DNV-RP-C208: Determination of Structural Capacity by Non-linear ...

RECOMMENDED PRACTICE

DET NORSKE VERITAS AS

The electronic pdf version of this document found through http://www.dnv.com is the officially binding version

DNV-RP-C208

Determination of Structural Capacity by Non-linear FE analysis Methods

JUNE 2013

© Det Norske Veritas AS June 2013

Any comments may be sent by e-mail to [email protected]

This service document has been prepared based on available knowledge, technology and/or information at the time of issuance of this document, and is believed to reflect the best ofcontemporary technology. The use of this document by others than DNV is at the user's sole risk. DNV does not accept any liability or responsibility for loss or damages resulting fromany use of this document.

FOREWORD

DNV is a global provider of knowledge for managing risk. Today, safe and responsible business conduct is both a licenseto operate and a competitive advantage. Our core competence is to identify, assess, and advise on risk management. Fromour leading position in certification, classification, verification, and training, we develop and apply standards and bestpractices. This helps our customers safely and responsibly improve their business performance. DNV is an independentorganisation with dedicated risk professionals in more than 100 countries, with the purpose of safeguarding life, propertyand the environment.

DNV service documents consist of among others the following types of documents:

— Service Specifications. Procedural requirements.

— Standards. Technical requirements.

— Recommended Practices. Guidance.

The Standards and Recommended Practices are offered within the following areas:

A) Qualification, Quality and Safety Methodology

B) Materials Technology

C) Structures

D) Systems

E) Special Facilities

F) Pipelines and Risers

G) Asset Operation

H) Marine Operations

J) Cleaner Energy

O) Subsea Systems

U) Unconventional Oil & Gas


Recommended Practice DNV-RP-C208, June 2013

Changes –

CHANGES – CURRENT

General

This is a new document.

Acknowledgement

This Recommended Practice is developed in a Joint Industry Project that was initiated through a pre-projectsupported by Statoil and DNV. The Joint Industry Project was sponsored by the following companies andinstitutions (in alphabetic order):

ConocoPhillips Skandinavia ASDet Norske Veritas ASMaersk Olie og Gas ASPetroleum Safety Authority Norway Statoil ASATotal E&P Norge AS

In addition to their financial support the above companies are also acknowledged for their technicalcontributions through their participation in the project.



Contents –

CONTENTS

1. Introduction.......................................................................................................................................... 51.1 Objective of the document .....................................................................................................................51.2 Validity ...................................................................................................................................................51.3 Definitions ..............................................................................................................................................51.4 Symbols...................................................................................................................................................6

2. Basic Considerations............................................................................................................................ 82.1 Limit state safety format .........................................................................................................................82.2 Characteristic resistance..........................................................................................................................92.3 Type of failure modes .............................................................................................................................92.4 Use of linear and non-linear analysis methods .......................................................................................92.5 Empirical basis for the resistance ...........................................................................................................92.6 Ductility ..................................................................................................................................................92.7 Serviceability Limit State .....................................................................................................................102.8 Permanent deformations .......................................................................................................................10

3. General requirements ....................................................................................................................... 113.1 Definition of failure ..............................................................................................................................113.2 Modelling .............................................................................................................................................113.3 Determination of characteristic resistance taking into account statistical variation .............................113.4 Requirement to the software .................................................................................................................123.5 Requirement to the user ........................................................................................................................12

4. Requirements to The FEM Analysis ............................................................................................... 134.1 General..................................................................................................................................................134.2 Selection of FE software.......................................................................................................................134.3 Selection of analysis method ...............................................................................................................134.4 Selection of elements ............................................................................................................................144.5 Mesh density .........................................................................................................................................154.6 Geometry modelling .............................................................................................................................154.7 Material modelling................................................................................................................................164.8 Boundary conditions .............................................................................................................................224.9 Load application....................................................................................................................................224.10 Application of safety factors.................................................................................................................234.11 Execution of non-linear FE analyses, quality control ...........................................................................234.12 Requirements to documentation of the FE analysis..............................................................................23

5. Representation of different failure modes ....................................................................................... 245.1 Design against tensile failure ................................................................................................................245.2 Failure due to repeated yielding (low cycle fatigue) ............................................................................265.3 Accumulated strain (“Ratcheting”).......................................................................................................305.4 Buckling................................................................................................................................................315.5 Repeated buckling.................................................................................................................................36

6. Bibliography ....................................................................................................................................... 37

Appendix A. Commentary.......................................................................................................................... 39

Appendix B. Examples................................................................................................................................ 42



Sec.1 Introduction –

1 Introduction

1.1 Objective of the document

This document is intended to give guidance on how to establish the structural resistance by use of non-linearFE methods. It deals with determining the characteristic resistance of a structure or part of a structure in a waythat fulfils the requirements to ultimate strength in DNV offshore standards.

Non-linear effects that may be included in the analyses are e.g., material and geometrical non-linearity, contactproblems etc.

The characteristic resistance should represent a value that meets the requirement that there is less than 5%probability that the resistance is less than this value. This definition of the characteristic resistance is similar towhat is required by many other structural standards using the limit state safety format and theserecommendations are expected to be valid for determination of capacities to be used with such standards.

The recommendations are foreseen to be used for cases where the characteristic resistance is not directlycovered by codes or standards. The objective is that analyses carried out according to the recommendationsgiven in this document will lead to a structure that meets the requirements to the minimum safety margin in thestandard.

This document is not intended to replace formulas for resistance in codes and standards for the cases wherethey are applicable and accurate, but to present methods that allows for using non-linear FE-methods todetermine resistance for cases that is not covered by codes and standards or where accurate recommendationsare lacking.

1.2 Validity

The document is valid for marine structures made from structural steels meeting requirements to offshorestructures with a yield strength of up to 500 MPa.

The recommendations presented herein are adapted to typical offshore steels that fulfil the requirementsspecified in DNV-OS-C101 /14/ or an equivalent offshore design standard. The specified requirements are madeunder the assumption that the considered structure is operating under environmental conditions that are withinthe specifications of the applied offshore standard. If the offshore unit is operating outside these specifications,the failure criterion presented in this Recommended Practice can only be utilized if it can be documented thatboth the weld and parent material have sufficient toughness in the actual environmental conditions.

This recommended practice is concerned only with failure associated with extreme loads, and failure due torepeated loading from moderate loads (fatigue) needs to be checked separately. See DNV-RP-C203 /17/.

1.3 Definitions

This Recommended Practise uses terms as defined in DNV-OS-C101. Additional terms are defined below:

characteristic resistance

the resistance that for a particular failure mode is meeting the requirement of having a prescribed probability that the resistance falls below a specified value (usually 5% fractile)

conservative load load that maintains its orientation when the structure deforms (e.g., gravity loads)

ductility the ability to deform beyond the proportionality limit without significant reduction in the capacity due to fracture or local buckling (originally ductility refers to the behaviour of the material, but is here also used for the behaviour of structures and structural details)

engineering shear strain

equivalent strain

follower load load that changes direction with the structure (e.g., hydrostatic pressure)

gross yielding yielding across the entire structural component e.g., a flange

low cycle fatigue the progressive and localised damage caused by repeated plastic strain in the material. Low cycle fatigue assessments are carried out by considering the cyclic strain level.

net area area of a cross-section or part of a cross-section where the area of holes and openings are subtracted

net section ratio the ratio between the net area and the gross area of the tension part of a cross-section

redundant structure a structure may be characterized as redundant if loss of capacity in one of its structural elements will lead to little or no reduction in the overall load-carrying capacity due to load redistribution

shake down shake down is a state in which a structure after being loaded into the elasto-plastic range will behave essentially linear for all subsequent cycles

�� = 2 ∙ �� = 2 ∙ �� = 2 ∙ ��

�� = 11 + � �12 �� − �� 2 + �� − � �2 + �� − �� 2� + 3�� 2 + �� 2 + �� 2�




1.4 Symbols

b span of plate

c flange outstand

C damping matrix

CFEM resistance knock down factor

D outer diameter of tubular sections

E modulus of elasticity

stress-strain curve parameter


external forces

internal forces

yield stress/yield strength

K Ramberg-Osgood parameter

eigenvalue for governing buckling mode

characteristic element size of smallest element

M mass matrix

N number of cycles to failure

design resistance

characteristic resistance

design action effect

characteristic action effect

t time, thickness

u displacement vector

strain

critical strain

engineering (nominal) strain

equivalent strain

fatigue ductility coefficient

εp_ult stress-strain curve parameter

εp_y1 stress-strain curve parameter

true (logarithmic) strain

fully reversible maximum principal hot spot strain range

fully reversible local maximum principal strain range

time step

material factor

partial factor for actions

reduced slenderness

Poisson’s ratio = 0.5 for plastic strain

density

principal stresses

representative stress

��1 ��2 ��

� !"

#$ #� %$ %�

� �&' �� ′

��') Δ�ℎ" Δ�, Δ� �- �� .̅ � 0 11,12 1#�




engineering (nominal) stress

fatigue strength coefficient

critical buckling stress

linearized buckling stress disregarding local buckling modes

linearized local buckling stress


true (Cauchy) stress




1� 1�′ 1�� 1�� 1��, 1�'2� 1�') 1),� 1��,$ 1��,$ 2



Sec.2 Basic Considerations –

2 Basic Considerations

2.1 Limit state safety format

A limit state can be defined as: “A state beyond which the structure no longer satisfies the design performancerequirements.” See e.g. /1/.

Limit states can be divided into the following groups:

Ultimate limit states (ULS) corresponding to the ultimate resistance for carrying loads.

Fatigue limit states (FLS) related to the possibility of failure due to the effect of cyclic loading.

Accidental limit states (ALS) corresponding to failure due to an accidental event or operational failure.

Serviceability limit states (SLS) corresponding to the criteria applicable to normal use or durability.

This Recommended Practice deals with limit states that can be grouped to ULS and ALS. It also addressesfailure modes from cyclic loading for cases that cannot adequately be checked according to the methods usedin codes for check of FLS. This is relevant for situations where the structure is loaded by a cyclic load at a highload level but only for a limited number of cycles (low cycle fatigue).

The safety format that is used in limit state codes is schematically illustrated in Figure 2-1 .

Figure 2-1Illustration of the limit state safety format

The requirement can be written as:

It can be seen from this figure that it is important that the uncertainty in the resistance is adequately addressedwhen the characteristic resistance is determined.

Sd ≤ Rd (1)

Sd = Sk γf Design action effect

Rd = Rk / γM Design resistance

Sk = Characteristic action effect

γf = Partial factor for actions

Rk = Characteristic resistance

γM = Material factor.

Sd< Rd




2.2 Characteristic resistance

The characteristic resistance should represent a value which will imply that there is less than 5% probabilitythat the resistance is less than this value. Often lack of experimental data prevents an adequate statisticalevaluation so the 5% must be seen as a goal for the engineering judgments that in such cases are needed.

The characteristic resistance given in design codes is determined also on the basis of consideration of otheraspects than the maximum load carrying resistance. Aspects like post-peak behaviour, sensitivity toconstruction methods, statistical variation of governing parameters etc. are also taken into account. In certaincases these considerations are also reflected in the choice of the material factor that will be used to obtain thedesign resistance. It is necessary that all such factors are considered when the resistance is determined by non-linear FE methods.

2.3 Type of failure modes

When steel structures are loaded to their extreme limits they will either fail by some sort of instability (e.g.,buckling) that prevents further loading or by tension failure or a combination of the two. For practical cases itis often necessary to define characteristic resistance at a lower limit in order to be able to conclude on structuralintegrity without excessive analysis. Examples of this can be limiting the plastic strain to avoid cyclic failurefor dynamically loaded structures or deformation limit for structural details failing by plastic strain incompression. See Section [3.1].

The following types of failure modes are dealt with in this Recommended Practice:

— tensile failure— failure due to repeated yielding (low cycle fatigue)— accumulated plastic strain— buckling — repeated buckling.

2.4 Use of linear and non-linear analysis methods

Traditionally, the ultimate strength of offshore structures are analysed by linear methods to determine theinternal distribution of forces and moments, and the resistances of the cross-sections are checked according todesign resistances found in design codes. These design resistance formulas often require deformations well intothe inelastic range in order to mobilise the code defined resistances. However, no further checks are normallyconsidered necessary as long as the internal forces and moments are determined by linear methods. When non-linear analysis methods are used, additional checks of accumulated plastic deflections and repeated yieldingwill generally be needed. These checks are important in case of variable or cyclic loading e.g., wave loads.

2.5 Empirical basis for the resistance

All engineering methods, regardless of level of sophistication, need to be calibrated against an empirical basisin the form of laboratory tests or full scale experience. This is the case for all design formulas in standards. Inreality the form of the empirical basis vary for the various failure cases that are covered by the codes, fromdetermined as a statistical evaluation from a large number of full scale representative tests to cases where thedesign formulas are validated based on extrapolations from known cases by means of analysis and engineeringjudgements. It is of paramount importance that capacities determined by non-linear FE methods build onknowledge that is empirically based. That can be achieved by calibration of the analysis methods toexperimental data, to established practise as found in design codes or in full scale experience.

2.6 Ductility

The integrity of a structure is also influenced by other factors than the value of the characteristic resistance. Theability of a structural detail to maintain its resistance in case of overload is highly influencing the resultingreliability of the structure. It is therefore necessary to consider not only the value of the resistance whendetermining the characteristic resistance, but also to judge how the load deflection relationship is for aparticular failure mode.

The check for ductility requires that all sections subjected to deformation into the inelastic range should deformwithout loss of the assumed load-bearing resistance. Such loss of resistance can be due to tensile failure,instability of cross-sectional parts or member buckling. The design codes give little explicit guidance on thisissue, with exception for stability of cross-sectional parts in yield hinges, which normally are covered byrequirement to cross-sectional class 1. See e.g. DNV OS-C101 /14/.

Steel structures generally behave ductile when loaded to their limits. The established design practise is basedon this behaviour, which is beneficial both with respect to simplifying the design process and improving theperformance of the structure. For a ductile structure, significant deflections may occur before failure and thusgive a collapse warning. Ductile structures also have larger energy absorption capabilities against impact loads.The possibility for the structure to redistribute stresses lessens the need for an accurate stress calculation duringdesign as the structure may redistribute forces and moments to be in accordance with the assumed static model.This is the basis for use of linear analyses for ULS checks even for structures which behave significantly non-linear when approaching their ultimate limit states.




2.7 Serviceability Limit State

Use of non-linear analysis methods may result in more structural elements being governed by the requirementsto the Serviceability Limit State and additional SLS requirements may be needed compared with design usinglinear methods. E.g., when plate elements are used beyond their critical load, out of plane deflections may needto be considered from a practical or aesthetic point of view.

2.8 Permanent deformations

All steel structures behave more or less non-linear when loaded to their ultimate limit. The formulas for designresistance in DNV Offshore Standard /14/ or similar codes and standards are therefore developed on the basisthat permanent deformation may take place before the characteristic resistance is reached.



Sec.3 General requirements –

3 General requirements

3.1 Definition of failure

In all analyses a precise definition of failure should be formulated. The failure definition needs to correspondwith the functional requirement to the structures. In certain cases like buckling failure it may be defined by themaximum load, while in other cases it need to be selected by limiting a suitable control parameter e.g., plasticstrain.

For Ultimate Limit States (ULS) and Accidental Limit States (ALS) the definition of failure needs to reflectthe functional requirement that the structure should not loose is load-carrying resistance during thedimensioning event. That may e.g., imply that in an ULS check the failure is defined as the load level wherethe remaining cycles in the storm that includes the ULS loadcase, will not lead to a progressive or cyclic failure.Alternatively a specific check for these failure modes can be carried out. See also [5.2]. Another example is incase of an ALS check for blast pressure, where one may consider the failure criterion to be the limitingdeflection for the passive fire protection.

Care should be made to ascertain that all relevant failure modes are addressed either directly by the analysis orby additional checks. Examples are local buckling, out of plane buckling, weld failure etc.

3.2 Modelling

It should be checked that the analysis tool and the modelling adopted represent the non-linear behaviour of allstructural elements that may contribute to the failure mechanism with sufficient accuracy. The model shouldbe suitable to represent all failure modes that are intended to be checked by the analysis. It should be made clearwhich failure modes the model will adequately represent and which failure modes that are excluded from theanalysis and that are assumed to be checked by other methods.

3.3 Determination of characteristic resistance taking into account statistical variation

When FE methods are used to determine the structural resistance it is necessary to take due account of thestatistical variation of the various parameters such that the results will be equal to or represent an estimate tothe safe side compared with what would be obtained if physical testing could be carried out.

The model should aim to represent the resistance as the characteristic values according to the governing code.In general that means 5% fractile in case a low resistance is unfavourable and 95% fractile in case a highresistance is unfavourable.

In cases where data of the statistical variation of the resistance is uncertain one needs to establish a selectionof the governing parameters by engineering judgement. The parameters should be selected such that it can bejustified that the characteristic resistance established meets the requirement that there is less than 5%probability that the capacity is below this value.

All parameters that influence the variability of the resistance need to be considered when establishing thecharacteristic resistance.

It is therefore necessary to validate the analysis procedure according to one of the following methods:

a) Selection of all governing parameters to be characteristic or conservative values.In this method all parameters that influence the result (key parameters) are selected to give results to thesafe side. E.g., element type, mesh size, material curve, imperfections, residual stresses etc. For structures or structural details where the resistance is dominated by the value of the yield stress, usingthe specified minimum yield stress according to offshore steel material standards will represent therequirements to the characteristic resistance. Other parameters with statistical variation that will influencethe resistance e.g., plate thickness should be selected as a safe estimate of the expected value in order tomeet the required statistical requirement for the resulting resistance. In cases of doubt a sensitivityassessment may be necessary.In some cases values are given in the codes for analysis of specific problems see e.g., [5.4.3].

b) Validation against code valuesIn this method a selected code case is used for calibration (denoted code calibration case). The case shouldrepresent the same failure mode that is to be investigated. The key parameters e.g., element type, mesh size,material curve, imperfections, residual stresses etc. should be selected so the analysis provide the resistancepredicted by the code for the code calibration case. The same parameters are then used when the resistanceof the actual problem is determined. If the analysis is calibrated against ordinary code values that meet the requirements to characteristicresistance then the resistance of the analysed structure also will meet the requirement.

c) Validation against testsIn this method one or more physical tests that are judged to fail in a similar way as the problem to beanalysed are selected for calibration (denoted test calibration case). First the key parameters e.g., elementtype, mesh size, material curve, imperfections, residual stresses etc. are varied so the analysis simulates thetest calibration case satisfactorily. (Giving the same or less resistance.) Then the actual problem is analysedusing the same key parameters. It should be ascertained that the statistical variation of the problem is duly



Sec.3 General requirements –

covered such that the requirements for determination of resistance by use of FE methods correspond to therequirements for determination of resistance from testing as given in Annex D of Eurocode 1990 /3/ or inISO 19902 /10/.

3.4 Requirement to the software

The software used shall be documented and tested for the purpose.

3.5 Requirement to the user

The user should be familiar with FE methods in general and non-linear methods in particular.

The analyst needs to understand the structural behaviour of the problem in question.

The user shall know the theory behind the methods applied as well as the features of the selected software.

When documenting structures to meet a code described reliability level with use of non-linear methods fordetermination of the resistance it is necessary the engineers understand the inherent safety requirements of thegoverning code.



Sec.4 Requirements to The FEM Analysis –

4 Requirements to The FEM Analysis

4.1 General

The term non-linear FE analysis covers a large number of analysis types for different purposes and objects. Thecontent of this section is written with analyses of steel structures in mind. The objective is to documentstructural capacity of the structure in a way that fulfils the requirements for determining characteristicresistance in accordance with DNV Offshore Standards and other similar standards, such as the Norsok N-series, /11/ to /13/, and the ISO 19900 suite of standards /9/.

4.2 Selection of FE software

The software must be tested and documented suited for analysing the actual type of non-linear behaviour. Thisincludes:

— non-linear material behaviour (yielding, plasticity)— non-linear geometry (Stress stiffening, 2nd order load effects).

Other types of non-linearity that may need to be included are:

— contact— temperature effects (e.g. material degradation, thermal expansion)— non-linear load effects (e.g. follower loads).

4.3 Selection of analysis method

4.3.1 Implicit versus explicit solver

Both implicit and explicit equation solvers may be used to solve the general equation system:

Where

M is the mass matrix

C is the damping matrix

u is the displacement vector

Fint is the internal forces

Fext is the external forces.

In dynamic analyses, explicit solvers are attractive for large equation systems, as the solution scheme does notrequire matrix inversion or iterations, and thus, are much more computational effective for solving one timestep than solvers based on the implicit scheme. However, unlike the implicit solution scheme, which isunconditionally stable for large time steps, the explicit scheme is stable only if the time step size is sufficientlysmall. An estimate of the time step required to ensure stability for beam elements is:

where Ls is the characteristic element size of the smallest element and C is the speed of sound waves in thematerial. Similar expressions exist for shells and solids.

This makes the explicit scheme well suited for shorter time transients as seen in for instance impact - orexplosion response analyses. For longer time transients the number of time steps will, however, be much largerthan needed for an implicit solution scheme. For moderately non-linear problems, implicit Newton Raphsonmethods are well suited, gradually incrementing the time and iterate to convergence for each time step.

4.3.2 Solution control for explicit analysis

Most explicit FE codes calculate the governing size of the time step based on equations similar to Equation (3).For problems of longer duration, one often wants to save analysis time by reducing the number of time steps.This can be done by accelerating the event or mass scaling. Accelerating the event reduces the simulation timeand thus computational time, the mass scaling increases the time step reducing the computational time, seeEquation (3).

The time saving methods are only useful if the inertia forces are small. Thus, it must be documented that thekinetic energy is small compared to the deformation energy (typically less than 1%) when explicit analyses areused to find quasi-static response.

Due to the typically large number of time steps in explicit analyses, the numerical representation of decimalnumbers is important for the stability of the solution. The software options to use high precision (“doubleprecision”) float are generally preferred.

(2)

(3)

-)3 �� + 4)5 �� + �� = ��

∆� = !"4 = !"70�




4.3.3 Solution control for dynamic implicit analysis

A large number of time integration procedures exists (e.g. The Newmark family of methods and the α-Method).For non-linear analyses they should be used in combination with Newton iterations. As a rule of thumb the timestep should not be larger than 1/10 of the lowest natural period of interest.

The most commonly used integration procedures can be tuned by selection of the controlling parameters. Theparameters should in most cases be selected to give an unconditionally stable solution.

For the α-method (HHT method) ref. /27/ the parameters α, β and γ can be selected by the user. The method isunconditionally stable if:

Selecting α less than 0 gives some numerical damping. In order to avoid “noise” from high frequency modes,parameters that give some numerical damping can be useful. Table 4-1 presents some combinations ofparameters that give unconditional stability.

4.3.4 Solution control for static implicit analysis

In case the dynamic effects are not important, the equation system to solve may be reduced to:

In such cases the implicit equation solvers are in general better suited, as the dynamic terms cannot be excludedin an explicit analysis.

Instead of time, applied load or displacement boundary conditions are normally incremented in a static solution.The selection of a load control algorithm for the analysis should be based on the expected response and needfor post peak-load results:

— A pure load control algorithm will not be able to pass limit points or bifurcation points when the inertiaeffects are not included.

— Using a displacement control algorithm, limit points and bifurcation points can be passed, but the analysiswill stop at turning points.

— For snap-back problems (passing turning point), or limit/bifurcation point problems that cannot be analysedusing displacement control, an “arc length” method is needed.

Figure 4-1Limit, Bifurcation and Turning points

4.4 Selection of elements

The selection of element type and formulation is strongly problem dependent. Points to consider are:

— shell elements or solid elements— elements based on constant, linear or higher order shape functions

(4)

Table 4-1 Combinations of α, β and γ for unconditional stability

α β γ Comment

0 0.25 0.5 Trapezoidal rule, no numerical damping

-0.05 0.2756 0.55 Numerical damping

-0.1 0.3025 0.6 Numerical damping

(5)

8 = 14 �1 − :�2 , � = 12 − : and − 13 ≤ : ≤ 0

�� = ��




— full vs. reduced, vs. hybrid integration formulations— number of through thickness integration points(shell)— volumetric locking, membrane locking and transverse shear locking— hourglass control/artificial strain energy (for reduced integration elements).

In general higher order elements are preferred for accurate stress estimates; elements with simple shapefunctions (constant or linear) will require more elements to give the same stress accuracy as higher orderelements. Constant stress elements are not recommended used in the area of interest.

Some types of elements are intended as transition elements in order to make the generation of the element mesheasier and are known to perform poorly. Typically 3-noded plates/shells and 4-noded tetrahedrons are oftenused as transition elements. This type of elements should if possible be avoided in the area of interest.

Proper continuity should be ensured between adjacent elements if elements of different orders are used in thesame model.

For large displacements and large rotation analyses, simple element formulations give a more robust numericalmodel and analysis than higher order elements.

Care should be taken when selecting formulations and integration rules. Formulations with (selective) reducedintegration rules are less prone to locking effects than full integrated simple elements; however the reducedintegration elements may produce zero energy modes (“hourglassing”) and require hourglass control. Whenhourglass control is used, the hourglass energy should be monitored and shown to be small compared to theinternal energy of the system (typically less than 5%).

4.5 Mesh density

4.5.1 General

The element mesh should be sufficiently detailed to capture the relevant failure modes:

— For ductility evaluations, preferably several elements should be present in the yield zone in order to havegood strain estimates.

— For stability evaluations, sufficient number of elements and degrees of freedom to capture relevantbuckling modes, typically minimum 3 to 6 elements dependent upon element type per expected half waveshould be used.

The element aspect ratio should be according to requirements for the selected element formulation in the areasof interest.

Care is required in transitioning of mesh density. Abrupt transitioning introduces errors of a numerical nature.

Load distribution and load type also have an influence on the mesh density. Nodes at which loads are appliedneed to be correctly located, and in this situation can drive the mesh design, at least locally.

4.5.2 Mesh refinement study

Often it will be necessary to run mesh sensitivity studies in order to verify that the results from the analyses aresufficiently accurate.

The analyst should make sure that the element mesh is adequate for representing all relevant failure modes. Inthe general case mesh refinement studies may be done by checking that convergence of the results are obtainede.g. by showing that the results are reasonably stable by rerunning the analysis with half the element size. Seeexample in Appendix [B.2].

4.6 Geometry modelling

Geometry models for FE analyses often need to be simplified compared to drawings of the real structure.Typically small details need to be omitted as they interfere with the goal of having a good regular element mesh.

The effect the simplifications may have on the result should be evaluated. Typical simplifications include:

— Cut-outs or local reinforcements are not included— Eccentricities are not included for beam elements or in thickness transitions in shell models— Weld material is not included— Welded parts are modelled as two parts and joined using contact surfaces.

For buckling analyses it is necessary to introduce equivalent geometric imperfections in order to predict thebuckling capacity correctly, see Section [5.4]. A common way to include the imperfections is to use one ormore of the structures eigenmodes and scale these such that the buckling capacity is predicted correctly for thecalibration model.

For problems where the geometry of the real structure deviates from the theoretical one, the analysis needs toreflect that possible geometrical tolerances may have impacts on the result. Example is fabrication tolerancesof surfaces transferring loads by contact pressure.




4.7 Material modelling

4.7.1 General

The selected material model should at least be able to represent the non-linear behaviour of the material bothfor increasing and decreasing loads (unloading). In some cases the material model also needs to be able toaccount for reversed loading.

The material model selected needs to be calibrated against empirical data (see [3.3]). The basic principle is thatthe material model needs to represent the structural behaviour sufficiently for the analysis to be adequatelycalibrated against the empirical basis.

4.7.2 Material models for metallic materials

For metallic materials time independent elasto-plastic models are often used. The main components in suchmodels are:

— A yield surface, defining when plastic strains are generated.von Mises plasticity is commonly used for metals. The model assumes that the yield surface is unaffectedby the level of hydrostatic stress.

— A hardening model defining how the yield surface changes for plastic strainsCommonly used are isotropic hardening (expanding yield surface) and kinematic hardening (translatingyield surface) or a combination of both.

— A flow rule (flow potential) defining the plastic strain increment a change in stress gives.The yield surface function is often used as a flow potential (associated flow).

The von Mises yield function is considered suitable for most capacity analyses of steel structures.

The hardening rule is important for analyses with reversed loading due to the Bauschinger effect. A materialmodel with kinematic (or combined kinematic/isotropic) hardening rule should be used in such analyses.

Figure 4-2The von Mises yield surface shown in the σ1-σ2 plane with isotropic (left) and kinematic (right) hardening models

Figure 4-3Isotropic vs Kinematic hardening




4.7.3 Stress strain measures

Stress and strain can be measured in several ways:

— From material testing the results are often given as “Engineering” stress-strain curves (calculated based onthe initial cross section of the test specimen).

— FE software input is often given as “True” stress-strain (calculated based on updated geometry).— Other definitions of strains are also used in FE formulations, e.g., the Green-Lagrange strain, and the Euler-

Almansi strain.

For small deformations/strains, all strain measures give similar results. For larger deformations/strains thestrain measure is important, e.g. the Green-Lagrange measure is limited to “small strains” only. Figure 4-4 shows a comparison of some strain measures. Limitations in the formulations on the use of the selected elementtype should always be noted and evaluated for the intended analysis.

Figure 4-4Comparison of some strain measures

The relationship between Engineering (Nominal) stress and True (Cauchy) stress (up to the point of necking) is:

The relationship between Engineering (Nominal) strain and True (Logarithmic) strain is:

The stress-strain curve should always be given using the same measure as expected by the software/ elementformulation.

4.7.4 Evaluation of strain results

As element strain in FE- analyses is an averaged value dependent on the element type and element size, thereported strain will always depend on the computer model. It is often necessary to re-mesh and adjust theanalysis model after the initial analyses are done in order to have a good model for strain estimates.

Strain extracted from element integration points are the calculated strain based on element deformations. MostFE software presents nodal averaged strains graphically. At geometry intersections the nodal average valuemay be significantly lower than the element (nodal or integration point) strain if the intersecting parts aredifferently loaded. When evaluating strain results against ductility limits, the integration point strains orextrapolated strains from integration points should be used.

4.7.5 Stress - strain curves for ultimate capacity analyses

When defining the material curve for the analysis, the following points should be considered:

— Characteristic material data should normally be used, see Section [3.3].— The predicted buckling capacity will depend on the curve shape selected, thus equivalent imperfection

calibration analyses and final analyses should be performed using the same material curves.

(6)

(7)

σtrue = σeng �1 + εeng �

εtrue = ln�1 + εeng �




— The extension of the yield zones and predicted stress and strain levels depend on the curve shape selected.Acceptance criteria should thus be related to the selected material curve, the curve need not represent theactual material accurately as long as the produced results are to the safe side.

— The stiffness of most steels reduces slightly before the nominal yield stress is reached; in fact yield stressis often given as the stress corresponding to 0.2% plastic strain.

— Some steels have a clear yield plateau; this is more common for mild steels than for high strength steels.— One should avoid using constant stress (or strain) sections in the material curves, due to possible numerical

instability issues.

Idealized material curves for steel according to European Standards EN-10025 /38/ and EN 10225 /39/ areproposed in Table 4-2 to Table 4-5 . These properties are assumed to be used with the acceptance criteria givenin this RP. Idealized material curves for steel materials delivered according to other codes e.g. DNV OffshoreStandards can be established by comparison with these curves. Figure 4-5 shows the parameters. Figure 4-6 shows the resulting curves for thicknesses less than 16 mm. The stress-strain values are given using theengineering stress-strain measure. Table 4-6 to Table 4-9 and Figure 4-7 show the corresponding true stress-strain values.

Alternative bi-linear curves may be used for buckling problems e.g. as shown in Figure 5-6 .

The curves should also be adjusted for temperature effects as appropriate.

Figure 4-5Parameters to define stress – strain curves

σ

ε,εp

σyield

σprop

σult

εp_y1

Ep1Ep2

E

εp_ultεp_y2

εp=0

σyield2




Table 4-2 Proposed non-linear properties for S235 steels (Engineering stress-strain)

S235

Thickness [mm] t ≤ 16 16 < t ≤ 40 40 < t ≤ 63

E [MPa] 210000

σprop/σyield 0.9

Ep1/E 0.001

σprop [MPa] 211.5 202.5 193.5

σyield [MPa] 235 225 215

σyield2 [MPa] 238.4 228.4 218.4

σult [MPa] 360 360 360

εp_y1 0.004

εp_y2 0.02

εp_ult 0.2

Ep2/E 0.0032 0.0035 0.0037


S355

Thickness [mm] t ≤ 16 16< t ≤ 40 40 < t ≤ 63

E [MPa] 210000

σprop/σyield 0.9

Ep1/E 0.001

σprop [MPa] 319.5 310.5 301.5

σyield [MPa] 355 345 335

σyield2 [MPa] 358.4 348.4 338.4

σult [MPa] 470 470 450

εp_y1 0.004

εp_y2 0.02

εp_ult 0.15

Ep2/E 0.0041 0.0045 0.0041


S420

Thickness [mm] t ≤ 16 16< t ≤ 40 40 < t ≤ 63

E [MPa] 210000

σprop/σyield 0.9

Ep1/E 0.001

σprop [MPa] 378.0 360.0 351.0

σyield [MPa] 420 400 390

σyield2 [MPa] 421.3 401.3 391.3

σult [MPa] 500 500 480

εp_y1 0.004

εp_y2 0.01

εp_ult 0.12

Ep2/E 0.0034 0.0043 0.0038





S460

Thickness [mm] t ≤ 16 16< t ≤ 25 25< t ≤ 63

E [MPa] 210000

σprop/σyield 0.9

Ep1/E 0.001

σprop [MPa] 414.0 396.0 373.5

σyield [MPa] 460 440 415

σyield2 [MPa] 461.3 441.3 416.3

σult [MPa] 540 530 515

εp_y1 0.004

εp_y2 0.01

εp_ult 0.1

Ep2/E 0.0042 0.0047 0.0052

Table 4-6 Proposed non-linear properties for S235 steels (True stress strain)

S235

Thickness [mm] t ≤ 16 16< t ≤ 40 40 < t ≤ 63

E [MPa] 210000 210000 210000

σprop [MPa] 211.7 202.7 193.7

σyield [MPa] 236.2 226.1 216.1

σyield2 [MPa] 243.4 233.2 223.0

σult [MPa] 432.6 432.6 432.6

εp_y1 0.0040 0.0040 0.0040

εp_y2 0.0198 0.0198 0.0198

εp_ult 0.1817 0.1817 0.1817


S355

Thickness [mm] t ≤ 16 16< t ≤ 40 40 < t ≤ 63

E [MPa] 210000 210000 210000

σprop [MPa] 320.0 311.0 301.9

σyield [MPa] 357.0 346.9 336.9

σyield2 [MPa] 366.1 355.9 345.7

σult [MPa] 541.6 541.6 518.5

εp_y1 0.0040 0.0040 0.0040

εp_y2 0.0197 0.0197 0.0197

εp_ult 0.1391 0.1391 0.1392




Figure 4-6Material curves according to Table 4-2 to Table 4-5 (Engineering stress-strain) (t< 16 mm)


S420

Thickness [mm] t ≤ 16 16< t ≤ 40 40 < t ≤ 63

E [MPa] 210000 210000 210000

σprop [MPa] 378.7 360.6 351.6

σyield [MPa] 422.5 402.4 392.3

σyield2 [MPa] 426.3 406.0 395.9

σult [MPa] 561.2 561.2 538.7

εp_y1 0.0040 0.0040 0.0040

εp_y2 0.0099 0.0099 0.0099

εp_ult 0.1128 0.1128 0.1128


S460

Thickness t ≤ 16 16< t ≤ 25 25< t ≤ 63

E [MPa] 210000 210000 210000

σprop [MPa] 414.8 396.7 374.2

σyield [MPa] 462.8 442.7 417.5

σyield2 [MPa] 466.9 446.6 421.2

σult [MPa] 595.4 584.3 567.8

εp_y1 0.0040 0.0040 0.0040

εp_y2 0.0099 0.0099 0.0099

εp_ult 0.0948 0.0948 0.0948




Figure 4-7Material curves according to Table 4-6 to Table 4-9 (True stress-strain) (t< 16 mm)

4.7.6 Strain rate effects

The proposed material curves in Section [4.7.5] can be used for strain rates up to 0.1 s-1. For impact loads higherstrain rates may be experienced, and the increased strength and reduced ductility may be considered.

The Cowper-Symonds (CS) model is one model often used to simulate strain rate effects:

As seen, the relative effect will be the same for all static stress (strain) levels. Thus the model must be calibratedfor the expected maximum stress (strain), otherwise the effect may be overestimated for large strains.

The constants C and p should be based on experiments and the maximum strain level expected. In lack of data,C = 4000 [s-1] and p = 5 is proposed for common offshore steel materials.

4.8 Boundary conditions

The selected model boundary condition needs to represent the real condition in a way that will lead to resultsthat are accurate or to the safe side.

Often it is difficult to decide what the most “correct” or a conservative boundary condition is. In such casessensitivity studies should be performed.

4.9 Load application

Unlike linear elastic analyses, where results from basic load cases can be scaled and added together, thesequence of load application is important in non-linear analyses. Changing the sequence of load applicationmay change the end response.

The loads should be applied in the same sequence as they are expected to occur in the condition/event to besimulated. E.g. for an offshore structure subjected to both permanent loads (such as gravity and buoyancy) andenvironmental loads (such as wind, waves and current); the permanent loads should be incrementally appliedfirst to the desired load level, then the environmental load should be incremented to the target level or collapse.

In some cases the initial load cases (e.g. permanent loads) may contribute positively to the load carryingcapacity for the final load case, in such cases a sensitivity study on the effect of reduced initial load should beperformed.

The analyst needs to evaluate if the loads are conservative (independent of structure deformation) or non-conservative (follow structure deformation) and model the loads correspondingly.

The number of time/load increments used to reach the target load level may also influence the end predictedresponse. Increment sensitivity studies should be performed to ensure that all failure modes are captured.

(8)

1$��IJ�& = 1"�I��& K1 + L�54M1� N




4.10 Application of safety factors

Applying load and resistance safety factors in a non-linear analysis can be challenging as application of safetyfactors on the capacity model side for one failure mode may influence the capacity of another failure model.One example of this is yielding vs. column buckling capacity.

In general it is more practical to prepare one capacity model representing the desired characteristic capacity forall failure modes to be analysed for, and then apply all the safety on the load side, defining a target load levelthat accounts for both load and resistance safety. Using this approach, the same model may be used for bothULS and ALS type of analysis without recalibration of the model:

where Rk is the characteristic resistance found from the analysis, and Sk is the characteristic load effect.

4.11 Execution of non-linear FE analyses, quality control

The following points should be considered in a quality control of non-linear FE analyses:

— boundary conditions— calibration against known values.— inertia effects in dynamic analyses— element formulation/ integration rule suited for the purpose— material model suited for the purpose— mesh quality suited for the purpose, mesh convergence studies performed for stress strain results.— equivalent imperfections calibrated for stability analyses— time/load increments sufficient small, convergence studies performed — numerical stability — reaction corresponds to input— convergence obtained for equilibrium iterations— hourglass control for reduced integration, hourglass energy remains small— sensitivity analysis both from idealisation and numerical points of views could be provided in particular

around singularities, for boundary conditions, etc.— reference recommendations in Standards, Codes or Rules that are applicable directly to the studied system,

or to a similar system with different dimensions— reference similar analyses for systems or subsystems that are validated from analytical or experimental

sources— evaluation of analysis accuracy based on performed sensitivity studies.

4.12 Requirements to documentation of the FE analysis

The analysis should be documented sufficiently detailed to allow for independent verification by a third party,either based on review of the documentation, or using independent analyses. The documentation should includedescription of:

— purpose of the analysis— failure criteria— geometry model and reference to drawings used to create the model— boundary conditions— element types— element mesh — material models and properties— loads and load sequence— analysis approach— application of safety factors— results— discussion of results— conclusions.

Sensitivity studies and other quality control activities performed in connection with the analyses should alsobe documented.

(9)#� > %� ∙ �!2I$ ∙ �-I�'�I,



Sec.5 Representation of different failure modes –

5 Representation of different failure modes

5.1 Design against tensile failure

5.1.1 General

An accurate analysis of tensile failure is demanding as numerous factors affect the problem and determinationof the results from the analysis is highly influenced on how the analysis is carried out.

The recommendations given in this document are not valid for failure that is related to unstable fracture due toeither insufficient material toughness, defects outside fabrication specifications or cracks. In such casesfracture mechanics methods need to be used.

In general accurate prediction of tensile failure needs to be made by analyses that are calibrated against tests ora known solution where the conditions for tensile failure are similar as in the structural detail beinginvestigated. This method is described in [5.1.2].

Simplified tensile failure criteria for the base material are presented in [5.1.3].

Welds are assumed to be made with overmatching material that ensures that plastic straining and eventualfailure takes place in the base material. Welds should therefore be checked according to ordinary code methodsbased on the forces carried by the welds. See [5.1.4].

5.1.2 Tensile failure resistance from non-linear analysis calibrated against a known solution

The most accurate method to check a structure against tensile failure is by calibrating the non-linear FE analysisagainst a known solution. In this method the following steps should be followed.

i) Select a test or a problem with known capacity (e.g. from a design code) as the reference object. Thereference object should have the similar conditions for tensile failure as the actual problem such as the typeof stress (axial, bending or shear) and the degree of triaxial stress state.

ii) Model and analyse the reference object following recommended modelling and analysis technique.

iii) Determine the selected strain parameter that is judged to best describe the problem (e.g. principal strain) atfailure for the reference object.

iv) Model the actual object using the same analysis technique as for the reference object i.e. mesh density,element type, material properties, etc.

v) Determine the capacity against tensile failure for the structure as the load corresponding to the load levelwhen the failure strain as determined in iii) is reached.

5.1.3 Tensile failure in base material. Simplified approach for plane plates

5.1.3.1 General

Tensile failure can be assessed by the following simplified procedure for some selected situations if a calibratedsolution as given in 5.1.2 is not attainable. The simplified check of tensile failure is a two-step check:

i) tensile failure due to gross yielding along the failure line (see [5.1.3.2])

ii) tensile failure due to cracks starting from local strain concentrations (see [5.1.3.3]).

Tensile failure in structures modelled by beam elements is best checked on the basis of the total deflection e.g.as given in DNV-RP-C204 /18/.

This method is valid for structures made with typical offshore steel that will meet requirements to ductility andtoughness. The structural details need to meet fabrication requirements for offshore steel structures.

The safety factors that should be used for tensile fracture according to this procedure should apply an additionalsafety factor γtf = 1.2 compared with standard material factor of the code.

The analyses should apply stress-strain relationship according to [4.7.2].

As the calculated plastic strain in the non-linear FE analysis will always involve inaccuracies depending uponmesh density, element type, material model etc., the analyst must assure that any deviation from a correct resultis to the safe side. A mesh convergence study can be used to check that satisfactory accuracy is obtained. If itis found impractical to perform the analysis with the required mesh density, larger elements can be used as longas the calculated capacity is reduced by e.g. applying a resistance knock down factor CFEM or by reducing thematerial key parameters. Such factors should be determined for the actual problem by means of a meshconvergence study.

5.1.3.2 Strain limits for tensile failure due to gross yielding of plates

This method assumes that a failure line across the plate being analysed can be identified. Along this failure linethe plastic strain can be calculated as an averaged linear strain (gross yield strain) that can be compared withcritical values given in Table 5-1 . With the gross yield strain is understood a plastic strain that is averaged overa length in the direction of the largest principal strain and is linearized (axial and a bending components) in thetransverse directions.




The strain shall be calculated as the maximum principal plastic strain along the likely failure line. The failureline should be assumed as a straight line.

Local cut outs or holes need to be included in the model if the ratio of the net area to the gross area of the tensionpart is less than specified in Table 5-1 . Local strains caused by non-loaded attachments like doubler plates orbrackets do not need to be considered.

The strains caused by tension stresses both due to axial tension, in-plane and out-of-plane bending and in-planeshear should be extrapolated based on the linearized distributed maximum principal stress to the plate edgecorners both transverse to the plate as well as through thickness. The strain can be taken as the average valueover a length lavg given in Equation (10) in the direction of the maximum principal strain.

Where:

Shear strain due to out-of-plane bending should be checked separately as averaged engineering shear across thefailure line. The critical engineering shear strains can be taken as equal to the maximum principal strains givenin Table 5-1 .

The linearized averaged plastic strain can be found by curve fitting a straight line using the method of leastsquares.

In cases where the direction of the maximum principal plastic strain significantly varies along the failure lineit is recommended to consider different failure line directions.

5.1.3.3 Strain limits for tensile failure due to local yielding in plane plates

The danger of cracks developing in ductile materials due to local concentrated yielding can be checked by thefollowing simplified procedure.

The local gross yielding can be checked by averaging the strain over a rectangular prismatic volume at thelocation with the largest strain. The volume should be taken through the thickness (t) of the plate and shouldextend from t to 5*t in the other directions. Where strain gradients are present due to changes in the crosssection or holes the length of the averaging volume should not be larger than a 25% of the length or width of anotch or 20% of the diameter of a hole for problems dominated by in-plain strains. In case of out-of-planebending the length and width of the averaging volume should be taken equal to the thickness. See Figure 5-1 .

(10)

tp = plate thickness

wp = plate width

Table 5-1 Critical strain and net area ratio for uniaxial stress state 1), 2)

Maximum principal plastic linearized strain 1)

S235 S355 S420 S460

Critical gross yield strain 0.05 0.04 0.03 0.03

Net section ratio 0.94 0.95 0.96 0.97

1) The strain can be calculated as average values over a length (in the direction of the principal strain) equal to the thickness for in-plane bending and up to 5 times the thickness for pure membrane strains.

2) Any strain due to cold-forming should be added to the calculated plastic strain considering the direction of the plastic strain due to cold forming

,IP = 163 �� − R�3 S)� ,IP ≥ 2 ��




Figure 5-1Example of rectangular prism for check of local strains

The strain can be calculated as the average value in the direction of the maximum principal strain and shouldbe linearized (axial and bending component) in the other two directions. The corner with the largest strainshould be compared with the critical strains given in Table 5-2 .

For cases with out-of-plane bending the engineering shear strain in the direction normal to the plate should bechecked against the critical values in Table 5-2 . The engineering shear strain can be taken as the averageengineering shear strain across the thickness for all cross-sections within the prismatic volume.

5.1.4 Failure of welds

The welds may or may not be represented with separate elements. For cases where the welds are not modelledthe check of the strength of welds should be based on stress resultants determined by integration of stressesfrom the closest elements and checked against ordinary code requirements e.g. EN 1993-1-8 /7/ or the relevantcode for the problem at hand.

If welds are modelled the linearized stress components (axial, bending, shear) should be determined fromintegration of the stresses in the elements representing the welds and checked against ordinary coderequirements e.g. EN 1993-1-8 /7/ or the relevant code for the problem at hand.

Normally it is required that in welded connections the welds are stronger than the base material (overmatch).See also Section [2.6].

5.2 Failure due to repeated yielding (low cycle fatigue)

5.2.1 General

Non-linear FE-analyses may imply that the structure is assumed to be loaded beyond proportionality limits.This means that the structure may be weakened against subsequent load cycles by repeated yielding leading toa possible cyclic failure. This is called low cycle fatigue and need to be treated different from how high cyclefatigue checks are carried out.

The fatigue damage due to loads that leads to repeated yielding, i.e. cyclic plastic strains, will be under-estimated if conventional linear elastic methods, such as those presented in DNV-RP-C203 ref. /17/, areapplied. The methodology presented in the following must therefore be applied if repeated yielding occurs.

The low cycle fatigue strength will be reduced for details that may include damage from high cycle fatigue. Forsuch cases the damage from high cycle fatigue should be added to the damage from low cycle fatigue. See[5.2.2].

Table 5-2 Critical local maximum principal plastic strain for uniaxial stress states 1)

Maximum principal plastic critical strain

S235 S355 S420 S460

Critical local yield strain 0.15 0.12 0.10 0.09

1) Any strain due to cold-forming should be added to the calculated plastic strain considering the direction of the plastic strain due to cold forming

t




5.2.2 Fatigue damage accumulation

The fatigue life may be calculated under the assumption of linear cumulative damage, i.e.

where D is the accumulated fatigue damage. ni is the number of cycles in block i and Ni the number of cyclesto failure at constant strain range ∆ε.

In cases where the fatigue damage from high cycle fatigue (HCF) is considerable the total damage is obtainedby summation, i.e. D(tot) = D(LCF) + D(HCF)

5.2.3 Determination of cyclic loads

Failure due to repeated yielding is associated with Ultimate Limit States (ULS) or Accidental Limit States(ALS). The cyclic loads should meet the same requirements as for a single extreme load when it comes topartial safety factors and selection of return periods.

Depending on the nature of the actual loads it may be necessary to carry out a check against failure due torepeated plastic straining. This check is necessary as non-linear analysis allows parts of the structure to undergosignificant plastic straining and the ability to sustain the defined loads may be reduced by the repeated loading.For offshore structures this is evident for environmental loads like waves and wind. When cyclic loads arepresent it is necessary to define a load history that will imply a probability of failure that is similar or less thanintended for static loads. See also [3.1].

The load-history for the remaining waves in a 10 000 year dimensioning storm investigated for southern NorthSea conditions have been found to have a maximum value equal to 0.93 of the dimensioning wave, a durationof 6 h and a Weibull shape parameter of 2.0. This applies for check of failure modes where the entire storm willbe relevant, such as crack growth. When checking failure modes where only the remaining waves after thedimensioning wave (e.g. buckling) need to be accounted for, a value of 0.9 of the dimensioning wave may beused /26/.

All the remaining cycles in the storm of the maximum wave action may be assumed to come from the samedirection as the dimensioning wave.

5.2.4 Cyclic stress strain curves

It is required that the cyclic stress-strain curve of the material is applied. The use of monotonic stress-straincurve must be avoided since it may provide non-conservative fatigue life estimates, especially for high strengthsteels. It is required that the welds are produced with overmatching material. Consequently the cyclic stress-strain properties of the base material should be used when assessing welded joints.

Unless the actual cyclic behaviour of the material is known the true cyclic stress strain curves presented inFigure 5-2 can be applied. Kinematic hardening, as illustrated in Figure 4-3 should be assumed. The curves aredescribed according to the Ramberg-Osgood relation:

The value of the coefficient K is given in Table 5-3 .

(11)

(12)

Table 5-3 Ramberg-Osgood parameters for base material

Grade K (MPa)

S235 410

S355 600

S420 690

S460 750

U = V ��W��

�=1 ,

� = 1� + X1YZ10 .




Figure 5-2The true cyclic stress-strain curve for common offshore steel grades

5.2.5 Low cycle fatigue of welded joints

5.2.5.1 Accumulated damage criterion

The number of cycles to failure, N, for welded joints due to repeated yielding is estimated by solving thefollowing equation

Where:

The parameters in Equation (13) are given in Table 5-4 for air and seawater with cathodic protection.

(13)

∆εhs is the fully reversible maximum principal hot spot strain range

E is the modulus of elasticity (material constant)

σf' is the fatigue strength coefficient (material constant)

εf' is the fatigue ductility coefficient (material constant)

Table 5-4 Data for low cycle fatigue analysis of welded joints

Environment σf ' (MPa) εf

'

Air 175 0.095

Seawater with cathodic protection 160 0.060

∆�ℎ"2 = 1�′� �2W�−0.1 + ��′ �2W�−0.5




Figure 5-3ε-N curves for welded tubular joints in seawater with cathodic protection and in air.

5.2.5.2 Derivation of hot spot strain for plated structures

It is recommended to derive the hot spot strain by applying the principles of the procedure given in Section 4of DNV-RP-C203 ref. /17/. The procedure in ref. /17/ is originally developed for assessing the hot spot stressof a linear elastic material in relation to high cycle fatigue assessments. However, by substituting maximumprincipal stresses with maximum principal strains it may also be applied for determining hot spot strains.

It is recommended to mesh with elements of size t*t in the hot spot region.

The strain gradient towards the hot spot may be steep because the cyclic plastic strains often will be localisedin a limited area near the hot spot. In order to reflect steep strain gradient in a good manner it is recommendedto use finite elements with mid side nodes, such as 8-noded shell elements or 20-noded brick elements.

For modelling with shell elements without any weld included in the model a linear extrapolation of the strainsto the intersection line from the read out points at 0.5t and 1.5t from the intersection line can be performed toderive hot spot strain. For modelling with three-dimensional elements with the weld included in the model alinear extrapolation of the strains to the weld toe from the read out points at 0.5t and 1.5t from the weld toe canbe performed to derive hot spot strain.

5.2.5.3 Derivation of hot spot strain for tubular joints

Reference is made to section on stress concentration factors in DNV-RP-C203 ref. /17/.

5.2.6 Low cycle fatigue of base material

5.2.6.1 Accumulated damage criterion

Despite the fact that the fatigue capacity of structures very often is governed by welded joints there aresituations where the origin of a fatigue crack is in the base material. This is often due to geometrical details,such as notches, that cause rise in the cyclic stress-strain level. A low cycle fatigue check of the base materialmay therefore be necessary.

As opposed to assessments of welded joints where the fatigue damage is determined by means of the cyclic hotspot strain, low cycle fatigue analysis of base material is based on the maximum principal strain range. Thestrain range is obtained from the local maxima of the considered detail.

The number of cycles to failure, N, for base material due to repeated yielding is estimated by solving thefollowing equation

(14)

∆�,2 = 1�′� �2W�−0.1 + ��′ �2W�−0.43




where

Values of the parameters in Equation (14) are given in Table 5-5 for air and seawater with cathodic protection.

Figure 5-4ε-N curve for low cycle fatigue of base material for tubular joint in seawater with cathodic protection and in air.

5.2.6.2 Derivation of local maximum principal strain

The maximum principal strain is obtained from the local maxima of the considered detail. The local strain statewill be underestimated if the finite element mesh is too coarse. A mesh sensitivity study should therefore becarried out to ensure that the applied strain is not underestimated. Reference is made to [4.5.2] regarding meshrefinement.

5.2.7 Shake down check

Structures loaded beyond the elastic range may alter their response behaviour for later cycles. However, if astructure is behaving essentially linear for all cyclic loads after the first few cycles following the dimensioningload, it will be said to achieve shake down and further checks of failure due to repeated yielding or buckling isnot necessary.

In the general case it is necessary to define a characteristic cyclic load and to use this load with appropriatepartial safety factors. It should be checked that yielding only takes place in the first few loading cycles and thatlater load repetitions only cause responses in the linear range. This may then serve as an alternative to a lowcycle fatigue check as described in [5.2.5].

It is necessary to show that the structure behaves essentially linear for all possible load situations and loadcycles.

5.3 Accumulated strain (“Ratcheting”)

For cases where the structure will be loaded by cyclic loads in a way that incremental plasticity may accumulateand in the end lead to tensile failure or excessive deformations the maximum accumulated strain needs to bechecked against the strain values in Section [5.1].

The criteria for excessive deformations may alternatively be determined on a case by case basis due to

∆εl is the fully reversible maximum principal hot spot strain range

E is the modulus of elasticity (material constant)

σf' is the fatigue strength coefficient (material constant)

εf' is the fatigue ductility coefficient (material constant)

Table 5-5 Data for low cycle fatigue analysis of base material

Environment σf' (MPa) εf

'

Air 175 0.091

Seawater with cathodic protection 160 0.057




requirements to the structural use or performance. Cases where accumulated strain may need to be checked canbe structures that are repeatedly loaded by impacts in the same direction or functional loads that change positionor angle of attack. Examples of the first are protection structures that are hit by swinging loads and the lattermay be wheel loads on stiffened plate decks.

5.4 Buckling

5.4.1 General

The buckling resistance of a structure or structural part is a function of the structural geometry, the materialproperties, the imperfections and the residual stresses present. When the buckling resistance is determined byuse of non-linear methods it is important that all these factors are accounted for in a way so that the resultingresistance meets the requirement to the characteristic resistance or is based on assumptions to the safe side.

Three different methods for carrying out the analysis are proposed in the following:

a) Linearized approach: Apply the FE method for assessing the buckling eigenvalues (linear bifurcationanalysis) and determine the ultimate capacity using empirical formulas,

b) Full non-linear analysis using code defined equivalent tolerances and/or residual stresses and

c) Non-linear analysis that is calibrated against code formulations or tests.

Either of these methods can be used to determine the resistance of a structure or part of a structure andrecommendations for their use are given in the following sections.

The proposed methods are valid for ordinary buckling problems that are realistically described by the FEanalysis. Care should be exercised when analysing complex buckling cases or cases that involve phenomenalike snap through, non-conservative loads, interaction of local and global stability problems etc.

5.4.2 Determination of buckling resistance by use of linearized buckling values

5.4.2.1 General

In order to establish the buckling resistance of a structure or part of the structure using linearized bucklingvalues (eigenvalues) the buckling resistance can be determined by following the steps:

i) Build the model. The element model selected for analysis need to represent the structure so that anysimplifications are leading to results to the safe side. If certain buckling failure modes are not seen asappropriate to be represented by the model their influence on the resulting resistance can be establishedaccording to [5.4.2.2].

ii) Perform a linear analysis for the selected representative load case SRep showing maximum compressive andvon-Mises stresses.

iii) Determine the buckling eigenvalues and the eigenmodes (buckling modes) by the FE analysis

iv) Select the governing buckling mode (usually the lowest buckling mode) and the point for determining thebuckling representative stress. The point for reading the representative stress is the point in the model thatwill first reach yield stress when the structure is loaded to its buckling resistance.

v) Determine the von-Mises stress at the point for the representative stress σRep from step ii).

vi) Determine the critical buckling stress as the eigenvalue for the governing buckling mode times therepresentative stress:

Determine the reduced slenderness as:

σki = kg σRep (15)

(16). = � ��1��




vii) Select empirically based buckling curve to be used based on the sensitivity of the problem with respect toimperfections, residual stresses and post buckling behaviour. Relevant buckling curves can be selectedfrom codes, but if not available the following may be used:

viii)Determine the buckling resistance Rd as:

Figure 5-5Examples of buckling curves showing sensitivity for imperfections etc. for different buckling forms

Empirical buckling curves are needed to account for the buckling resistance reduction effects fromimperfections, residual stresses and material non-linearity. The effect is illustrated in Figure 5-5 . For allbuckling forms the usable buckling resistance is less than the critical stress for reduced slenderness less than1.2. Above this value, plates with possibility of redistributing stresses to longitudinal edges may reach buckling

Table 5-6 Buckling curves

Type of buckling κ

Column and stiffened plate and plate without redistribution possibilities

Plate with redistribution possibilities

Shell buckling Curves to be selected from specific shell buckling codes such as

DNV-RP-C202 /16/ or Eurocode EN-1993-1-6 /6/

(17)

α = 0.15 for strict tolerances and low residual stresses

0.3 for strict tolerances and moderate residual stresses

0.5 for moderate tolerances and moderate residual stresses

0.75 for large tolerances and severe residual stresses

(18)

1] + 7]2 − .2

≤ 1.0

= 1.0 for . ≤ 0.673

. − 0.22.2 for . > 0.673

] = 0.5 �1 + :�. − 0.2 � + .2�

#$ = a�� %#��-1#�

0

0,2

0,4

0,6

0,8

1

1,2

1,4

Critical stress (Euler)

Plate

Column

Shell

reduced slenderness

bu

ckli

ng

fa

cto

r κ




capacities above the critical value, column buckling problems will be less than the critical value, but approachthe critical value for large slenderness. Shell buckling is more sensitive to imperfections and the differencebetween the buckling capacities that may be exploited in real shell structures are considerably less than thecritical value also for large slenderness.

Members will buckle as columns for cross-section classes 1, 2 and 3 with exception of tubular sections exposedto external hydrostatic pressure. For definition of cross-sectional classes see DNV-OS-C101 Appendix A /14/.

5.4.2.2 Correction for local buckling effects

There may be cases where a reliable FE representation of local buckling phenomena is not feasible. This mayfor instance be torsional buckling of stiffener or local stability of stiffener flange and web. For such cases theeigenvalue analysis should be carried out without the local buckling modes represented and the interaction oflocal and global buckling may be accounted for in a conservative manner by linear interaction as shown inEquation (19).

σkig is the linearized buckling stress when local buckling modes are disregarded and σkil is the linearized localbuckling stress.

5.4.3 Buckling resistance from non-linear analysis using code defined equivalent tolerances

The buckling resistance of a structure or part of a structure can be determined by performing non-linearanalyses where the effects of imperfections, residual stresses and material non-linearity is accounted for by useof a defined material stress-strain relationship and the use of empirically determined equivalent imperfections.The defined equivalent imperfections will include effects from real life imperfections, but will in general bedifferent in shape and size. This method is only valid for buckling problem similar to the cases where theequivalent imperfections are given in Table 5-7 . For other cases see [5.4.4].

The material model to be used with the equivalent imperfections is shown in Figure 5-6 or with the modelsproposed in [4.7.5].

Figure 5-6Material model for analysis with prescribed equivalent imperfections

(19)

kilkigki σσσ111 +=

Strain

Stress

E

1

1

E/100




It is required that an eigenvalue analysis is carried out to determine the relevant buckling modes. Usually thepattern from the buckling can be used as the selected pattern for the imperfections, but in certain cases e.g. whenthe shape of the buckling load differ from the deflected shape from the actual loads it may be necessary toinvestigate also other imperfection patterns.

It may be useful to divide the imperfections into local and global imperfections as shown in Figure 5-7 . Thevalues in Table 5-7 apply to the total imperfection from local and global imperfection patterns. Sensitivityanalyses may be required for cases that are particularly imperfection sensitive.

Table 5-7 Equivalent imperfections

Component Shape Magnitude

Member bow L/300 for strict tolerances and low residual stressesL/250 for strict tolerances and moderate residual stressesL/200 for moderate tolerances and moderate residual stressesL/150 for large tolerances and severe residual stresses

Longitudinal stiffener girder webs

bow L/400

Plane plate between stiffeners

buckling eigenmode

s/200

Longitudinal stiffener or flange outstand

bow twist 0.02 rad




Figure 5-7Example of local (left) and global (right) imperfections for stiffened panel

5.4.4 Buckling resistance from non-linear analysis that are calibrated against code formulations or tests

Buckling resistance can be found by non-linear methods where the effect of imperfections, residual stressesand material non-linearity is accounted for by use of equivalent imperfections and/or residual stresses bycalibrating the magnitude of the imperfections (and, or the residual stresses) to the resistance of a known casethat with regard to the stability resistance resembles the buckling problem at hand.

The following procedure assumes that an equivalent imperfection is accounting for all effects necessary toobtain realistic capacities:

i) Prepare a model that is intended to be used for the analysis.

ii) Perform an eigenvalue analysis to determine relevant buckling modes.

iii) Select the object for calibration and prepare a model using the same element type and mesh density asintended for the model to be analysed.

iv) Perform eigenvalue analysis of the calibration object and determine the appropriate buckling mode for thecalibration object.

v) Determine the magnitude of the equivalent imperfection that will give the correct resistance for thecalibration object.

vi) Define an equivalent imperfection for the most relevant failure mode for the problem under investigationbased on the results from the calibration case.

The definition of the equivalent imperfection may in certain cases not be obvious and it will then be requiredto check alternative patterns for the equivalent imperfections.

Usually an imperfection pattern according to the most likely buckling eigenmode will be suitable for use.Exceptions may be cases where the pattern of the deflected shape due to the loads differ from the shape of thebuckling eigenmodes. In cases of doubt several patterns may be needed.

Example of the use of this procedure is included in the Appendix [B.3].

5.4.5 Strain limits to avoid accurate check of local stability for plates and tubular sections yielding in compression.

5.4.5.1 General

For cases where compressed parts of the cross-section (as a flange) are experiencing plastic strain incompression, but one wants to avoid an accurate stability analysis of the local buckling effects the stability canbe assumed to be satisfactory if the plastic strain are limited to the values given below. The requirements arevalid for plates that are loaded in the longitudinal direction and supported on one or both of their longitudinaledges, and for tubular sections.

Plates supported on both longitudinal edges:

Where b is distance between longitudinal supports and t is plate thickness

(20)

�&' ≤ L2.7 L�SM 2 − 0.0016M �355�� S)� 0 < �&' < 0.10




Plates supported on one edge (flange outstand)

Where c is the plate outstand and t is plate thickness

Tubular sections without hydrostatic pressure:

The strain shall be calculated as plastic strain and may be taken as the average value through a cross-section ofthe compressed plated for element length no less than 2 times the plate thickness. Material properties shouldbe according to [4.7].

For structural parts meeting requirements to cross-sectional class 3 or 4 no plastic strain due to compressivestresses can be allowed without an accurate buckling analysis.

For definition of cross-sectional classes see DNV-OS-C101 /14/.

5.5 Repeated buckling

For cases where buckling of parts of the structure may occur before the total capacity of the entire structure isreached, it is necessary to investigate if the buckling may cause reduced capacity against cyclic loads. Whensignificant cyclic loads are present one should limit the capacity to the load level that corresponds to the firstincident of buckling or a cyclic check needs to be carried out. See [5.2.3] for determination of cyclic loads.

For cyclic loads following an extreme wave or wind load, it is considered acceptable to disregard failure dueto repeated buckling of the following cases:

— Buckling of the individual plates in a stiffened plate structure if the plate span to thickness ratio is less than120.

— Member buckling if all parts of the cross-section meet requirements to cross-sectional Class 1 and thereduced member slenderness as a column is above 0.5.

Failure due to low cycle fatigue according to [5.2] needs to be checked also for these cases.

It should be noted that structural parts that are yielding in tension may buckle when unloaded. If cyclic loadsleads to yielding in tension one must check against buckling through the entire dimensioning load cycle.

In certain cases sufficient capacity may be proved by disregarding the structural part that suffer buckling in thecyclic capacity checks.

(21)

(22)

�&' = c0.29 L�&M2 − 0.0016e �355�� S)� 0 < �&' < 0.10

�&' = c8.5 L �UM2 − 0.0016e 355�� S)� 0 < �&' < 0.10



Sec.6 Bibliography –

6 Bibliography

/1/ ISO 2394, General principles on reliability for structures, Second edition 1998-06-01

/2/ API RP 2A Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms – Working Stress Design, Errata and supplement 3 October 2007

/3/ EN 1990, Eurocode - Basis of structural design, April 2002

/4/ EN 1993-1-1 Eurocode 3 Design of steel structures. Part 1-1 General rules and rules for buildings

/5/ EN 1993-1-5, Eurocode 3 - Design of steel structures - Part 1-5: Plated structural elements, October 2006

/6/ EN 1993-1-6, Eurocode 3 - Design of steel structures - Part 1-6: Strength and Stability of Shell Structures, February 2007

/7/ EN 1993-1-8, Eurocode 3 - Design of steel structures - Part 1-8: Design of Joints, 2005/AC:2009

/8/ AISC 360-05, Specification for Structural Steel Buildings, March 9 2005

/9/ ISO 19900 Petroleum and natural gas industries – General requirements for offshore structures. First edition 2002-12-01

/10/ ISO 19902 Petroleum and natural gas industries – Fixed steel offshore structures, First edition 2007-12-01

/11/ Norsok Standard N-001, Integrity of offshore structures, Edition 7, June 2010

/12/ Norsok Standard N-004, Design of steel structures, Revision 4, February 2004

/13/ Norsok Standard N-006, Assessment of structural integrity for existing offshore load-bearing structures, Edition 1, March 2009

/14/ DNV-OS-C101, Design of Offshore Steel Structures, General (LRFD Method), April 2011

/15/ DNV-RP-C201 Buckling Strength of Plated Structures, October 2010

/16/ DNV-RP-C202 Buckling Strength of Shells

/17/ DNV-RP-C203 Fatigue Design of Offshore Steel Structures October 2012

/18/ DNV-RP-C204 Design against Accidental Loads, October 2010

/19/ ECCS publication No. 125, Buckling of Steel Shells. European Design Recommendations, 5th Edition, J.M. Rotter and H. Smith Editors.

/20/ DNV-RP-F110 Global Buckling of Submarine Pipelines Structural Design due to High Temperature/High Pressure, October 2007

/21/ DNV-SINTEF-BOMEL: Ultiguide, Best practice for use of non-linear analysis methods in documentation of ultimate limit state for jacket type offshore structures, April 1999.

/22/ Skallerud, Amdahl: Nonlinear analyses of offshore structures, Research studies press ltd., 2002 (ISBN 0-86380-258—3)

/23/ Corrocean ASA: Design of offshore facilities to resists gas explosion hazards. Engineering handbook. Oslo 2001.

/24/ ASME Boiler & Pressure Vessel Code 2013 Edition July 1, 2010 VIII Division 2, Alternative Rules

/25/ EN 13445-3:2012 Unfired pressure vessels Part 3

/26/ Hagen, Ø, Solland, G. Mathisen, J. Extreme storm wave histories for cyclic check of offshore structures OMAE 2010-20941

/27/ H. M. Hilber, T. J. R. Hughes and R. L. Taylor: Improved numerical dissipation for time integration algorithms in structural dynamics, Earthquake engineering and structural dynamics, 5 (1977), page 283-292.

/28/ Skallerud, Eide, Amdahl, Johansen: On the capacity of tubular T-joints subjected to severe cyclic loading. Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering - OMAE, v 1, n Part B, p 133-142, 1995.

/29/ Weignad, Berman: Behaviour of butt-welds and treatments using low-carbon steel under cyclic inelastic strains, Journal of Constructional Steel Research, v 75, p 45-54, August 2012.

/30/ Boge, Helland, Berge: Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering - OMAE, v 4, p 107-115, 2007.



Sec.6 Bibliography –

/31/ Scavuzzo, Srivatsan, Lam: Fatigue of butt welded steel pipes. American Society of Mechanical Engineers, Pressure Vessels and Piping Division (Publication) PVP, v 374, p 113-143, 1998, Fatigue, Environmental Factors, and New Materials.

/32/ Belytschko, Liu, Moran, Nonlinear Finite Elements and Continua and Structures, John Wiley&Sons, Ltd., November 2009

/33/ Maresca, Milella, Pino: A critical review of triaxiality based failure criteria, IGF 13 Cassino 27 e 28 Maggio, 1997

/34/ Kuhlmann: Definition of Flange Slenderness Limits on the Basis of Rotation Capacity Values, Journal of Constructional Steel Research, 14 (1989) 21-40

/35/ Gardner, Wang, Liew: Influence of strain hardening on the behavior and design of steel structures, International Journal of Structural Stability and Dynamics Vol. 11. No. 5 (2011) 855-875

/36/ DNV-OS-F101 Submarine Pipeline Systems, August 2012

/37/ Heo, Kang, Kim, Yoo, Kim, Urm: A Study on the Design Guidance for Low Cycle Fatigue in Ship Structures. 9th Symposium on Practical Design of Ships and Other Floating Structures. Germany. 2004.

/38/ EN-10025 Hot rolled products of structural steels. Part 2, 3, 4 and 6

/39/ EN-10225 Weldable structural steels for fixed offshore structures - Technical delivery conditions



App.A Commentary –

APPENDIX A COMMENTARY

A.1 Comments to [4.1] General

The element model selected for analysis needs to represent the structure so that any simplifications are leadingto results to the safe side. This is especially important for the selection of boundary conditions and therepresentation of the load. The analyst needs to assess the possibility that simplification may lead to anoverrepresentation of the resistance. An example may be the representation of neighbouring elements that alsoare subjected to buckling. In the case that the stiffness of the adjoining structure is uncertain it is recommendedto use boundary condition corresponding to simple support. If there are uncertainties with respect tosimplification in load it is recommended to vary the load pattern and perform alternative analyses to check theeffect.

The requirements to characteristic resistance in other codes for offshore structures like ISO 19902 /10/ aresimilar and the analysis carried out according to the recommendations in this RP is expected to fulfil therequirements also in this code.

A.2 Comments to [4.4] Selection of elements

Guidance on selection of suitable elements for non-linear analysis can be found in text books e.g. /32/.

A.3 Comments to [4.7.5] Stress - strain curves for ultimate capacity analyses

The proposed stress-strain curves are based on steel according to /38/ and /39/. The curve is also applicable tomaterials according to DNV Offshore Standard /14/.

A.4 Comments to [5.1.1] General

There is not a universal method available that can be used for predicting tensile failure for practical engineeringapplications by FE methods.

The value of the acceptable strain will be governed by:

— stress triaxiality

— load history

— cold deformation.

— material properties

— material inhomogeneity

— different material properties of materials being joined. (Even material with the same strength specificationmay differ due to statistical variance if not from same batch)

— presence of defects.

The calculated strain values will be a function of:

— element type

— element density

— material properties

— flow rules

— sequence of load modelling.

The acceptable strain values can therefore not be given with large accuracy without consideration of theconditions of the actual problem. This RP proposes to either use a simplified tensile failure criterion or tocalibrate a problem specific criterion according to a specified procedure.

There are several models describing the local phenomenon of tensile failure. Common for most of these is thatthe strain and stress state during the entire loading sequence until failure is considered important for describingthe damage process properly. Unfortunately, a high degree of complexity is a common feature of many of themodels, and the theoretical and practical knowledge required to perform a FE analysis based on these criteriais judged not to be suited for engineering purposes.

Base material has in general better toughness properties than weld material. It is therefore regarded as gooddesign practice to ensure that large plastic deformation occurs in the parent material and not in the weld. Thisis normally the case for full penetration welds where the overmatching material ensures limited plasticdeformation in the weld. Weld material may however contain defects of considerable size. In such cases afracture mechanics assessment is necessary in order to determine if fracture in the weld may be the governingfailure mode.




A.5 Comments to [5.1.3] Tensile failure in base material. Simplified approach for plane plates

Dominant structural steel design codes like Eurocode 3 /7/ and AISC /8/ apply a larger material factor fortensile failure when the capacity is based on the tensile strength. In order to determine a resistance by use ofnon-linear FE methods a similar increase in the material factor should be included since the material curve usedis including the strain hardening.

A.6 Comment to [5.2.3] Determination of cyclic loads

The check against cyclic failure should be carried out with the use of a dimensioning load history that has theprescribed probability of occurrence as required for a single extreme load. For environmental loads like waveand wind it should be established a dimensioning storm that the structure is required to survive. It would be inline with check for other failure modes to check the structure for one single storm from each of the criticaldirections, but without adding the calculated damage from different directions.

The load history for the remaining waves in a 10 000 year dimensioning storm investigated for southern NorthSea conditions have been found to have a maximum value equal to 0.93 of the dimensioning wave, a durationof 6 h and a Weibull shape parameter of 2.0. This applies for check of failure modes where the entire storm willbe relevant, such as crack growth.

When checking failure modes where only the remaining waves after the dimensioning wave (e.g. buckling)need to be accounted for, a value of 0.9 of the dimensioning wave may be used ref. /26/.

The load history for the remaining waves in a 100 year dimensioning storm investigated for southern North Seaconditions have been found to have a maximum value equal to 0.95 of the dimensioning wave, a duration of 6h and a Weibull shape parameter of 2.0. The largest remaining waves after the dimensioning wave (e.g. forcases like buckling) the largest wave is found as 0.92 of the dimensioning wave.

A.7 Comment to [5.2.4] Cyclic stress strain curves

The cyclic stress-strain curves are only intended for low cycle fatigue analysis. The use of monotonic stressstrain curve in low cycle fatigue analysis may provide non-conservative results and must therefore be avoided.

The cyclic stress strain curves presented in Table 5-3 are based on cyclic behaviour of similar steels reportedin reference /37/. In order to account for uncertainties in material behaviour the curves are based onconservative assumptions. A steel grade similar to S235 was not reported in /37/. Here, the same exponent of10 in the Ramberg-Osgood relation was assumed. K was assessed by assuming a strain value of approximately0.005 when the stress has approached the monotonic stress level of 235MPa.

A.8 Comment to [5.2.5.1] Accumulated damage criterion

Laboratory test results presented in references /28/ to /31/ make up the basis for the established ε-N curve forwelded joints. The proposed mean and design curve for air along with the laboratory test data is presented inFigure A-1 . Note that some of the results presented in the figure are not obtained directly from the referredarticles. In some cases further analysis and interpretation was needed to obtain the data on a proper format.

The mean curve is established based on judgement. The results reported by Weigans and Berman ref. /29/ areobtained from testing of dog-bone specimens cut out from a butt welded plate. These results have thereforebeen weighted less than results from ref. /28/ and ref. /30/ which is based on full scale testing of tubular joints.The fatigue test results presented in ref. /31/ are from pipes with wall thickness of less than 10 mm. The fatiguestrength of welded joints is to some extent dependent on the wall thickness and since the thickness of structuralelements normally is significantly larger than this the results have been weighted less.

Because the fatigue test data come from several different sources it was not found reasonable to establish thestandard deviation from a regression analysis. Instead, a standard deviation of 0.2 in log N scale is assumed forconstructing the design curve in air. A standard deviation of 0.2 is identical to what is used in high cycle fatigue(DNV-RP-C203 ref. /17/). It is a general opinion within the body of fatigue expertise that the statisticaldeviation in fatigue test results, decreases with decreasing fatigue life. Hence, assuming a standard deviationvalue of 0.2 should be conservative.

The high cycle fatigue design curve in DNV-RP-C203 is defined as the mean curve minus two standarddeviations. In order to account for limited test data, the design curve has been established by subtracting threestandard deviations. Three standard deviations on log N corresponds to a factor of 103·0.2 ≈ 4, i.e the designcurve is below the mean curve by a factor of approximately four on fatigue life.

The design curve for seawater with cathodic protection is constructed by reducing the fatigue life by a factorof 2.5. This is identical to the reduction used in DNV-RP-C203 for fatigue lives less than 106.

Due to limited test data the proposed model does not take into account that the fatigue strength decreases withincreasing thickness. It is however believed that this effect is less pronounced in low cycle fatigue comparedto high cycle fatigue, ref. DNV-RP-C203. In order to avoid non-conservative results it is recommended not toapply the proposed curves for thicknesses above 60 mm. For larger thicknesses it is recommended to multiplythe strain amplitude (∆ε/2) with the thickness correction factor used in DNV-RP-C203. The reference thicknessis set equal to 60 mm. In case that low-cycle fatigue is to be considered together with high cycle fatigue it maybe more practical to use the same reference thickness in both checks.




Figure A-1 Mean and design curve for welded joints along with laboratory test results.

A.9 Comments to [5.2.7] Shake down check

When a structure is loaded beyond linear limits the response for subsequent cycles will be changed. It istherefore necessary to investigate the behaviour through the full cycles also for the next cycles. See e.g. /22/for more guidance.

A.10 Comments to [5.4.1] General

The modelling of geometrical imperfections, out-of-straightness etc. is crucial for achieving a credible and safeestimate of the buckling and ultimate strength limits. The less redundant the structure is the more important itwill be to model the geometrical deviations from perfect shape in a consistent way using the eigenmode,postbuckling shapes, combinations thereof or similar. For redundant structures the sensitivity of the ultimateload bearing capacity to the size of the geometrical imperfections will be negligible. In such cases the triggeringof the governing modes rather than accounting for actual tolerance size will be most important for the analyses.

Guidance on analysis of stability problems may be found in e.g. /19/.

A.11 Comments to [5.4.5] Strain limits to avoid accurate check of local stability for plates and tubular sections yielding in compression.

The strain limits for plates are established from analysis of flanges meeting rotational capacities according tocross-section class 1 and 2 and by comparison with tests. See /34/ and /35/. Strain limits are also compared withrecommendations given in the DNV Offshore Standard for submarine pipelines /36/.



App.B Examples –

APPENDIX B EXAMPLES

B.1 Example: Strain limits for tensile failure due to gross yielding of plane plates (uniaxial stress state)

B.1.1 T-section cantilever beam

Gross yielding check of a T-section cantilever beam, subjected to axial and shear force and moment loading,is presented in this example. The finite element software ABAQUS is used to perform the analyses.

The geometry and boundary conditions of the beam are shown in Figure B-1 . Loading is applied to a referencepoint coinciding with the neutral axis of the beam cross section, using kinematic coupling between cross sectionand reference point. The beam is modelled using 4-noded shell elements with reduced integration (S4R) withmesh size of 16 mm x 16 mm. Material grade is S355, modelled according to Section [4.7.5].

The magnitude of the applied forces and moments are given by axial force Nx, shear force Py = − 0.15Nx andbending moment Mz = − 0.45Nx.

The loading and boundary conditions result in a stress state dominated by uniaxial stress. Hence, the criterionpresented in Section [5.1.3.2] is applied for assessing the beam.

Figure B-1 Geometry and boundary conditions for cantilever beam

According to the criterion presented in Section [5.1.3.2], the strain should be calculated as the linearizedmaximum principal plastic strain along the likely failure line and checked against the limit for the critical strain.The limit is a critical gross yield strain of 0.04 for this example.

Figure B-2 shows a contour plot of the maximum principal plastic strain and the chosen failure line, the 3rd

element column from the clamped end. For the chosen failure line the maximum principal plastic strain isobtained from integration points and linearized using the method of least squares.

Based on the finite element analysis results and the linearization the critical load is determined to be betweenNx= 489 kN and Nx= 500 kN. The maximum principal plastic strain distribution and corresponding linearizeddistribution for load level Nx=489 kN are shown in Figure B-3 and analysis results are shown in Table B-1 .

Table B-1 Analysis results

Nx Maximum linearized

[kN] principal plastic strain

489 3.9·10-2

500 4.2·10-2

3000

A

A

460

300

Section A

500N

M

P40

8



App.B Examples –

Figure B-2 Maximum principal plastic strain contour plot, with chosen failure line highlighted

Figure B-3 Maximum principal plastic strain and linearized maximum principal plastic strain distributions for web

The design resistance will be found as the characteristic resistance divided by the appropriate material factors.

The selected element mesh is tested to be accurate meaning that a FEM knock down factor CFEM can be takenas 1.0 in this case.

B.1.2 T-section cantilever beam with notch

Check for tensile failure of a T-section cantilever beam with a notch in the free edge of the web is presented inthis example. The geometry and boundary conditions are shown in Figure B-4 . The model, loading and analysissetup and procedure are the same as in [B.1.1], except the size of the mesh which in this case is 25% of thenotch height, i.e. 25 mm x 25 mm.

In addition to the gross yielding criterion presented in [5.1.3.2], the local tensile failure criterion presented in[5.1.3.3] must be applied when assessing the beam.

Chosen failure line



App.B Examples –

Figure B-4 Geometry and boundary conditions for cantilever beam with notch

For the gross yielding two likely failure lines were chosen; one at mid-notch and one at the notch cornerdisplaying the highest strain values, see Figure B-5 . The maximum principal plastic strain is obtained fromintegration points and linearized using the method of least squares. Both failure lines must comply with thecriterion of an allowable maximum principal plastic linearized strain of 0.04. In addition, the local strain,according to [5.1.3.3], must not exceed the critical strain value of 0.12. In this case the mesh size falls withinthe defined volume criteria. Hence, the local strain value is taken as the maximum principal plastic strain in theelement with the largest strain.

Figure B-5 Maximum principal plastic strain contour plot, with chosen failure lines highlighted

Based on the finite element analysis results the linearization line 1 is found to be the critical failure line, withcritical load determined to be between Nx = 310 kN and Nx = 315 kN. The maximum principal plastic straindistributions and corresponding linearized distributions for both failure lines at load level Nx = 310 kN areshown in Figure B-6 and analysis results are shown in Table B-2 .

Table B-2 Analysis results

Nx[kN]Maximum linearized principal plastic strain

Line 1 Line 2 Largest element strain

310 3.8·10-2 3.0·10-2 7.6·10-2

315 4.1·10-2 3.3·10-2 8.1·10-2

100

200

3000

A

A

500500 460

300

Section A

40

8

N

M

P



App.B Examples –

Figure B-6 Maximum principal plastic strain and linearized maximum principal plastic strain distributions for web

A convergence study is used to ensure that satisfactory accuracy is obtained. When convergence is reached thecritical load is determined between Nx = 305 kN and Nx = 310kN, resulting in a FEM knock down factor

The calculations in the convergence study are performed for a fixed volume, i.e. the volume used in thepresented mesh. For the local strain criteria this is illustrated in Figure B-7 and Figure B-8 .

The design resistance will be found as the characteristic resistance divided with the appropriate materialfactors.

Figure B-7 Element mesh used for convergence study. Elements used in local strain check circled in red

4��- = 305kN315kN = 0.968



App.B Examples –

Figure B-8 Maximum principal plastic strain for three mesh densities, the left mesh is presented in this example

B.2 Example: Convergence test of linearized buckling of frame corner

A symmetric frame of beams with I-section is analysed. The frame with boundary conditions is shown in FigureB-9 and Figure B-10 . The loading is applied as a displacement of the web at one end of the frame, u2,applied =− 0.01 m. Three different mesh densities and two element types are included in a convergence study, to ensurea sufficiently refined mesh. See Figure B-11 . The element types used are 4 node rectangular shell elements and8 node rectangular shell elements.

The analyses are performed using the FEM-software ABAQUS.

Figure B-9 Geometry of test example

a

4000

2000

2000

R = 1500

4000

b = 500

500

480

Section A

A

A

20 20

15



App.B Examples –

Figure B-10 Displacement/boundary conditions



App.B Examples –

Figure B-11 Top: coarse mesh. Middle: fine mesh. Bottom: very fine mesh



App.B Examples –

For the eigenvalue analyses and the linear analyses elastic material properties were used and for the bucklingcapacity analyses non-linear material properties were used. Details are shown in Table B-3 .

The loading is applied as displacement on the web at one end of the frame, as shown in Figure B-10 . Hence,the eigenvalue defines the displacement corresponding to linearized buckling.

A convergence study is performed by analysing 6 cases and the resulting buckling displacements are listed inTable B-4 . From these results all combinations of mesh size and element type except the coarse 4 nodecombination, seems to be sufficiently refined. However, the stress results wanted are also highly dependent onthe mesh refinement, and a fine mesh in the area where high stress values are reached is preferable. An analysisusing the very fine mesh is time consuming, hence the mesh size and element type combination chosen is the4 node elements with fine meshing.

In summary the convergence test has shown that case number 2 and case 4 will produce sufficiently accurateresults of the linearized buckling value. Case 2 is preferred as the analysis is more efficient compared to case4. The increased mesh refinement of case 3, 5 and 6 will not significantly improve the accuracy for the actualproblem solution.

B.3 Example: Determination of buckling resistance by use of linearized buckling values

B.3.1 Step i) Build model

The same problem as shown in Figure B-9 will be used in this example and the boundary conditions are as inFigure B-10 . The material properties are shown in Table B-5 .

The analysis follows the steps as given in [5.4.2]. Step i) is completed as the model from the example in [B.2].

B.3.2 Step ii) Linear analysis of the frame

The results from a linear analysis are shown in Figure B-12 and Figure B-13 for the von-Mises and membranecompression stresses respectively.

The linear analysis is performed with the same applied displacement as in the eigenvalue analysis u2,applied = − 0.01m, equivalent to an applied load i y-direction SRep= 75.7 kN.

Table B-3 Material properties

Density, ρ 7850 kg/m3

Young’s modulus, E 210 GPa

Poisson’s ratio, ν 0.3

Table B-4 Convergence study of frame

Case number

Mesh size Element type Linearized buckling displacement [m]

1 Coarse 4-node 0.0653

2 Fine 4-node 0.0624

3 Very fine 4-node 0.0618

4 Coarse 8-node 0.0616

5 Fine 8-node 0.0615

6 Very fine 8-node 0.0615





Yield strength, σY 355 MPa



App.B Examples –

Figure B-12 Distribution of von-Mises stress from linear analysis



App.B Examples –

Figure B-13 Distribution of compressive stress from linear analysis (minimum in-plane principal stress)

B.3.3 Step iii) Determine the buckling eigenvalues

Eigenvalue analysis is performed to find the buckling modes and eigenvalues of the frame. The first eigenvalueis kg= 6.24, and the corresponding buckling mode shape is shown in Figure B-14 .



App.B Examples –

Figure B-14 First buckling mode

B.3.4 Step iv) Select the governing buckling mode and the point for reading the representative stress

The lowest buckling mode is judged to be a realistic buckling shape for this case and is selected.

The reference stress is taken as the maximum von-Mises stress in the structural part subjected to buckling.

B.3.5 Step v) Determine the von-Mises stress at the point for the representative stress σRep from step ii)

Stress from linear analysis:

σ Rep = 97.4 MPa

B.3.6 Step vi) Determine the critical buckling stress

The critical buckling stress for the governing buckling mode is determined as:

MPa

The reduced slenderness is determined as:

B.3.7 Step vii) Select empirically based buckling curve

The buckling curve used here is taken from Table 5-6. The curve selected is the one for column and stiffenedplate and plate without redistribution possibilities as it is judged that the corner plate cannot redistribute stressesin a way so the plate curve could be used.

1�� = � 1#� = 608

.̅ = � �i1�� = 0.76

a = 1] + j]2 − .̅2 ≤ 1.0



App.B Examples –

α is set to 0.3 for the following calculations.

B.3.8 Step viii) Determine the buckling resistance Rd

With then the buckling factor is

Assuming a material factor , the buckling resistance is

B.4 Example: Determination of buckling resistance from non-linear analysis using code defined equivalent tolerances

B.4.1 Description of model

The same problem as shown in Figure B-9 will be used in this example and the boundary conditions are as inFigure B-10 . The material properties are shown in Table B-5 and the material model is shown in Figure B-15 .

Figure B-15 Material model for analysis with material non-linearity

A non-linear analysis (using the arc-length method) is performed, where the effects of imperfections, residualstresses and material non-linearity is accounted for by use of a defined material stress-strain relationship andthe use of empirically determined equivalent imperfections. The shape of the governing buckling mode is takenas the lowest buckling mode as shown in Figure B-14 , and is used as the pattern for the equivalent imperfection.The magnitude of the equivalent imperfection δ is calculated using the tolerances given in Table 5-7 .

The analysed frame can be considered equivalent to a component of longitudinal stiffener or flange outstand,hence the magnitude is given as

where c is half the width of the flange. Two values of c are analysed, the largest c; , where is the distance between where the webs cross in the corner of the frame and the midpoint of the

flange curvature, and an average c; , where is the width of the flange outside thecurved area. See Figure B-9

] = 0.5 [1 + :�.̅ − 0.2� + .̅2]

.̅ = 0.76 a = 0.767

#$ = a�� %#��-1#�

�- = 1.15

#$ = 0.767 ∙ 355 ∙ 0.07571.15 ∙ 97.4 = 0.184 MN

n = 0.02 rad = 0.02& &I = I I = 0.975 J

&IP = I + S22 S = 0.5 J



App.B Examples –

Figure B-16 Stress distribution for non-linear analysis with initial imperfection at maximum applied force

B.4.2 Results

The stress distribution for the non-linear analysis with initial imperfection is shown in Figure B-16. Figure B-17 displays the force-displacement curves for the displaced end of the frame for the linear analysis and the force-displacement corresponding to the critical buckling stress where imperfections are taken into consideration ascalculated in Section [B.3], and from the non-linear analyses.

nI = 0.0195 nIP = 0.0122

op



App.B Examples –

Figure B-17 Force-displacement from non-linear analyses, linear analysis and the calculated critical value

B.5 Example: Determination of buckling resistance from non-linear analysis that are calibrated against code formulations or tests

B.5.1 Step i): Prepare model

A conical transition subjected to external hydrostatic pressure and axial tension is chosen for this analysis. Thegeometry of the conical transition and the calibration object is shown in Figure B-18 . The applied loading isdefined as a hydrostatic pressure and an axial tension .

The boundary conditions are modelled using constraints with kinematic coupling between a reference point in thecross-section centre and the nodes on the circumference of the conical transition ends. At the bottom alltranslations and rotations of the reference point are constrained and the top reference point is constrained in thehorizontal plane (x- and z-direction). Load and boundary conditions and element mesh are shown in Figure B-19 .The conical transition is modelled using 4-noded shell elements (S4R). Material properties are listed in Table B-6 .

Figure B-18 Geometry of conical transition (on top) and calibration object (bottom), dimensions in mm

� = 1.01MPa W� = 58.4MN



App.B Examples –

Figure B-19 Left: Load and boundary conditions. Right: Element mesh





Yield strength, σY 420 MPa

Density water, ρw 1030kg/m3



App.B Examples –

B.5.2 Step ii): Determine relevant buckling modes

Eigenvalue analysis is performed to find the buckling modes for the conical transition. The first relevantbuckling mode (with positive eigenvalue) is mode 3, shown in Figure B-20 .

Figure B-20 Buckling mode shape for conical transition



App.B Examples –

B.5.3 Step iii): Select object for calibration and prepare model

The calibration object is selected as a cylinder. The diameter and wall thickness are equal to the lowercylindrical part of the conical transition, while the length is chosen as 2/3 of the conical transition length (lowerpart, conical part and a part of the top part). The load and boundary conditions, element type and mesh densityused is the same as for the model of the conical transition, see Figure B-21 .

Figure B-21 Left: Load and boundary conditions. Right: Element mesh



App.B Examples –

B.5.4 Step iv): Determine the appropriate buckling mode for the calibration object

Eigenvalue analysis is performed to find the buckling modes for the calibration object. These buckling modes

are compared to the buckling modes found for the conical transition and a mode with similar pattern is selected.

Figure B-22 shows the first cylinder buckling mode. This shows a similar pattern to the buckling mode of the

conical transition Figure B-20 , hence this is determined to be an appropriate buckling mode.

Figure B-22

Buckling mode shape for cylinder

B.5.5 Step v): Determine magnitude of the equivalent imperfection

To determine the magnitude of the equivalent imperfection a non-linear analysis of the cylinder with

imperfections is performed. The imperfection shape from the chosen buckling mode was transferred to the non-

linear analysis, and the same load and boundary conditions as for the eigenvalue analysis were applied. The

material model shown in Figure 5-6 is used for the non-linear analysis.

The imperfection is scaled so the buckling capacity of the cylinder is equal to the buckling capacity for cylinders

given in N-004 /12/. To obtain this capacity the magnitude of the imperfection was found to be 40 mm.

B.5.6 Step vi): Perform non-linear analysis of the model with imperfections

A non-linear analysis of the conical transition with imperfections is performed. The load and boundary

conditions remain the same, and the material model and magnitude of the calibrated imperfection from Step v

is used. The load proportionality factor for this case is shown in Figure B-23 . The maximum load proportionality

factor is . Thus the buckling capacity of the conical transition subjected to the given load

combination is; hydrostatic pressure and an axial tension . Figure B-24 shows the

von Mises stress at maximum load on the deformed conical transition.

!r�JI� = 0.936 � = 0.95MPa W� = 54.7MN



App.B Examples –

Figure B-23 Load proportionality factor for conical transition with initial imperfection

Figure B-24 Deflected shape showing von Mises stress at maximum load deformations scaled with a factor of 10



App.B Examples –

B.6 Example: Low cycle fatigue analysis of tubular joint subjected to out of plane loading.

This example presents a low cycle fatigue analysis of a tubular T-joint subjected to an out-of-plane fullyreversible load of ± 60 kN. The objective of the analysis is to estimate the design life based on therecommendations in Section [5.2.5]. The assumed geometry and dimensions are given in Table B-7 and FigureB-25 .

Figure B-25 Geometry of test example, dimensions in mm

It is assumed that the cyclic stress-strain behaviour is well described by the Ramberg-Osgood relationship:

The values for the Ramberg-Osgood parameters are presented in Table B-8 for the chord and the brace.

In order to obtain the cyclic strains a finite element analysis was carried out using the FEM-software ABAQUS.An 8-noded shell element (S8R) model was established with load and support conditions as shown in FigureB-26 .

The chord was constrained at each end for all translational and rotational degrees of freedom. The out-of-planeload was applied by means of a reference point located at the cross-section centre of the brace end. Thisreference point is connected to the circumference of the brace end by means of kinematic coupling. The loadwas applied using three steps as illustrated in Figure B-27 .

Table B-7 Dimensions

[mm]

Chord diameter D = 300

Chord thickness T = 15.9

Chord length L = 1800

Brace diameter d = 160

Brace thickness t = 11.5

Brace length l = 500

Table B-8 Ramberg-Osgood parameters

K’ [MPa]

n’

Chord 731.7 0.096

Brace 699.5 0.108

� = 1� + X 1Y′ Z 1� ′



App.B Examples –

Figure B-26 Boundary and loading conditions for tubular joint

Figure B-27 Load steps

Figure B-28 (a) shows an overview of the finite element mesh. Figure B-28 (b) shows a close-up of the brace-chord intersection area. The finite element mesh in the hot spot region is in accordance with the recommendedpractice DNV-RP-C203 for tubular joints. In this example the element nodes coincide with the specifiedextrapolation points (a and b). Hence, nodal values are applied in the extrapolation procedure for calculatingthe hot spot strain range. The distance from the hot spot to the first extrapolation point, a is obtained by meansof Equation (23). The distance to the second extrapolation point, b is obtained by Equation (24).

(23)

(24)

I = 0.2√'� = 6.1mm

S = t#36 = 13.1mm



App.B Examples –

Figure B-28 Left: (a) Meshed model, Right: (b) Close-up of brace-chord intersection area

Figure B-29 shows the principal strain range due to the out-of-plane cyclic loading.

Figure B-29 Maximum principal strain range

The hotspot strain range is obtained according to the following procedure:

i) Establish the total strain ranges ( etc.) by subtracting the minimum strain values of loadstep 2 by the maximum values of load step 3. In ABAQUS this is done by using the “Create Field Output”option.

ii) Extract the principal strain ranges at the two extrapolation points, i.e, at distances a and b.

iii) The hotspot strain range is calculated by means of the following equation:

The hot spot strain range along with values at distances a and b are presented in Table B-9 .

(25)

Table B-9 Hotspot strain range

Nodal value saddle 0.0036 0.0021 0.0048

Δ�� , Δ�� , Δ��

Δ�ℎ"

Δ�ℎ" = Δ�I − X IS − IZ �Δ�S − Δ�I�

vwp vwx vwyz



App.B Examples –

Air environment is assumed. Hence, the characteristic design life due to the cyclic loading is obtained bysolving the following equation, ref. Section [5.2.5]:

B.7 Example: Low cycle fatigue analysis of plate with circular hole.

In this example a low cycle fatigue analysis of a plate with a circular hole subjected to cyclic displacement of1.0mm is presented. The objective of the analysis is to estimate the design life based on recommendations inSection [5.2.6]. The dimensions of the plate are presented in Figure B-30 . The plate material is of grade S355and the cyclic stress-strain curve is obtained from Table 5-3 .

Figure B-30 Geometry of considered specimen

The maximum principal strain range is obtained by performing a finite element analysis with the FEM-softwareABAQUS. The finite element analysis was performed with 8-noded shell elements with reduced integration(S8R). The material modelling is according to specifications in Section [4.7.5]. The boundary conditions andcyclic displacement is applied as illustrated in Figure B-32 . Note that a total of 6 complete load cycles isspecified in the analysis in order to see how the stress strain hysteresis curve developed.

Figure B-31 Geometry and loading

∆εhs2 = σf′E �2N�−0.1 + εf′ �2N�−0.5

→ W = 1270 cycles



App.B Examples –

Figure B-32 Load steps

The maximum principal strain range is obtained according to the following procedure:

i) Perform a strain convergence study. The mesh around the hole is refined until the strain value in therelevant nodal point converges. Based on the convergence study it was found sufficient to use 48 elementsaround the hole.

ii) Establish the strain component ranges ( , etc.) by subtracting the strain component valuesof load step 12 from the values of load step 13. Hence, the stress/strain output from the last cycle is used asbasis for calculating the design fatigue life. In ABAQUS the strain range is obtained by using the “CreateField Output” option.

iii) Calculate the maximum principal strain range based on the strain component ranges.

iv) Calculate the design fatigue life based on the seawater with cathodic protection curve.

Figure B-33 shows the maximum principal strain range due to the specified cyclic loading of the plate. Thehysteresis loop in the location adjacent to the hole with the highest cyclic strain is plotted in Figure B-34 . Themaximum principal strain range of ∆εl = 0.011 obtained from the last cycle in the finite element analysis is usedas basis for calculating the design fatigue life. Hence, by solving Equation (13) in Section 5.2.6 a design fatiguelife of N = 139 is obtained.

Figure B-33 Equivalent strain range

Δ�� , Δ�� , Δ��



App.B Examples –

Figure B-34 Stress versus strain in y – direction (parallel to the 1st principal strain direction).

DNV-RP-C208: Determination of Structural Capacity by Non-linear ...

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