way cyclic loading), then the cyclic effects will be minor. An example is a TLP supportedby piles, where the cyclic axial load amplitude at the pile top is always less than the permanenttensile force exerted by the tethers. On the other hand, if the cyclic load amplitude is larger thanthe permanent load such that two-way cyclic loading may occur during the design storm, thencyclic loading effects may be significant, especially in overconsolidated clays. An example is alight jacket at a site consisting of overconsolidated clay and exposed to severe storms.

Typically, the mean cyclic shear strength representative for a 6-hour North Sea design storm is 5-15% lower than the reference static strength for normally consolidated clay. The cyclicdegradation effect increases with increasing overconsolidation ratio (OCR), such that for a claywith OCR close to 40, the mean cyclic shear strength will be 30-40% lower that the staticstrength. The coefficient of variation of cyclic shear strength is expected to be 20-30% higherthan the coefficient of variation of static shear strength.

NGI (NGI files) estimated the uncertainty in the cyclic shear strength of two different clayssupporting a gravity structure with deep skirts. The uncertainty was estimated by Monte-Carlosimulation with Latin hypercube sampling of the random variables that influence the cyclicstrength. The mean cyclic shear strength was 30-35% lower than the reference static strength andhad a coefficient of variation of 10-18% depending on the soil type and mode of failure. The soil




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considered at the site was very uniform, and pore pressure accumulation occurred over the 70-year lifetime of the platform. These numbers are an example only and should not be construed astypical for a gravity structure on clay.

No systematic studies have been performed to evaluate the cyclic loading effects on pile capacity.The few deterministic case studies that have been published (e.g. Karlsrud and Nadim, 1990)suggest that the cyclic axial pile capacity in clay may be 60 - 80% of the static capacity for two-way cyclic loading (e.g., a light jacket) and 85 - 110% of the static capacity for one-way cyclicloading (e.g., a heavy jacket or a TLP). A cyclic capacity greater than static capacity means thatthe strength increase due to rapid rate of loading is more dominant than the strength degradationdue to cyclic loading.

Jostad et al. (1994) describe a procedure for establishing the cyclic capacity of a jack-up spud canon clay which accounts for the spud can fixity. The cyclic lateral and overturning momentcapacities are significantly less than the corresponding static capacities because for these types ofloads, the soil supporting the spud can is subjected to pure two-way cyclic loading. On the otherhand, the cyclic vertical capacity is usually greater than the static vertical capacity because it is aone-way cyclic load and the rate effects are more important than cyclic degradation. Jostad et al.(1994) suggest the curves shown in Figure 7. 1 for evaluating the cyclic effects on the capacity ofa spud can at a clay site in the North Sea under a 50-year storm.

The bearing capacity of spud cans on dense sand was studied by NGI (NGI Report 524085-8,1993). Even for a large spud can, there will be little accumulation of pore pressures during theentire storm. However, during one load cycle, the situation will be partially drained. Due todilation, the undrained or partially drained capacity of dense sand is significantly higher than itsdrained capacity. Therefore, in the NGI study it is suggested that the cyclic effects are ignoredfor a spud can on dense sand.

Figure 7. 1 Normalized cyclic capacity for a jack-up spud can at a North Sea claysite (Jostad et al., 1994)




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7.3.4 Uncertainty in Calculation Model

The uncertainty in calculation model, with bias and coefficient of variation, depends on the pro-blem at hand. The mean and coefficient of variation of the model uncertainty to be used in thereliability analysis are often difficult to assess. Quantification of the model uncertainty, wherepossible, should be based on the following information:

• Literature review• Comparisons of model test results with calculations• Survey of expert opinions (when available)• Relevant case studies and back-calculation

In probabilistic geotechnical analyses, it is recommended to apply a bias and coefficient ofvariation to each of the geotechnical parameters entering the formulation.

Table 7. 7 Uncertainty in the calculation models for bearing capacity of gravitystructures and axial pile capacity (20th edition of API RP2A, 1993)*

Foundationtype Soil type

Probabilitydistribution

function

Recommendedmean value

(bias)

Recommendedcoefficient of

variation

Gravity,

Suctionanchor

Dense sand (un-drained or partiallydrained)Clay (undrained)Other soils

Normal/Lognormal

Normal/BetaNormal

1.1 - 1.5**

1.0 - 1.1 1.0 - 1.3

10 - 15%

5%10 - 15%

Pile, loadedin compres-sion

ClaySilicate sandOther soils

NormalNormal ?

1.0 - 1.21.3?

15%40%

?

Pile, loadedin ten-sion***

ClaySilicate sandOther soils

NormalNormal ?

1.0 - 1.21.1 - 1.3

?

15%30%

?

* Model uncertainty on API RP2A for piles only** Bias could be even higher for highly dilative dense sands*** Only skin friction contributes in tension, no end bearing

Table 7. 7 lists the NGI recommendations for uncertainty in the calculation model for bearingcapacity of gravity and piled platforms.

An extensive study of the uncertainty in different calculation models for axial capacity of piles isprovided in NGI Report 514166-1 (1994).




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An extensive study of model uncertainty for shallow types of foundation (e.g. gravity structure,anchors, spud can) was done in NGI Report 51411-8 (1988). Since then Andersen and his co-workers presented a number of model tests that enabled a quantification of model uncertainty(e.g., Andersen et al., 1989, 1993). These lead to the best estimate of the model uncertaintyvalues in Table 7. 7.

In Table 7. 7 a bias greater than 1 means that the calculation model is conservative (i.e., it isexpected to underpredict the actual capacity), and a bias less than 1 means that the calculationmodel is expected to overpredict the actual capacity.

7.3.5 Uncertainty in Load Effects

The uncertainties in the load effects may be aleatory as well as epistemic. The uncertainties inthe loads, storm characteristics and load effects used as input to the geotechnical analyses can bevery important. Previous studies at NGI and at Veritas Research showed that the uncertainties inthe loads and storm characteristics can predominate the geotechnical reliability and probability ofnon-performance of a concept.

Table 7. 8 Geotechnical analyses where uncertainties in environmental loads aremost significant

Structure Geotechnical analysis Comments

Jack-up(Nadim andLacasse, 1993;Nadim et al., 1993)

×Bearing capacity×Hydraulic stability×Soil reactions×Load-displacement (soilspring stiffness)

Soil type and size of footingdetermine whether short term orlong term loads are morecritical. Combined static andcyclic loads are important.Distribution of loads amongfootings is needed.

Piled ×Axial pile capacity×Lateral pile capacity (rigidity matrix)

Only worst characteristic stormis needed. Distribution of loadsamong piles is needed.

Gravity,

Suction anchor

×Bearing capacity×Hydraulic stability×Soil reactions×Load-displacement (soilspring stiffness)

In most cases, long term effects(design storm or lifetime) needto be considered.Combined static and cyclicloads are important.

The geotechnical analyses in Table 7. 8, exclusive of response to earthquake loading, were foundto be most affected by the environmental loads.

In 1991, NGI carried out an expert opinion survey on the uncertainties in predicted extreme mud-line forces and long term distribution of mudline forces (NGI Report 514165-4). Based on thesurvey, a global coefficient of variation on the maximum mudline wave forces due to theenvironment between 15 and 30%, with 20 or 25% as a best estimate, could be expected for




177

gravity, jack-up, and jacket structures. The respondents who used the results of recent researchtended to estimate lower uncertainties than the respondents basing their estimates on presentpractice.

Until more information becomes available, the uncertainties obtained should be used asguidelines for geotechnical analyses. Parametric studies should be included to illustrate theeffect of a change in the uncertainty in the load effects on the geotechnical component.

A dialogue among the specialists (environment, hydrodynamics, structures and geotechnics)involved in the design of an offshore structure will enhance the understanding of theuncertainties in the load effects and should contribute to safer and more cost-effective designs.

7.4 Geometry Data

Geometrical quanties describe the shape, size and overall arrangement of structures, componentsand cross sections. The variability of most of the geometrical quantities can be considered smallor negligible, in comparison with the variability associated with load effects and materialproperties. These geometrical quantities may be assumed to be non-random and as specified onthe drawings.

Where the deviation of certain of the geometrical quantities from prescibed values may have asignificant effect on structural behaviour and resistance of a structure, the geometrical quantitiesshould be considered either as random variables or as parameters of the random variables whichdescribe actions or structural properties.

Eccentricities, inclinations and curvature affecting columns and walls are the most usualgeometric quantities to be taken into account as basic variables. Eccentricities and inclinationsmay be significant for fatigue life evaluation while curvature effects may be important todetermine the buckling capacity of columns.

When structural members are rolled in the steel mill, the shaping rollers will gradually wear,resulting in variations in the cross-sectional dimensions of the rolled member. Also, the rollingmill practice tends to adjust the rollers in such a way as to obtain the maximum amount ofmaterial within the tolerance limits. Variations in dimensions of about 5000 H-shapes rolled atseveral European steel mills are given by ECCS (1976). The relative variation in height andflange width is of course small as compared to the variation in thickness. It can be seen that thereis a tendency for the flanges to be thinner, and the web to be thicker than the nominal values.Coefficients of variations for the plate thickness are dependent on the actual thickness as shownin Table 7. 9. A normal distribution is assumed.

Table 7. 9 Variations in thicknesses of steel plates

Plate thickness (mm) Normal distribution Coefficient of variation (%)12.7 1.825.4 1.050.8 0.7




178

The statistical evaluation of measurements on unstraightened IPE 160 test columns of differentlengths is also given by ECCS (1976). A normal distribution for the central bow of the columnsis assumed. The mean value is given as 0.00085 times the column length with a coefficient ofvariation equal 23%. It should be noted that the measurements are performed on single columnsbefore they are placed into a structure. The deviations in fabricated structures are expected to belarger than this, cfr. fabrication requirements on tolerances such as NPD (1994). For structuralmembers subjected to corrosion, a long-term degradation of material thickness is to be expected.

7.5 Fatigue Data

7.5.1 Welded Connections and Base Material Steel Structures

S-N data for welded connections and base steel material are given in the DNV ClassificationNote No 30.2 (1984). The data are given for air, seawater with cathodic protection, and seawaterand free corrosion. It should be kept in mind that the major part of the fatigue test data areobtained in the region 10 105 7− cycles to failure. Outside this range the uncertainties in the S-Ncurves may be significantly larger than that given in the design code. S-N data for welded steelstructures in air are shown in Table 7. 1. S-N data for cathodically protected structures inseawater are the same as those for number of cycles less than 107 cycles from Table 7. 1. Forthese data, a cutoff value at 2 108⋅ is assumed.

Table 7. 10 S-N data for welded steel structures in air.

Eq. (7. 6); DNV Classification Note No. 30.2 (1984).

N ≤ 107 N > 107

Class Log a Log s Log a' m Log a' mB 15.3697 0.1821 15.01 4.0 17.01 5.0C 14.0342 0.2041 13.63 3.5 16.47 5.0D 12.6007 0.2095 12.18 3.0 15.63 5.0E 12.5169 0.2509 12.02 3.0 15.37 5.0F 12.2370 0.2183 11.80 3.0 15.00 5.0F2 12.0900 0.2279 11.63 3.0 14.72 5.0G 11.7525 0.1793 11.39 3.0 14.32 5.0W 11.5662 0.1846 11.20 3.0 14.00 5.0T 12.6606 0.2484 12.16 3.0 15.62 5.0

The intercept Log a defines the mean S-N line. Log s is the standard deviation of the residuals inlogN. Log a' is the characteristic value of the intercept, defined as the mean minus two standarddeviations. Log denotes log base 10.

Table 7. 11 Crack growth parameters. DNV Classification Note No. 30.2 (1984)

m lnC in units (N,mm)




179

Mean value Standard deviationWelds in air 3.1 −29.84 0.55Welds exposed to seawater 3.5 −31.01 0.77

Crack growth data such as C and m in Paris' crack growth formula, Eq. (7. 7) can be found inthe DNV Classification Note No 30.2 (1984), see Table 7. 11, which quotes (fixed) values for mand mean values and standard deviations for lnC. Threshold values as function of mean stress,strength of steel and environment can be found in British Standard Institution (1991).

C is assumed to be lognormally distributed, hence lnC is normally distributed. Note that the meanvalue of C is not equal to the exponential function value of the mean value of lnC. Formulas forthe mean value and standard deviation of C in terms of the mean value and standard deviation oflnC can be found on page 86 of Madsen et al. (1986). Reference is also made to Section 7.7 ofthis guideline.

7.5.2 High Strength Steel

It will normally be conservative to use the S-N data quoted in the DNV Classification Note No.30.2 also for high strength steel with a yield strength in the range 500-700 MPa. The initiationperiod will be longer for high strength steel than for traditional steels for which the S-N data arederived. Thus the degree of conservatism will be larger for higher capacity curves denoted B andC than for curves more relevant for crack growth from more heavily notched areas as for examplethe F2 curve.

7.5.3 Cast and Forged Steels

S-N curve B in the DNV Classification Note No 30.2 may be used for cast materials. It should benoted, however, that in case of weld repair the classification is reduced to that of class C and it isrecommended to use the corresponding data for reliability analysis provided that more relevant S-N data are not available.

S-N curve B in the DNV Classification Note No. 30.2 may be used for forged material.

7.5.4 Welded Connections and Base Material Aluminium Structures

S-N data for aluminium structures can be found in ECCS (1992).

7.5.5 Chains

A design S-N curve is given in API RP 2FP1 (1993b). The data used as background for designcurves for chains are shown in Figure 7. 2. A regression line through the data gives a mean curveas N*Rm =a, with m=3.36, and a=2367. R is defined as the ratio between the load range and theminimum guaranteed breaking strength, and N is the number of cycles to failure. Assuming thatthe data are lognormally distributed, the standard deviation of the residuals of logN is log s=0.403. The data are assumed to apply both for chains with studs and studless chains. The dataare assumed to apply for one chain link only. It may be difficult to achieve a satisfactory




180

corrosion protection system for chains; thus, the S-N data should be reduced by a factor 2 on lifebased on DNV Classification Note No. 30.2. The S-N data are relevant for one chain link. Forfatigue analysis of a long chain the series system effect is included in the design throughreliability analysis of the chain as a system where the reliability of single links is input to thecalculation.

logR

logN

Figure 7. 2 S-N Data for Chain Links

For reliability analysis the Miner damage ratio is modelled as lognormal with mean 1.0 andCOV=0.30.

7.5.6 Wire Ropes

7.5.6.1 General

A design S-N curve is given in API RP 2FP1(1993b) based on data. The S-N data are given as themean curve minus two standard deviations, or as lower bound values. Here, the same S-N dataapplies for six/multi strand wires and spiral strand wires. Different design curves have beengiven for the six/multi strand and the spiral strand wires in the proposed 1995 edition of the API




181

RP 2FP1. The curve for spiral strand data is above that for six/multi strand data. It is suggestedto base the reliability analysis for wire ropes on the last available data which are assumedincluded in the design curve for spiral strand in API RP 2FP1 (1995). It is assumed that also thesedata are lower bound data or data corresponding approximately to the mean minus two standarddeviations as normally used for design.

7.5.6.2 Scatter in S-N Data

The scatter in test data can be expressed by the standard deviation, log s, as measured along thelog N axis, when assuming that the test data in log N are normal distributed. Note in this contextthat, although log s is the standard deviation of logN, s=10log s is not the standard deviation of N.

Tests reported in the DEWA project (Det Norske Veritas, 1985) for 32 mm six strand wires fromtwo manufacturers tested at a pretension level 17.5% of the wires minimum breaking strengthresulted in log s = 0.27. Taking data from more manufacturers, as well as different levels ofpretension, a larger scatter and a higher standard deviation may be considered likely. Assumingthat 95% of the test data should fall within an upper and a lower scatter band, a log s value equalto 0.30 corresponds to a factor of 15 on life between the upper and lower scatter bands; and a logs value equal to 0.35 corresponds to a factor 25 on life. It is not likely that the scatter in the datais larger than this, and the lower value is probably more correct than the highest value which maybe included for sensitivity evaluation. It is assumed that all failures at the transitions to thesockets are deleted from the data base. (I.e., it is a prerequisite that the design of the transitionbetween the wire and the socket is such that the probability of a fatigue failure at this region issmall.) These values may be used in reliability analysis of wires. (Note that joint industryprojects on wires have been performed in Great Britain lately and that these data may be used forreliability analysis when they are available.)

7.5.6.3 Length Effect

A long wire has a larger number of fatigue initiation points than a short wire. The actual wiresare longer than that the test data are based on. Knowing the scatter for one fabrication, this effectcould be accounted for in a reliability analysis and also in a deterministic design procedure. It isnoted that the scatter in test data from one manufacturer of wires is as compared with the generalscatter in S-N data used for the reliability study, ref. test data from DEWA (DNV, 1985). It isfurther noted that there is approximately full correlation in load effect along the length of a wire.Hence, it may be sufficient to use the S-N data with the proposed scatter without any additionalreduction in strength due to length effects.

7.5.6.4 Effect of Mean Load Level

The S-N data in API are given for a mean load of 30% of the breaking strength. The actual meanload level for traditional anchoring systems is expected to be less than this. This is thusconservative as the fatigue capacity is reduced by increased mean load, cfr. the DEWA project(DNV, 1985).

7.5.6.5 Threshold Value




182

There is a threshold value in the S-N data also for wires. This value has been given as 13% of thebreaking strength by HSE (1991). However, this is not included in the design data and if notincluded in the reliability analysis either, this will lead to conservative results.

7.5.6.6 Load Spectrum

It has been demonstrated by tests that a preloading of a wire is beneficial for the fatigue life, ref.data presented by HSE (1991). Thus, the fatigue damage obtained under a fatigue spectrum loadis normally less than that for constant amplitude testing. Thus, it is considered conservative touse S-N data where the major part of the data are obtained from constant amplitude testing whenused for design with a spectrum load. For a reliability analysis, the Miner damage ratio may beassumed lognormally distributed with mean value 1.0 and COV=0.20.

7.5.7 Stress Concentration Factors

Parametric formulas for prediction of stress concentration factors at simple tubular joints aregiven in the literature: Kuang et al. (1975), Gibstein (1978), Wordsworth and Smedley (1978),and Efthymiou (1988). These formulas are derived on the basis of experiments and finite elementanalyses. By comparison of measurements with predictions by these models, Hellier et al. (1990)have estimated mean values and standard deviations of the ratio between measured and predictedSCF for the individual models, see Table 7. 12, which covers SCF data for T and Y joints. Such aratio between a measured and a predicted quantity is a (random) model uncertainty factor. Thereported mean values are in the range 0.81-1.01 and the reported COV's in the range 0.13-0.25.Mean values deviating from 1.0 indicate the presence of a bias in the predictions, and the meanvalues are therefore sometimes denoted bias factors. In Table 7. 12, axial load, in-plane bending,and out-of-plane bending have been considered.

Table 7. 12 Measured-to-predicted SCF's in Tubular T and Y joints

based on data from Hellier et al. (1990) (Normal distribution)

Type of load(Number ofspecimens)

SCF range Distributionparameter

Gibstein Kuang Smedley UCL

Axial load(24)

3.3-13.7 Mean valueSt. dev.

0.970.22

0.920.17

0.850.11

0.790.13

In-planebending (12)


0.890.19

0.960.20

0.810.19

0.700.14

Out-of-planebending (25)


0.980.18

1.010.25

0.910.16

0.810.11

Note in this context that the bias θ of an estimator �x of a quantity x is in general defined as theaverage error of estimate, i.e., the deviation of the expected value of a statistical estimate and thequantity it estimates, hence

[ ] [ ]θ = − = −E x x E x x� � (7. 2)




183

in which x is the sought-after quantity and �x is the estimate of x. Reference is made to Benjaminand Cornell (1970), Snedecor and Cochran (1989) and Efron and Tibshirani (1993). Accordingly,the bias factor is defined as

f E xxθ = �

��

��(7. 3)

This factor can be applied as a factor on predicted quantities to correct for their bias andthus obtain quantities with correct expectation for use in calculations. Note, however, thatin engineering contexts, the bias factor is sometimes referred to just as the bias.

7.5.8 Miner-Palmgren Hypothesis of Cumulative Damage

For reliability of steel structures the Miner damage ratio may be assumed lognormal with meanvalue 1.0 and COV=0.30, Wirsching (1984). This also applies to chains. For wires, reference ismade to Section 7.5.6. A summary of data is given in Table 7. 13.

Table 7. 13 Cumulative damage

Component Distribution Mean value Coefficient ofvariation

Welded steelstructure

Lognormal 1.0 0.30

Chain Lognormal 1.0 0.30Wire Lognormal 1.0 0.20

7.6 Non-Destructive Examination

7.6.1 Background

Non-Destructive Examination (NDE) is used to detect and size defects in structures. In thissection NDE methods used to localise defects in welded structures are described. Cracks andcorrosion in form of general corrosion and corrosion pits are of major concern for offshorestructures. Cracks may be due to fatigue and are then most frequently found at the surface attransitions from weldment to base material (at undercuts and notches) and less frequently in theinterior of the connections. For general references, see Silk et al. (1989), Førli (1989), CIRIA(1989), and Førli (1990).

Magnetic Particle Inspection (MPI) is the method most frequently used for detection of surfacecracks. However, the development of use of Eddy Current has been promising and may partlysubstitute the use of MPI. MPI and Eddy Current can be used to detect cracks and the surfacelength can be determined from these measurements. For sizing of crack depths other methodshave to be resorted to, such as the AC Potential Drop method (ACPD) and the Time of FlightDiffraction Technique (TOFD). For a description of ACPD, see Hansen and Rangnes (1981), andDower (1985). For a description of TOFD see Hawher (1984) and Gruber and Jackson (1985).The cleaning of the actual area is the main contribution to the cost of performing MPI. Normallythe paint has to be removed by grinding, sandblasting, or brushing. After an MPI, it is important




184

to reestablish a good painting such that corrosion of the inspected area is avoided. (Such post-inspection corrosion has formed a severe problem for existing structures, cfr. semisubmersibles.)Both alternating current and direct current can be used for MPI, but alternating current ispreferred as this method leads to a more efficient magnetic field at the surface where fatiguecracks are most likely to occur. The efficiency of the NDE techniques is described by probabilityof detection (POD) curves as are defined in Section 7.6.2.

Eddy current may be used to detect fatigue surface cracks without removal of the painting. Thus,the use of Eddy Current becomes less costly than use of MPI. Eddy Current is based onelectromagnetic induction. A coil placed in a sensor subjected to alternating current produces amagnetic field in the material. This magnetic field introduces eddy current which results inmagnetic flux in the material. This work against the coil-produced magnetic field and ismeasured electronically. As the magnitude of the eddy current is a function of the materialpermeability and electrical conductivity, imperfections in the material, like cracks and pits, willbe discovered. One of the main problems during development of the method has been to adaptthe method to variations in the weldment, the heat effected zone and the base material. Also themethod has been rather sensitive to rough surfaces and variations in paint thicknesses. Theseproblems have been partly solved and the method can supplement and to a large extent replaceMPI on painted areas provided that the weld surfaces are not too rough. On welded surfaces themethod is less reliable. The method can be used on plates in deck structures and similarstructures, but is not recommended to be used on sharp transitions like that around thecircumference of brackets and cutouts. Another problem with eddy current may be the largenumber of spurious indications. According to Rudlin and Dower (1991) as much as close to 50percent of the inspected components resulted in indications of defects which were not reallypresent. For practical problems this indicates that a number of components inspected by eddycurrent might have to be investigated further by MPI. This may reduce the advantage of the eddycurrent as compared with MPI.

Ultrasonics can be used to detect internal defects, also surface cracks and size the defects.Ultrasonics can also be used for measurements of wall thicknesses and localizing corrosion pits.Reference is made to a joint industry project on chains carried out at Det Norske Veritas for thereliability of ultrasonics on chains (Harbitz, 1980). Some of the data are also presented inLereim (1985).

7.6.2 POD Data

Probability of detection data (POD) for the nondestructive examination techniques mostfrequently used for inspection of cracks can be described by the following expression

( )P a ba

x

( ) = −

+

1 1

10

(7. 4)

where x0 and b are given in Table 7. 14 for a 95 percent confidence level. This is one of manymodels for POD curves. The crack length at 90% probability of detection is also shown in thetable. The recommended values are derived from investigations by Førli (1990, 1991) andKauppinen and Sillanpaa (1990).




185

The POD data for magnetic particle inspection below water and above water, as well as withrough surface and ground surface, are shown in Figure 7. 3. Also the POD for Eddy Current isshown in Figure 7. 3. Note that under field conditions, the curves will not approach a probabilityof detection of 1.0 for increasing crack length a, but some value near 0.95, dependent on manyfactors.

Table 7. 14 POD data for NDE

x0 (mm) b a0 90. (mm)

MPI Under Water 2.950 0.905 33.

MPI above water;ground test surface

4.030 1.297 22.

MPI above water; notground test surface

8.325 0.785 137.

Eddy current 12.28 1.790 42.

-

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,80

0,90

1,00

0 20 40 60 80 100 120

Crack length in mm

Prob

abili

ty o

f det

ectio

n

MPI under w ater

MPI above w ater; ground

MPI above w ater; not ground

Eddy current

Figure 7. 3 Probability of detection vs. crack length under various laboratoryconditions




186

7.7 Example Application

7.7.1 Physical Problem

This example will evaluate the fatigue capacity of a welded detail applying both the S-N fatiguecapacity and the Fracture Mechanics (FM) fatigue capacity. Different types of loading conditionswill be considered, as well as the effect of inspection updating applying both MPI inspectionmethods with and without previous grinding, as well as Eddy Current inspection.

a

2c

t = 22 mmW = 2000 mm

Figure 7. 4 Fatigue sensitive detail investigated

7.7.2 Deterministic Analysis

The butt weld is located in non-corrosive environment, and having the S-N fatigue capacitydefined through the C-curve, see DNV Classification Note 30.2. The crack growth is assumed tooccur from a surface defect at an undercut of the weld (transient to base material).

The surface weld defect is modelled as a two-dimensional semi-elliptical surface crack. TheParis' equation is applied to describe the fatigue capacity, accounting for a threshold on the stressintensity.




187

In the modelling of the long-term stress range that the structural detail is exposed to, threeloading conditions are considered: (1) Pure membrane loading, (2) combined 50% membrane and50% bending loading, and (3) combined 30% membrane and 70% bending loading. The type ofloading will not influence the estimated S-N fatigue capacity, but will greatly influence both theestimated FM-fatigue capacity as well as the effect of the inspection updating. The loading isnormal to the plane of the crack.

The long-term stress range distribution is assumed to be described through a Weibull distributionhaving a shape parameter equal to h=0.90 and a highest stress range out of n0

810= stress cyclesequal to ∆σ0 346= (N / mm2 ) , giving a Weibull scale parameter of, [DNVC Fatigue Assessmentof Ship Structures],

qn h= =

∆σ0

01 13 6

(ln ). )/ ( N / mm2 (7. 5)

The parameters of the C-curve for non-corrosive environment are given in Table 7. 10.

The S-N fatigue capacity is given by

D n qa

mh

Sq

qa

mh

Sq

nm h

nm h

= +�

��

�

��

�

��

�

�� + +

�

��

�

��

�

��

�

��

�

�

�

��

17

27

1

1 10

2

2 101 1Γ ; ;γ (7. 6)

where

S107 = Stress range for which change of slope of S-N curve occura m1 1, = S-N fatigue parameters for N <107 cyclesa m2 2, = S-N fatigue parameters for N >107 cyclesγ(; ) = Incomplete Gamma function, to be found in standard tablesΓ(; ) = Complementary Incomplete Gamma function, to be found in standard tables

The accumulated S-N fatigue damage over 20 years is found to be D = 0 33. . Based on a usagefactor of η =1 0. , the equivalent S-N fatigue design life is 60 years.

The FM approach is based on Paris' equation,

[ ]

[ ]

dadN

C K a c

dcdN

C K a c

am

cm

= ⋅

= ⋅

∆

∆

( , )

( , )(7. 7)

where a is the crack depth and 2c is the crack length. The crack growth rate is zero for stressintensities lower than ∆Kth . The crack growth parameters in Table 7. 11 are used for the analyses.

The initial crack depth is calibrated in order to achieve an equivalent fatigue capacity for the FMfatigue model as the S-N fatigue model, where FM failure is assumed to occur for through-




188

thickness crack. Assuming an initial aspect ratio a c/ .= 0 15, an equivalent initial crack depth of0 05. mm is estimated for pure membrane loading condition.

In Table 7. 15 both the S-N and FM estimated fatigue lives are estimated for different loadingconditions, assuming an annual number of stress cycles equal to 5 0 106. ⋅ . It is seen that the FMfatigue life is sensitive to the stress distribution over the weld, while the S-N fatigue life is onlyinfluenced by the hot spot stress at the weld surface.

Table 7. 15 Calculated Fatigue Lives

Mem. Stress Ratio SN Fatigue Life FM Fatigue Life1.0 60 years 60 years0.5 60 years 80 years0.3 60 years 92 years

7.7.3 Inspection Updating

The influence of inspection updating is to be investigated for the given detail, where both MPIand Eddy Current inspection procedures are to be considered. As the effect of both theseinspection methods are dependent on the crack length (the failure margin is modelled as afunction of the crack depth), the effect of inspection updating is highly dependent on the aspectratio, or the crack-depth/crack-length relationship.

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0Crack Depth

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Asp

ect R

atio

a/c

Smem / Stot = 1.0

Smem / Stot = 0.5

Smem / Stot = 0.3




189

Figure 7. 5 Aspect ratio of the surface weld defect over the fatigue life dependingfor different stress distributions over the cross section

The aspect ratio of the crack as a function of the crack size is, however, dependent on both theinitial geometry of the surface crack and the type of stress distribution the crack is exposed toover the weld thickness (the fraction of membrane/bending stress). In Figure 7. 5, the aspect ratioof the surface weld defect over the fatigue life of the crack is derived for different loadingconditions applying PROFRAC. The loading conditions are defined as the ratio of membranestress relative to the total stress level (absolute sum of membrane stress and bending stress atweld surface).

In the modelling of the effect of inspection updating, the POD for NDE as given in Table 7. 14 isapplied for MPI inspection with and without previous grinding, and Eddy Current inspection. Thedetection ability of the three inspection methods can be represented through a minimumdetectable crack size (length). The cumulative distribution functions for the detectable crack sizeis defined through the respective POD curves, for which equivalent Lognormal distributions arefitted. The minimum detectable crack size distribution for the different inspection methods aresuggested defined through lognormal distributions with parameters as quoted in Table 7. 16.

Table 7. 16 Statistics of Minimum Detectable Crack Size (Length in mm)

Inspection Method Mean(mm)

Standard deviation(mm)

Defined lower limit(mm)

MPI with grinding: 11.1 36.7 0.0MPI without grinding: 85.8 909.0 0.0Eddy Current: 20.6 29.7 0.0

It is seen from the data that an MPI inspection not including grinding gives poor inspectionresults. It is further noticed that even though the Eddy Current inspection has a mean detectablecrack size about twice that of the MPI with grinding, less uncertainties are associated with thisinspection procedure (a lower standard deviation).

In order to investigate the effect of the different inspection means, a stochastic modelling of theweld containing the surface weld defect was introduced and a reliability analysis carried out.Uncertainties were associated with the modelling of the material parameters of the weld, theequivalent initial crack depth defining the surface weld defect, the semi-elliptical aspect ratio ofthe weld, and the equivalent stress intensity level at the edge of the crack due to an externalnominal stress level.

In Figure 7. 5, the estimated reliability of the weld against through-thickness crack is estimated asfunction of the number of years the structural detail is in service. It is seen that the reliability ofthe weld is decreasing quite rapidly with time.

After 10 years of service the weld is inspected with the above described inspection methods andit is assumed that cracks have not been detected. It is seen that an estimated higher reliabilityagainst through thickness cracking is achieved after the inspection. The inspection method givingthe highest effect on the updated fatigue reliability after inspection is MPI inspection wheregrinding has been carried out prior to inspection. This can also be seen from the modelled




190

minimum detectable crack size (length) for the different inspection procedures, where MPIinspection with grinding has the lowest mean detectable crack size.

The effect of the inspection updating is also highly dependent on the estimated crack size at thetime of inspection. If this crack size is too small, the probability of detecting an existing crack isreduced, which again reduces the influence of not detecting any cracks at an inspection on theestimated fatigue reliability after inspection.

As the goodness of the inspection method applied is dependent on the crack length at the time ofinspection and the failure margin for the crack is a function of the crack depth (and crack length),the aspect ratio of a crack ( / )a c plays an important part in all fatigue inspection updatinganalyses. The necessity of having proper tools for fatigue crack growth evaluation over lifetimefor all fatigue inspection updating analyses is therefore stressed.

So far the crack detection probability data have mainly been provided under laboratoryconditions. Under field conditions, a remodelling is usually necessary in order to include humanreliability aspects of the inspector.

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30Life (Years)

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Relia

bilit

y In

dex

Bet

a

Reliability Curve

No Inspection

MPI with grinding

MPI not grinding

Eddy Current Inspection

Figure 7. 6 Updated fatigue reliability over the life-time as a function of inspectionmethod applied.




191

Addendum

Fatigue material parameters for the S-N and FM fatigue models are usually achieved fromexperiments on a logarithmic scale. The description of the uncertainties on these materialparameters are therefore also defined on a logarithmic scale, e.g. standard deviations are definedfor log10 a in the S-N fatigue model and for ln C in the FM fatigue model. As the stochasticdistribution of the logarithm of these parameters usually is modelled as normally distributed, theparameters themselves, as they appear in the S-N and FM fatigue model, will be lognormal.

In order to avoid confusion in the transformation of mean values and standard deviation of thelogarithm of these parameters from experiment to the equivalent mean values and standarddeviation of the Lognormal distribution of the parameters themselves, the following transformingequations are included:

Assuming the log base 10 of a, i.e., log10 a , to be normally distributed, then the variable a itselfwill be lognormally distributed, having mean and standard deviation:

µµ σ

aa a=

+ ⋅10

12

10 2log logln

; σ µ σa a

a= −10 110 2ln log (7. 8)

If the mean and standard deviation of the lognormally distributed variable a are known, theequivalent mean and standard deviation of the normal distribution of log10 a are,

µ µ σlog loglog( ) lna a a= − ⋅12

10 2 ; σ σ µ µlog log( ) log( )a a a a= + −2 2 2 (7. 9)

The same relationships hold for logarithms with any other base, for example the naturallogarithm, by replacing the base number 10 in Eq. (7. 8) by the pertinent base number, i.e., e forthe natural logarithm. This leads to the following transformation of mean values and standarddeviations between the lognormally distributed variable a and the normally distributed variablelna which is the natural logarithm of a:

µ µ σa a a= + ⋅exp( )ln ln12

2

; σ µ σa a a= ⋅ −2 2 1(exp( ) )ln (7. 10)

µ µ σln lnln( )a a a= − 12

2

; σ σ µ µln ln( ) ln( )a a a a= + −2 2 2 (7. 11)

To make the comparison complete, the following transformation of mean value and standarddeviation between the normally distributed 10th logarithm and the natural logarithm of a variablea exists,

µ µlog

ln

lnaa=

10; σ σ

logln

lnaa=

10 (7. 12)

µ µln loglna a= ⋅10 ; σ σln loglna a= ⋅10 (7. 13)




192

7.8 Geotechnical Example

7.8.1 Physical Problem

This example presents the deterministic and probabilistic analyses of an offshore pile foundationat two times in the platform lifetime.

1) In 1975, before platform installation, when limited information and limited methods of interpretation of the soil data were available.

2) In 1993, after a reinterpretation of the available data using the geotechnical improvements attained in the interim, a reanalysis of the loads, and an analysis of the installation records.

The reanalysis in 1993 was prompted because the operators hoped to increase the gravity loadson deck.

The structure consists of a steel jacket installed in 110 m of water in the North Sea. The jacketwas installed in 1976. The jacket rests on four pile groups at each corner. Each pile groupconsists of six piles (Figure 7. 7). The piles in the pile groups are 60" diameter, with wallthicknesses of 3” and 2.5”. The four intermediate pin piles along axes A and B were 48"diameter, with 2.125” and 1.75” wall thicknesses. At mudline, the jacket has rectangular basearea of about 50 m by 50 m.

Soil Profile

The soil profile consists of mainly stiff to hard clay layers, with relatively thinner layers of verydense sand in between.

In 1975, two soil borings were done at the jacket location, Borings B1 and B2 as shown in Figure7. 7. The two borings indicated comparable soil profiles, although the horizon and the thicknessof the sand layers differed. The soil profiles from Borings B1 and B2 are shown in Figure 7. 8.The soil characteristics in Figure 7. 8 were derived from the "standard" types of tests in commonuse at the time (torvane, pocket penetrometer, unconfined compression test, unconcolidatedundrained test), and an interpretation of the results based on the judgement and experience of thegeotechnical consultant at the time. The friction angle of the dense sand was based on the resultsof consolidated drained triaxial compression tests on recompacted specimens. The friction anglefor the specimens compacted to the highest density possible was measured between 38 and 40degrees.




193

Figure 7. 7 Foundation layout




194

Figure 7. 8 Soil profile from site investigation in 1975




195

Figure 7. 9 Soil profiles for axial pile capacity calculations, 1975-analysis

Figure 7. 9 gives two soil profiles used in 1975 to obtain the deterministic axial pile capacity.Below 20 m, the two borings B1 and B2 showed a wide variation in the soil parameters. Theundrained shear strength in the clay below 20 m was considerably higher, if based on Boring B2.




196

The depth and thickness of the second dense sand layer differed also for the two design profilesof 1975. No advanced laboratory tests enabled one to estimate more appropriate values for thesoil parameters. The parameters and the soil layering can therefore involve considerableuncertainty and include the effects of sampling disturbance.

During pile installation, records were made of the blow count during driving. The individualblow counts for the piles in Leg B1 are presented in Figure 7. 10. No instrumentation of thedriving operation was done. The installed pile lengths are between 36 and 45 m.

Figure 7. 10 Pile driving records, Pile P1 and Pile P6. Leg A5




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Figure 7. 11 Stratigraphy inferred from pile driving and reanalysis of existing data




198

Figure 7. 12 Example of soil profiles for axial pile capacity calculations,

1993-analysis




199

The pile driving records were evaluated by a consultant in 1993 and used to adjust the soilstratigraphy. The result of this "educated" adjustment is shown in Figure 7. 11.

A reevaluation of the borings and laboratory test results using normalized soil characteristics,new soil samples and the running of more advanced laboratory tests (direct simple shear tests,consolidated undrained triaxial tests) led to adjusted soil shear strengths in the stiff to firm clay.As shown in Figure 7. 12, a fairly narrow range of soil strengths are suggested. The full curverepresents pessimistic values, the dotted line the best estimate values. No reevaluation of thefriction angle of the sand was done, although ideally, this should have been done.

7.8.2 Analysis Methods

The deterministic analyses were done with the API RP2A recommended practice in use at thetime of the analysis. The design requirement was a factor of safety of 1.50 under extremeloading and 2.0 under operation loading. The axial pile capacity is a summation of the skinfriction on the pile shaft and end bearing on the pile tip.

The probabilistic analyses were done with first-order reliability method (FORM), where thedeterministic axial pile capacity model was formulated in terms of random variables in eachlayer. In the present example, only the results of the analysis of the capacity of the most loadedpile are considered.

7.8.3 Model Description

Soil Parameters

Table 7. 17 gives examples of the uncertainties associated with the soil parameters in two of themore important soil layers. The selected coefficients of variation reflect uncertainties in thelaboratory test results, possible measurement errors, spatial variability and the uncertainty indegradation due to cyclic loading. Cyclic degradation is important for an overconsolidated claysubjected to a fairly high ratio of cyclic loading. The effect of cyclic loading is expected to beminor for the dense sand.

Very little data were available for the different soil parameters. The mean and coefficients ofvariation were obtained as follows:

Submerged unit weight, γ ′:No measurements were available. The mean value and coefficient of variation were based onexperience acquired for similar soils where many measurements have been taken. For stiff clays,the mean submerged unit weight is 8.5 to 9.0 kN/m3 (as stiffness increases); for very dense sand,the mean submerged unit weight is normally 10 kN/m3. A COV of 5% is a common value forscatter in submerged unit weight.

Depth, z:The layer thickness can vary. Since only two borings are available, the values used in theanalysis are uncertain. The mean layer thickness is based on the measured values from the siteinvestigations. The COV of 10% is based on engineering judgment.

The position and thickness of Layer 7 were quite uncertain in 1975. For this reason the COVwas increased for this layer from 10 to 20%.




200

Undrained shear strength in stiff clay, su:In 1975, the undrained shear strength was based on punctual measurements from index strengthtests, known to give a relatively poor estimate of the undrained shear strength. The data pointsare shown in Figure 7. 9, which explain the high COV.

In 1993, the undrained shear strength profile was based on

(1) results of consolidated-undrained triaxial compression tests at effective stresses relevantfor the in situ values (Figure 7. 12)

(2) a recalculation of the soil shear strength based on the normalised strength ratio forsimilar clays within the same geographical area and with similar geological history.

This led to two soil strength profiles (Figure 7. 12) and COV’s of 10 or 15%.

Friction angles, φ′ and δ and coefficient of earth pressure, K, in very dense sand:Very little information was available for the very dense sand layers. A friction angle, φ′, of 40°(and soil friction angle δ of 35°) is typical for a very dense sand. In 1975 there was little knownabout this angle and the COV was set to 15%. In 1993, considerable research contributed toreducing this COV to about 5%. Lacasse and Goulois (1989) collated the opinion of 40international experts who suggested that the uncertainty about the mean is quite small.

For the coefficient of earth pressure, K, values are again undocumented, but based onengineering judgment, experience and the results of the Lacasse and Goulois (1989) study.

Pile capacity parameter in clay, α:The prediction of the axial capacity of a pile in clay is done with the friction parameter, α, timesthe undrained shear strength. The mean value is based on the API RP2A guideline. The COV isbased on engineering judgment and the experience gathered for piles in stiff clay.

Pile capacity parameter in sand, flim:The mean value of flim is specified by the API RP2A design guidelines. The decrease in the COVof flim from 25 to 15% between 1975 and 1993 reflects the understanding acquired over the yearon pile friction in sand and the results of the expert opinion pooling summarised in Lacasse andGoulois (1989).

Loads

The extreme axial load for the most loaded pile (Leg A5) was 19 MN in 1975, based on theknowledge available at the time. The cyclic component due to the design storm representedabout 40 % of the axial load, and the static component due to the submerged weight, the deck andloads on deck represented about 60 % of the axial load. In 1993 the characteristic load wasadjusted to 20 MN, based on obeservations of waves at the site and in the North Sea in generaland other internal studies done by the operator. The cyclic component then represented 35 % ofthe axial load and the static component represented 65 % of the axial load. No details areavailable on the load calculations and updating betweeen 1975 and 1993.




201

The coefficient of variation was taken as 15 % in 1975 and 10 % in 1993, reflecting the changein knowledge with increased research and the increased proportion of the gravity load on the totalaxial load. A Gumbel distribution was selected for the load.

Table 7. 17 Examples of uncertainty in soil parameters in Layers 5, 7 and 8

1975 and 1993 analyses (Figure 7. 9 and Figure 7. 12)Layer Variable Coefficient of variation PDF

1975-analyses 1993-analyses

5 γ' 5 % 5 % N z 10 % 10 % N

su 25 % 15 % LN α 10 % 10 % LN

7 γ' 5 % 5 % N z 20 % 10 % N

K 15 % 10 % N δ 15 % 5 % N flim 25 % 15 % N

8 γ' 5 % 5 % N z 10 % 10 % N

su 25 % 10 % LN α 10 % 10 % LN Nc 15 % 15 % N

γ' = submerged unit weight z = depth to bottom of layersu = undrained shear strength α = skin friction factorNc = bearing capacity factor K = coefficient of lateral earth pressureδ = soil-pile friction angle flim = limiting skin friction (sand)

PDF = probability distribution function (N = normal, LN = lognormal)

Model Uncertainty

In the probabilistic pile capacity analysis, two model uncertainty variables can be considered ineach layer, the first applying to the side friction calculation and the second to the end bearingcalculation. The duality of model uncertainty in each layer is important because side friction andend bearing are two different resistance mechanisms which are modelled by different equations.Having a model uncertainty in each layer is required because the soil type can vary from onelayer to the other and different resistance mechanisms need then to be considered. The modeluncertainty variables were taken as normally distributed.

The pile has a length to diameter ratio of less than 20. There should not be any length effect.

In a dense to very dense sand, the uncertainties due to the calculation model can be very large,and the bias is believed to show a lot of conservatism in the API RP2A method. The




202

uncertainties are believed to be far greater for piles in sand than for piles in clay. The modeluncertainty values assumed for sand are based on the study by Lacasse and Goulois (1989),which assembles the opinions of 40 experienced specialists on the capacity of offshore piles, andthe results of the survey of the regulatory agencies done at NGI and the results of several researchprojects at NGI on the behaviour of piles in sand and clay.

Table 7. 18 Model uncertainty used in probabilistic analyses

Soil Side friction End bearinglayer Mean COV Mean COV

12345678

1.001.001.101.001.001.001.101.00

0.100.100.150.150.150.150.150.15

--------------

1.20

--------------

0.15

Side Friction in ClayThe API RP2A 20th edition model for side friction has not been shown to either underpredict oroverpredict the soil resistance in overconsolidated clays, see American Petroleum Institute(1993d). The mean of model error was therefore kept at 1.00. The coefficient of variation wastaken as 0.15. This was assumed for all clay layers, except for the top 7.5 m layer, where slightlyless uncertainty is expected for a softer clay. These uncertainties are rather large, but reflect theexisting lack of knowledge when operating with high undrained shear strength and high (butuncertain) overconsolidation ratios.

Side Friction in SandBias (conservatism) probably increases as the relative density of the sand increases. The APIRP2A approach is believed to represent approximately a mean resistance for loose to mediumdense sands (bias factor of 1.00). For dense sands, the bias factor was selected as 1.10, based onthe expected relative density and the effective stresses in situ. The values selected tend to be onthe lower bound of the values that could be used. The coefficient of variation of 0.15 reflects theuncertainty in the model when the relative density is not very well known. This uncertainty isalso rather large, but reflects the existing lack of knowledge for piles in sand, and the lack ofrelevant model tests to establish model uncertainty.

End Bearing in SandFor end bearing in very dense sand, the existing calculation model is believed to be conservative.For this reason, the mean of the model uncertainty was taken as 1.20, with a coefficient ofvariation of 0.15 to reflect the lack of good reference pile load tests with comparable pile size asused offshore.

7.8.4 Results

The results of the analyses are summarized Table 7. 19. In 1975, only deterministic calculationswere carried out. The 1975 probabilistic calculations were run in 1994 for the purpose of thisexample calculation.




203

Table 7. 19 Results of 1975 and 1993 deterministic and probabilistic analyses

Pile P2 in Leg A5 (penetration depth = 40.8 m)Soil Profile Deterministic factor

of safetyReliability index β Annual probability of

failure1975 1.73 2.06 2 0 10 2. ⋅ −

1993 1.39 2.41 08 10 2. ⋅ −

7.8.5 Discussion and Conclusions

Figure 7. 13 illustrates schematically the results of the reliability analysis of the most loaded pilefor the offshore jacket used in this example. The newer deterministic analysis gave a low safetyfactor (FS), a situation of major concern since the safety factor was below the minimum requiredfactor of safety under extreme loads of 1.50. However, the added information reduced theuncertainty in both soil and load parameters. The pile with a safety factor of 1.39 is nominallysafer than the pile was believed to be in 1975 where the safety factor was 1.73. The probabilisticanalyses showed that the pile, although with a lower safety factor, had higher safety margin thanperceived at the time of design. The lower uncertainty in the parameters in the newer analysiscaused a reduction in the nominal probability of failure (Pf ) by a factor of 2.5.




204

Figure 7. 13 Illustration of safety factor and probability of failure of most loadedpile in example jacket

The factor of safety is therefore not a sufficient indicator of safety margin because theuncertainties in the analysis parameters affect probability of failure, but these uncertainties donot intervene in the deterministic calculation of safety factor.

This example illustrates well that it is increasingly important to adopt rational and"documentable" design approaches that inform of and account for the uncertainties in theanalysis parameters. This is especially true when "novel" design or design procedures areinvolved. Only reliability analyses can provide the designer with insight in the inherent risk levelof a design or when comparing different solutions. The probabilistic approach is therefore anecessary complement to the conventional deterministic approach and they provide addedknowledge to help decision-making in the presence of uncertainty.

Predictions of foundation behaviour are uncertain because of spatial variation of soil properties,limited site exploration and observations, limited calculation models, uncertainties in theparameters obtained by various testing methods and not the least, uncertainties in the loads. Inmany cases, reliability analyses will enable improved concept optimization or site investigationplanning by pointing out the most important uncertainties. A design with safety factor or partialsafety coefficients does not enable such optimization. However, a probabilistic solution does notimprove faulty or insufficient input. If the deterministic model is weak, the probabilistic modelwill also be weak.

As for deterministic calculations, the essential components of reliability estimates in geotechnicsare (1) a clear understanding of the physical aspects of the geotechnical behaviour to model and(2) the experience and engineering judgement that enter into all decisions at any level, whetherfor parameter selection, choice of most realistic analysis model, or decision-making on theviability of a concept. The most important contribution of reliability concepts to offshore geo-technical engineering is increasing the engineer's awareness of the existing uncertainties and theconsequences of these.

Other benefits include: (1) ability to assess or reassess structures for extended life, (2) ability fornew designs to benefit from optimization and reduction of inherent conservatism, and (3) abilityto set cost-effective criteria that can rationally distinguish between manned structures, structuresde-manned during adverse weather and unmanned structures (Vugts and Edwards, 1992).Finally, but none the least, reliability analyses provide an improved basis for discussing safetyissues between geotechnics, structures, hydrodynamics and environment specialists, regulatoryparties and management. The use of probability theory should improve the design codes, andhence lead to more cost-effective structures.




205

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—A—AC potential drop method, 183Autocorrelation function, 169

—B—Bias, 168, 172, 173, 175, 176, 182, 183, 201, 202Buckling, 165, 166, 177

—C—Chains, 179, 183, 184Characteristic load, 200Characteristic value, 164, 165, 178Confidence

level, 184Crack growth, 179, 186, 187, 190Cumulative distribution function, 189Current, 173, 184, 185

—D—Damage, 180, 182, 183, 187Design code, 166, 167, 168, 178, 204Design life, 187

—E—Earthquake load, 176Eddy current, 183, 184, 185, 186, 188, 189, 210Engineering judgment, 199, 200Estimation

probability paper, 169Estimator, 182Excitation process

earthquake, 176Expert opinions, 175, 176, 200




162

—F—Failure criterion, 168Failure mode, 166, 168Fatigue, 177, 178, 180, 181, 182, 183, 184, 186, 187,

188, 189, 190, 191damage, 182, 187

Fatigue crack growth, 190FORM (First-order reliability method), 199Fracture toughness, 165

—H—Human reliability, 190

—I—Inspection, 184, 185, 186, 187, 188, 189, 190

—L—Limit state, 163Load, 163, 165, 169, 171, 173, 174, 176, 177, 179,

181, 182, 200, 201, 202, 203, 208, 210earthquake, 176permanent, 173wave, 169

Lognormal distribution, 164, 169, 171, 189Lot, 201LRFD (Load and Resistance Factor Design), 168, 205,

207, 208

—M—Magnetic particle inspection, 183, 184, 185, 186, 188,

189Material strength, 168Model, 163, 167, 175, 176, 182, 187, 191, 199, 201,

202, 204, 208error, 202uncertainty, 167, 175, 176, 182, 201, 202

—N—Nondestructive examination, 184

—O—Observations, 204

—P—Poisson's ratio, 164Probability of detection, 184, 185Probability of failure, 203, 204Probability paper, 169

—R—Reliability method, 199

first-order (FORM), 199Response, 176

—S—Safety factor, 199, 203, 204Safety margin, 203, 204Simulation, 169, 173State, 163, 207Stress concentration factor, 182Stress range, 187System

series, 180

—T—Time-of-flight diffraction technique, 183

—U—Ultrasonics, 184Updating, 186, 187, 188, 189, 190, 200

—V—Variogram, 169

—W—Wave force, 176Weibull distribution, 165, 187Wind, 173Wire ropes, 181

—Y—Yield strength, 164, 165, 168, 179Young's modulus, 164

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