DNVGL­RP­C208
Edition September 2019
Amended January 2020
Determination of structural capacity by
non­linear finite element analysis methods
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RECOMMENDED PRACTICE
DNV GL recommended practices contain sound engineering practice and guidance.
©
DNV GL AS September 2019
Any comments may be sent by e­mail to rules@dnvgl.com
This service document has been prepared based on available knowledge, technology and/or information at the time of issuance of this
document. The use of this document by others than DNV GL is at the user's sole risk. DNV GL does not accept any liability or responsibility
for loss or damages resulting from any use of this document.
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FOREWORD
Changes ­ current
This document supersedes the June 2013 edition of DNV­RP­C208.
Changes in this document are highlighted in red colour. However, if the changes involve a whole chapter,
section or subsection, normally only the title will be in red colour.
Amendments January 2020
Topic
Reference
Description
Buckling
[5.4.2.1]
Formula (17) for critical buckling stress (
Examples
[8.1.1]
Formula for bending moment (
) corrected.
) corrected.
Changes September 2019
This document is a republished version of the September 2016 edition. No changes have been made to the
content of this document.
Main changes September 2016
On 12 September 2013, DNV and GL merged to form DNV GL Group. On 25 November 2013 Det Norske
Veritas AS became the 100% shareholder of Germanischer Lloyd SE, the parent company of the GL Group,
and on 27 November 2013 Det Norske Veritas AS, company registration number 945 748 931, changed its
name to DNV GL AS. For further information, see www.dnvgl.com. Any reference in this document to “Det
Norske Veritas AS”, “Det Norske Veritas”, “DNV”, “GL”, “Germanischer Lloyd SE”, “GL Group” or any other
legal entity name or trading name presently owned by the DNV GL Group shall therefore also be considered a
reference to “DNV GL AS”.
• Sec.4 Requirements to finite element analysis
—
—
—
—
In
In
In
In
[4.3.4] new section has been added. Previous text rearranged and moved to [4.5].
[4.5] title has been changed and text rearranged. Comment on drilling stiffness added.
[4.6] new material curves have been added, modified text regarding strain rate.
[4.9] new text on contact modelling has been added.
• Sec.5 Representation of different failure modes
— In [5.1] criteria have been revised.
— In [5.2] thickness effect has been included.
• Sec.7 Commentary (previously Appendix A)
—
—
—
—
—
In
In
In
In
In
[7.3] new material curves have been added.
[7.6] new comment has been added.
[7.7] new comment has been added.
[7.8] text revised and mean curves have been added.
[7.11] new comment has been added.
Recommended practice — DNVGL­RP­C208. Edition September 2019, amended January 2020
Determination of structural capacity by non­linear finite element analysis methods
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CHANGES – CURRENT
Page 3
Changes ­ current
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• Sec.8 Examples
—
—
—
—
—
In
In
In
In
In
[8.1]
[8.6]
[8.7]
[8.8]
[8.9]
example has been updated.
new example has been added.
new example has been added.
new example has been added.
numbers have been updated.
• App.A Structural models for ship collision analysis
— App.A Structural models for ship collision analysis has been added.
Editorial corrections
In addition to the above stated changes, editorial corrections may have been made.
Recommended practice — DNVGL­RP­C208. Edition September 2019, amended January 2020
Determination of structural capacity by non­linear finite element analysis methods
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Changes ­ current
Acknowledgements
This recommended practice is prepared based on results from two joint industry projects. The first joint
industry project was sponsored by the following companies and institutions (in alphabetic order):
ConocoPhillips Skandinavia AS
Det Norske Veritas AS
Mærsk Olie og Gas AS
Petroleum Safety Authority Norway
Statoil ASA
Total E&P Norge AS
A follow­up project was sponsored by the following companies and institutions (in alphabetic order):
ConocoPhillips Norge
DNV GL AS
DYNAmore Nordic AB
EDRMedeso AS
Force Technology Norway AS
Lundin Norway AS
Maersk Olie og Gas A/S
Petroleum Safety Authority Norway
Rambøll
Statoil ASA
Total E&P Norge
In addition to their financial support, the above companies are also acknowledged for their technical
contributions through their participation in the project.
Recommended practice — DNVGL­RP­C208. Edition September 2019, amended January 2020
Determination of structural capacity by non­linear finite element analysis methods
DNV GL AS
Page 5
Acknowledgements................................................................................. 5
Section 1 Introduction............................................................................................ 9
1.1 General............................................................................................. 9
1.2 Objective...........................................................................................9
1.3 Scope................................................................................................ 9
1.4 Validity..............................................................................................9
1.5 Definitions.......................................................................................10
Section 2 Basic considerations.............................................................................. 13
2.1 Limit state safety format................................................................ 13
2.2 Characteristic resistance.................................................................14
2.3 Types of failure modes................................................................... 14
2.4 Use of linear and non­linear analysis methods............................... 14
2.5 Empirical basis for the resistance................................................... 15
2.6 Ductility.......................................................................................... 15
2.7 Serviceability limit states............................................................... 15
2.8 Permanent deformations................................................................ 15
Section 3 General requirements............................................................................ 16
3.1 Definition of failure.........................................................................16
3.2 Modelling strategy.......................................................................... 16
3.3 Modelling accuracy......................................................................... 16
3.4 Determination of characteristic resistance taking into account
statistical variation............................................................................... 16
3.5 Requirement to the software.......................................................... 17
3.6 Requirements to the user............................................................... 17
Section 4 Requirements to finite element­analysis............................................... 18
4.1 General........................................................................................... 18
4.2 Selection of software for finite element analysis............................ 18
4.3 Selection of analysis method.......................................................... 18
4.4 Geometry modelling........................................................................20
4.5 Mesh............................................................................................... 21
4.6 Material modelling.......................................................................... 22
4.7 Boundary conditions....................................................................... 28
4.8 Load application..............................................................................29
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Determination of structural capacity by non­linear finite element analysis methods
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Page 6
Contents
Changes – current.................................................................................................. 3
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CONTENTS
4.11 Execution of non­linear finite element analyses, quality control... 31
4.12 Requirements to documentation of the finite element analysis..... 31
Section 5 Representation of different failure modes............................................. 32
5.1 Design against tensile failure......................................................... 32
5.2 Failure due to repeated yielding (low cycle fatigue)....................... 37
5.3 Accumulated strain (ratcheting)..................................................... 42
5.4 Buckling.......................................................................................... 43
5.5 Repeated buckling.......................................................................... 49
Section 6 Bibliography.......................................................................................... 51
6.1 Bibliography.................................................................................... 51
Section 7 Commentary.......................................................................................... 53
7.1 Comments to [4.1] General............................................................. 53
7.2 Comments to [4.5.2] Selection of element...................................... 53
7.3 Comments to [4.6.6] Recommendations for steel material
qualities (low fractile).......................................................................... 53
7.4 Comment to [4.6.8] Strain rate effects............................................54
7.5 Comments to [5.1.1] General.......................................................... 55
7.6 Comments to [5.1.3] Tensile failure in base material ­ simplified
approach for plane plates.....................................................................55
7.7 Comments to [5.1.5] Failure of welds............................................. 56
7.8 Comment to [5.1.6] Simplified tensile failure criteria in case low
capacity is unfavourable.......................................................................56
7.9 Comment to [5.2.3] Determination of cyclic loads........................... 58
7.10 Comment to [5.2.4] Cyclic stress strain curves..............................58
7.11 Comment to [5.2.6] Low cycle fatigue of base material................. 58
7.12 Comment to [5.2.5.1] Accumulated damage criterion.................... 58
7.13 Comments to [5.2.7] Shake down check........................................59
7.14 Comments to [5.4.1] General........................................................ 59
7.15 Comments to [5.4.5] Strain limits to avoid accurate check
of local stability for plates and tubular sections yielding in
compression..........................................................................................60
Section 8 Examples............................................................................................... 61
8.1 Example: Strain limits for tensile failure due to gross yielding of
plane plates (uniaxial stress state)...................................................... 61
8.2 Example: Convergence test of linearized buckling of frame
corner................................................................................................... 67
Recommended practice — DNVGL­RP­C208. Edition September 2019, amended January 2020
Determination of structural capacity by non­linear finite element analysis methods
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Page 7
Contents
4.10 Application of safety factors......................................................... 30
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4.9 Contact modelling........................................................................... 29
8.5 Example: Determination of buckling resistance from non­linear
analysis that are calibrated against standard formulations or tests...... 78
8.6 Example: Buckling check of jacket frame structure during deck
installation............................................................................................ 85
8.7 Example: Joint of rectangular hollow section (RHS) and circular
hollow section (CHS) under tension loading.........................................99
8.8 Example: Check of stiffened plate exposed to blast loads............. 119
8.9 Example: Low cycle fatigue analysis of tubular joint subjected to
out of plane loading........................................................................... 140
8.10 Example: Low cycle fatigue analysis of plate with circular hole...145
Appendix A Structural models for ship collision analyses................................... 148
A.1 Element library of offshore supply vessels................................... 148
Changes – historic.............................................................................................. 149
Recommended practice — DNVGL­RP­C208. Edition September 2019, amended January 2020
Determination of structural capacity by non­linear finite element analysis methods
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Contents
8.4 Example: Determination of buckling resistance from non­linear
analysis using standard defined equivalent tolerances......................... 75
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8.3 Example: Determination of buckling resistance by use of
linearized buckling values.................................................................... 71
1.1 General
This document is intended to give guidance on how to establish structural resistance by use of non­linear
finite element (FE) methods. It deals with determining the characteristic resistance of a structure or part of a
structure in a way that fulfils the requirements to ultimate strength in DNV GL standards.
Non­linear effects that may be included in the analyses are material and geometrical non­linearity, contact
problems, 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 characteristic resistance is similar to
what is required by many other structural standards that use the limit state safety format. Recommendations
in this document are expected to be valid for determination of capacities to be used with such standards.
1.2 Objective
The objective of this recommended practice is that analyses carried out according to the recommendations
given in this document will lead to a structure that meets the requirements to the minimum safety margin in
the governing structural standard.
This document is not intended to replace formulas for resistance in design standards for the cases where they
are applicable and accurate, but to present methods that allow for using non­linear FE­methods to determine
resistance for cases that is not covered by traditional standards.
1.3 Scope
This recommended practice is meant to supplement structural design standards for offshore steel structures
and gives recommendation on how to determine the structural capacity by the use of non­linear finite
element analysis.
1.4 Validity
The document is valid for marine structures made from structural steels meeting requirements to offshore
structures with yield strength of up to 500 MPa.
The recommendations presented herein are adapted to typical offshore steels that fulfil the requirements
specified in DNVGL­OS­C101 /9/ or an equivalent offshore design standard. The specified requirements are
made under the assumption that the considered structure is operating under environmental conditions that
are within the 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 that both 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. Failure due to
repeated loading from moderate loads (fatigue) needs to be checked separately. See DNVGL­RP­C203 /11/.
Recommended practice — DNVGL­RP­C208. Edition September 2019, amended January 2020
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SECTION 1 INTRODUCTION
1.5.1 Definition of terms
This recommended practice use terms as defined in DNVGL­OS­C101 /9/. The following additional terms are
defined below:
Table 1­1 Definition of terms
Term
Definition
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 the 5%
fractile
conservative load
load that maintains its orientation when the structure deforms, e.g. gravity loads
dimensioning event
the extreme load or sequence of loads that are the most unfavourable with respect to the
structural capacity
the ability to deform beyond the proportionality limit without significant reduction in the
capacity due to fracture or local buckling
ductility
Note: 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
expected resistance
the resistance having 50% probability of being exceeded
follower load
load that changes direction with the structure, e.g. hydrostatic pressure
gross yielding
yielding across larger parts of a structural detail.
low­cycle fatigue
the progressive and localised damage caused by repeated plastic strain in the material
Note: low­cycle 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 in which loss of capacity in one of its structural elements will lead to little or no
reduction in the overall load­carrying capacity due to load redistribution
shake down
a state in which a structure after being loaded into the elasto­plastic range will behave
essentially linear for all subsequent cycles
1.5.2 Symbols
b
span of plate
Recommended practice — DNVGL­RP­C208. Edition September 2019, amended January 2020
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1.5 Definitions
flange outstand, speed of sound
C
damping matrix
CFEM
resistance knock down factor
D
outer diameter of tubular sections
E
modulus of elasticity
Ep1
stress­strain curve parameter
Ep2
stress­strain curve parameter
Fext
external forces
Fint
internal forces
fy
yield stress/yield strength
K
Ramberg­Osgood parameter
kg
eigenvalue for governing buckling mode
Ls
characteristic element size of smallest element
lyz
length of yielding zone
M
mass matrix
N
number of cycles to failure
Rd
design resistance
Rk
characteristic resistance
Sd
design action effect
Sd
characteristic action effect
t
time, thickness
u
displacement vector
ε
strain
εcr
critical strain
εeng
engineering (nominal) strain
εeq
equivalent strain
εcrg
gross yielding strain limit
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c
fatigue ductility coefficient
εp_ult
stress­strain curve parameter
εp_y1
stress­strain curve parameter
εtrue
true (logarithmic) strain
Δεhs
fully reversible maximum principal hot spot strain range
Δεl
fully reversible local maximum principal strain range
Recommended practice — DNVGL­RP­C208. Edition September 2019, amended January 2020
Determination of structural capacity by non­linear finite element analysis methods
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time step
γM
material factor
γf
partial factor for actions
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Δt
reduced slenderness
ν
Poisson’s ratio = 0.5 for plastic strain
ρ
density
σ1,σ2
principal stresses
σRep
representative stress
σeng
engineering (nominal) stress
fatigue strength coefficient
σki
critical buckling stress
σkig
linearized buckling stress disregarding local buckling modes
σkil
linearized local buckling stress
σprop
stress­strain curve parameter
σtrue
true (Cauchy) stress
σult
stress­strain curve parameter
σyield
stress­strain curve parameter
σyield2
stress­strain curve parameter
1.5.3 Verbal forms
Table 1­2 Definition of verbal forms
Term
Definition
shall
verbal form used to indicate requirements strictly to be followed in order to conform to the document
should
verbal form used to indicate that among several possibilities one is recommended as particularly suitable,
without mentioning or excluding others
may
verbal form used to indicate a course of action permissible within the limits of the document
Recommended practice — DNVGL­RP­C208. Edition September 2019, amended January 2020
Determination of structural capacity by non­linear finite element analysis methods
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Page 12
2.1 Limit state safety format
A limit state can be defined as: A state beyond which the structure no longer satisfies the design
performance requirements. 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 addresses
failure modes from cyclic loading for cases that cannot adequately be checked according to the methods used
in standards for check of FLS. This is relevant for situations where the structure is loaded by a cyclic load at a
high load level, but only for a limited number of cycles (low­cycle fatigue).
The safety format that is used in limit state standards is schematically illustrated in Figure 2­1.
Figure 2­1 Illustration of the limit state safety format
The requirement can be written as:
Sd ≤ Rd
Sd
=
(1)
design action effect
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SECTION 2 BASIC CONSIDERATIONS
=
design resistance
Sk
=
characteristic action effect
=
partial factor for actions
=
characteristic resistance
=
material factor
Rk
It can be seen from Figure 2­1 that it is important that the uncertainty in the resistance is adequately
addressed when the characteristic resistance is determined.
2.2 Characteristic resistance
The characteristic resistance should represent a value which will imply that there is less than 5% probability
that the resistance is less than this value. Often, lack of experimental data will prevent an adequate
statistical evaluation so the 5% probability level shall be seen as a goal for the engineering judgments to be
made in such cases.
The characteristic resistance given in design standards is determined also on the basis of consideration of
other aspects than the maximum load carrying resistance. Aspects like post­peak behaviour, sensitivity
to construction methods, statistical variation of governing parameters etc. are also taken into account. In
certain cases these considerations are also reflected in the choice of material factor that will be used to
obtain the design resistance. It is necessary that all such factors are considered when the resistance is
determined by non­linear FE methods.
2.3 Types 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
it is often necessary to define characteristic resistance at a lower limit in order to be able to conclude on
structural integrity without excessive analysis. Examples of this can be to limit the plastic strain to avoid
cyclic failure for dynamically loaded structures, or to set a deformation limit for structural details that fail by
plastic strain in compression. See [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 non­linear analysis methods
Traditionally, the ultimate strength of offshore structures are analysed by linear methods to determine the
internal distribution of forces and moments, and the resistances of the cross sections are checked according
to design resistances found in design standards. These design resistance formulas often require deformations
well into the inelastic range in order to mobilise the standard defined resistances. However, no further checks
are normally considered necessary as long as the internal forces and moments are determined by linear
methods. When non­linear analysis methods are used, additional checks of accumulated plastic deflections
and repeated yielding will generally be needed. These checks are important in case of variable or cyclic
loading, e.g. wave loads.
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Rd
All engineering methods, regardless of level of sophistication, need to be calibrated against an empirical basis
in the form of laboratory tests or full scale experience. This is the case for all design formulas in standards.
In reality the form of the empirical basis vary for the various failure cases that are covered by the standards
from determined as a statistical evaluation from a large number of full scale representative tests to cases
where the design formulas are validated based on extrapolations from known cases by means of analysis and
engineering judgements. It is of paramount importance that capacities determined by non­linear FE methods
build on knowledge that is empirically based. That can be achieved by calibration of the analysis methods to
experimental data, to established practice as found in design standards 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.
The ability of a structural detail to maintain its resistance in case of overload is highly influencing the
resulting reliability of the structure. It is therefore necessary to consider not only the value of the resistance
when determining the characteristic resistance, but also to judge how the load deflection relationship is for a
particular failure mode.
The check for ductility requires that all sections subjected to deformation into the inelastic range should
deform without loss of the assumed load­bearing resistance. Such loss of resistance can be due to tensile
failure, instability of cross­sectional parts or member buckling. The design standards give little explicit
guidance on this issue, with exception for stability of cross­sectional parts in yield hinges, which normally are
covered by requirement to cross­sectional class 1. See e.g. DNVGL­OS­C101 /9/.
Steel structures generally behave ductile when loaded to their limits. The established design practice is based
on this behaviour, which is beneficial both with respect to simplifying the design process and improving
the performance of the structure. For a ductile structure, significant deflections may occur before failure
and thus give 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 during design as the structure may redistribute forces and moments to be in accordance with the
assumed static model. This is the basis why linear analyses can be used for ULS checks even for structures
which behave significantly non­linear when approaching their ultimate limit states.
2.7 Serviceability limit states
Use of non­linear analysis methods may result in more structural elements being governed by the
requirements to the serviceability limit state and additional SLS requirements may be needed compared with
design using linear methods. When plate elements are used beyond their critical load, for example, out­of­
plane deflections may need to be considered from a practical or aesthetic point of view.
2.8 Permanent deformations
All steel structures behave more or less non­linear when loaded to their ultimate limit. The formulas for
design resistance in DNVGL­OS­C101 /9/ or similar standards are therefore developed on the basis that
permanent deformation may take place before the characteristic resistance is reached.
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2.5 Empirical basis for the resistance
3.1 Definition of failure
In all analyses a precise definition of failure should be formulated. The failure definition needs to correspond
with the functional requirement to the structures. In certain cases like buckling failure it may be defined by
the maximum load, while in other cases it need to be selected by limiting a suitable control parameter e.g.,
plastic strain.
For ultimate limit states (ULS) and accidental limit states (ALS) the definition of failure needs to reflect
the functional requirement that the structure should not lose its load­carrying resistance during the
dimensioning event. That may e.g., imply that in an ULS check the failure is defined as the load level where
the remaining cycles in the storm that includes the ULS load case, 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 in case of an ALS check for blast pressure, where one may consider the failure criterion to be the
limiting deflection for the passive fire protection.
Care should be made to ascertain that all relevant failure modes are addressed either directly by the analysis
or by additional checks. Examples are local buckling, out of plane buckling, weld failure etc.
3.2 Modelling strategy
It should be checked that the analysis tool and the modelling adopted represent the non­linear behaviour
of all structural elements that may contribute to the failure mechanism with sufficient accuracy. The model
should be suitable to represent all failure modes that are intended to be checked by the analysis. It should
be made clear which failure modes the model will adequately represent and which failure modes are excluded
from the analysis and are assumed to be checked by other methods.
3.3 Modelling accuracy
All FE­analyses provide results that are based on simplified models of the actual structural behaviour. It is
the responsibility of the analyst to control the accuracy of the analysis. This may be achieved by means of
sensitivity studies, calibration and other methods.
3.4 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 the
statistical variation of the various parameters such that the results will be equal to or represent an estimate
to the 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
standard. In general that means 5% fractile in case a low resistance is unfavourable and 95% fractile in case
a high resistance is unfavourable.
In cases where data of the statistical variation of the resistance is uncertain one needs to establish a
selection of the governing parameters by engineering judgement. The parameters should be selected such
that it can be justified 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 the
characteristic resistance.
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SECTION 3 GENERAL REQUIREMENTS
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
the safe 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,
using the specified minimum yield stress according to offshore steel material standards will represent
the requirements to the characteristic resistance. Other parameters with statistical variation that will
influence the resistance e.g., plate thickness should be selected as a safe estimate of the expected value
in order to meet the required statistical requirement for the resulting resistance. In cases of doubt a
sensitivity assessment may be necessary.
b)
In some cases values are given in the standards for analysis of specific problems see e.g., [5.4.3].
Validation against design standard values
In this method a selected standard case is used for calibration (denoted standard calibration case).
The case should represent the same failure mode that is to be investigated. The key parameters e.g.,
element type, mesh size, material curve, degree of triaxial stress state, imperfections, residual stresses
etc. should be selected so the analysis provide the resistance predicted by the standard for the standard
calibration case. The same parameters are then used when the resistance of the actual problem is
determined.
c)
If the analysis is calibrated against ordinary standard values that meet the requirements to characteristic
resistance then the resistance of the analysed structure also will meet the requirement.
Validation against tests
In this method one or more physical tests that are judged to fail in a similar way as the problem to
be analysed are selected for calibration (denoted test calibration case). First the key parameters e.g.,
element type, mesh size, material curve, imperfections, residual stresses etc. are varied so the analysis
simulates the test calibration case satisfactorily. (Giving the same or less resistance.) Then the actual
problem is analysed using the same key parameters. It should be ascertained that the statistical
variation of the problem is duly covered such that the requirements for determination of resistance by
use of FE methods correspond to the requirements for determination of resistance from testing as given
in Annex D of Eurocode 1990 /2/ or in ISO 19902 /6/.
3.5 Requirement to the software
The software used shall be documented and tested for the purpose.
3.6 Requirements to the user
The user should be familiar with FE methods in general and non­linear 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 standard described reliability level with use of non­linear methods
for determination of the resistance, it is necessary that the engineer understand the inherent safety
requirements of the governing standard.
The use of this standard presupposes and does not replace the application of industry knowledge, experience
and know­how. It should solely be used by competent and experienced organizations, and does not release
the organizations involved from exercising sound professional judgment.
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It is therefore necessary to validate the analysis procedure according to one of the following methods:
4.1 General
The term non­linear FE analysis covers a large number of analysis types for different purposes and objects.
The content of this section is written with analyses of steel structures in mind. The objective is to document
structural capacity of the structure in a way that fulfils the requirements for determining characteristic
resistance in accordance with DNV GL standards and other similar standards, such as the Norsok N­series
and the ISO 19900 suite of standards.
4.2 Selection of software for finite element analysis
The software shall be tested and documented, and be suited for analysing the actual type of non­linear
behaviour. This includes:
— non­linear material behaviour (yielding, plasticity)
nd
— non­linear geometry (stress stiffening, 2 order load effects).
Other types of non­linearity that may need to be included are:
— contact problems
— temperature effects (e.g. material degradation, thermal expansion)
— non­linear 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:
(2)
where
M is the mass matrix,
C is the damping matrix
u is the displacement vector,
Fint is the internal forces and
Fext is the external forces
In dynamic analyses, explicit solvers are attractive for large equation systems, as the solution scheme does
not require matrix inversion or iterations, and thus, are much more computational effective for solving
one time step than solvers based on the implicit scheme. However, unlike the implicit solution scheme,
which is unconditionally stable for large time steps, the explicit scheme is stable only if the time step size is
sufficiently small. An estimate of the time step required to ensure stability for beam elements is:
(3)
where Ls is the characteristic element size of the smallest element and c is the speed of sound waves in the
material. Similar expressions exist for shells and solids.
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SECTION 4 REQUIREMENTS TO FINITE ELEMENT­ANALYSIS
4.3.2 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 non­linear analyses they should be used in combination with Newton iterations. As a rule of
thumb the time step 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. The
parameters should in most cases be selected to give an unconditionally stable solution.
For the α­method (HHT method)/16/ the parameters
unconditionally stable if:
α, β and γ can be selected by the user. The method is
(4)
Selecting α less than zero gives some numerical damping. In order to avoid “noise” from high frequency
modes, parameters that give some numerical damping can be useful. Table 4­1 presents some combinations
of parameters that give unconditional stability.
Table 4­1 Combinations of α, β and γ for unconditional stability
α
β
γ
0
0.25
0.5
­0.05
0.2756
0.55
Numerical damping
­0.1
0.3025
0.6
Numerical damping
Comment
Trapezoidal rule, no numerical damping
4.3.3 Solution control for static implicit analysis
In case the dynamic effects are not important, the equation system to solve may be reduced to
(5)
In such cases the implicit equation solvers are in general better suited, as the dynamic terms cannot be
excluded in 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 need for post peak­load results.
— A pure load control algorithm will not be able to pass limit points or bifurcation points when the inertia
effects are not included.
— Using a displacement control algorithm, limit points and bifurcation points can be passed, but the analysis
will stop at turning points.
— For snap­back problems (passing turning point), or limit/bifurcation point problems that cannot be
analysed using displacement control, an “arc length” method is needed.
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This makes the explicit scheme well suited for shorter time transients as seen in for instance impact ­ or
explosion response analyses. For longer time transients the number of time steps will, however, be much
larger than needed for an implicit solution scheme. For moderately non­linear problems, implicit Newton
Raphson methods are well suited, gradually incrementing the time and iterate to convergence for each time
step.
4.3.4 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 time and thus computational time, the mass scaling increases the time step reducing the
computational time, see Equation (3).
The time saving methods only give accurate results if the inertia forces are small. Thus, it must be
demonstrated that the kinetic energy is small compared to the deformation energy (typically less than
1%) when explicit analyses are used to find quasi­static response. Since the analyses still will be dynamic,
bifurcations points will not be identified. At static capacity, the kinetic energy will increase rapidly if the load
is increased further.
Due to the typically large number of time steps in explicit analyses, the numerical representation of decimal
numbers is important for the stability of the solution. The software options to use high precision (“double
precision”) float are generally preferred.
4.4 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 because they interfere with the goal of having a good, regular
element mesh.
The effect these simplifications may have on the final result should be evaluated. Typical simplifications
include:
—
—
—
—
cut­outs 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 the
buckling capacity correctly, see [5.4]. A common way of including such imperfections is to use one or more
of the structure’s eigenmodes and scale these such that the buckling capacity is predicted correctly for the
calibration model.
For problems where the geometry of the model deviates from the real structure, the analysis needs to reflect
that possible geometrical tolerances may have impacts on the result. An example is fabrication tolerances of
surfaces transferring loads by contact pressure.
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Figure 4­1 Limit, bifurcation and turning points
4.5.1 General
In general, structural parts welded together should be meshed using a continuous mesh.
Connections and constraints such as bonded contact or kinematic coupling etc. should not be used for welded
details in areas of interest unless the accuracy on stress and strain results is documented and accounted for
when evaluating the results.
4.5.2 Selection of element type
Selection of element type and formulation is strongly problem dependent. Items to consider are:
—
—
—
—
—
—
—
—
shell elements or solid elements
elements based on constant, linear or higher­order shape functions
full vs reduced, v. 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)
drilling rotation stiffness /artificial strain energy (for shell elements).
warping stiffness (shell elements).
In general higher­order elements are preferred for accurate stress estimates; elements with simple shape
functions (constant or linear) will require more elements to give the same stress accuracy as higher­order
elements. Constant stress elements are not recommended used in areas of interest.
For large­displacement and large­rotation analyses, simple element formulations give a more robust
numerical model and analysis than higher­order elements.
Some types of elements are intended as transition elements in order to make the generation of the element
mesh easier and are known to perform poorly. Typically 3­noded plates/shells and 4­noded tetrahedrons are
often used as transition elements. These types of elements should if possible be avoided in areas of interest.
Proper continuity should be ensured between adjacent elements if elements of different orders are used in
the same model.
Care should be taken when selecting formulations and integration rules. Formulations with (selective)
reduced integration rules are less prone to locking effects than fully integrated simple elements. The reduced
integration elements may, however, produce zero energy modes and will require hourglass control. When
hourglass control is used, the hourglass energy should be monitored and shown to be small compared to the
internal energy of the system, typically less than 5%.
Rotational stiffness normal to the shell element surface is normally not part of the shell element formulation.
Thus, an additional stiffness (drilling rotation constraint) to the local degree of freedom must be added to
certain shell element nodes when using implicit equation solvers to avoid singularity. The drilling rotation
constraint can produce a significant amount of artificial energy when used in large­deformation analyses
and the deformation resistance will increase. Similar to the hourglass energy, this artificial energy should be
monitored and controlled. In explicit analyses, the drilling stiffness is not needed for numerical stability, and
one solution can be to scale down or remove the drilling stiffness if present in the default settings.
4.5.3 Mesh density
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 have
good strain estimates.
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4.5 Mesh
The element aspect ratio should be according to requirements for the selected element formulation in
the areas of interest. Typically, an aspect ratio close to unity is required in and nearby areas with large
deformations.
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 applied
need to be correctly located, and in this situation can drive the mesh design, at least locally.
4.5.4 Mesh refinement study
Often it will be necessary to run mesh sensitivity studies in order to verify that the results from the analyses
are sufficiently accurate.
The analyst should make sure that the element mesh is adequate for representing all relevant failure modes.
In the general case mesh refinement studies may be done by checking that convergence of the results
are obtained e.g. by showing that the results are reasonably stable by rerunning the analysis with half
the element size. See example in [8.2]. Note that geometric sharp corners represent singularities where
convergence will never be obtained.
4.6 Material modelling
4.6.1 General
The selected material model should at least be able to represent the non­linear behaviour of the material
both for increasing and decreasing loads (unloading). In some cases the material model also needs to be able
to account for reversed loading, see [5.2].
The material model selected needs to be calibrated against empirical data (see [3.4]). The basic principle
is that the material model needs to represent the structural behaviour sufficiently for the analysis to be
adequately calibrated against the empirical basis.
4.6.2 Material models for metallic materials
For metallic materials time independent elasto­plastic models are often used. The main components in such
models are:
— A yield surface, defining when plastic strains are generated.
von Mises plasticity is commonly used for steel materials. The model assumes that the yield surface is
unaffected by the level of hydrostatic stress.
— A hardening model defining how the yield surface changes for plastic strains.
Commonly used are isotropic hardening (expanding yield surface) and kinematic hardening
(translating yield surface) or a combination of both.
— A flow rule (flow potential) defining the relation between the plastic strain increment and the stress
increment.
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 material
model with kinematic (or combined kinematic/isotropic) hardening rule should be used in such analyses.
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— For stability evaluations, sufficient number of elements and degrees of freedom to capture relevant
buckling modes, typically minimum 3 to 6 elements dependent upon element type per expected half wave
should be used.
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Figure 4­2 The von Mises yield surface shown in the σ1­σ2 plane with isotropic (left) and
kinematic (right) hardening models
Figure 4­3 Isotropic vs Kinematic hardening
4.6.3 Stress­strain measures
Stress and strain can be measured in several ways:
— From material testing the results are often given as “Engineering” stress­strain curves (calculated based
on the initial cross section of the test specimen).
— FE software input is often given as “True” stress­strain (calculated based on updated geometry)
— Other definitions of strains are also used in FE formulations, e.g., the Green­Lagrange strain, and the
Euler­Almansi strain.
For small deformations/strains, all strain measures give similar results. For larger deformations/strains the
strain measure is important, e.g. the Green­Lagrange measure is limited to “small strains” only. Figure 4­4
shows a comparison of some strain measures. Limitations in the formulations on the use of the selected
element type should always be noted and evaluated for the intended analysis.
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The relationship between engineering (nominal) stress and true (Cauchy) stress (up to the point of necking)
is:
(6)
The relationship between engineering (nominal) strain and true (logarithmic) strain is:
(7)
The stress­strain curve should always be given using the same measure as expected by the software/
element formulation.
4.6.4 Evaluation of strain results
As element strain in FE analyses is an averaged value dependent on the element type and element size, the
reported strain will always depend on the computer model. It is often necessary to re­mesh and adjust the
analysis 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.
Most FE software presents nodal averaged strains graphically. At geometry intersections the nodal average
value may be significantly lower than the element (nodal or integration point) strain if the intersecting parts
are differently loaded. When evaluating strain results against deformation limits, the integration point strains
or extrapolated strains from integration points should be used.
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Figure 4­4 Comparison of some strain measures
When defining the material curve for the analysis, the following points should be considered:
— Characteristic material data should normally be used, see [3.4].
— 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.
— 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 the actual 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 stress
is 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 EN­10025 /26/ and EN­10225 /27/
are proposed in [4.6.6] and [4.6.7] for analyses to represent low fractile and mean characteristic values
respectively. These properties are assumed to be used with the acceptance for criteria for tensile failure
given in this recommended practice. Idealized material curves for steel materials delivered according to other
standards e.g. DNV GL standards can be established by comparison with these curves. The curves are given
as true stress­strain values.
Alternative bi­linear curves may be used for buckling problems, e.g. as shown in Figure 5­7.
The curves should also be adjusted for temperature effects as appropriate. (See e.g. /33/).
4.6.6 Recommendations for steel material qualities (low fractile)
The material should be modelled as a combination of a stepwise linear and a power law with a yield plateau
as shown in Figure 4­5, given in true stress and strain parameters. Graphs of the material curves shown as
engineering stress and strain are given in the commentary, see [7.3].
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4.6.5 Stress­strain curves for ultimate capacity analyses
For Part 4 as shown in Figure 4­5, the relation between stress and strain is given as shown in Equation (8).
(8)
Values for the material parameters for selected steel grades are given in Table 4­2 to Table 4­6.
Table 4­2 Proposed properties for S235 steels (true stress strain)
S235
Thickness [mm]
t≤ 16
16< t ≤ 40
40< t ≤ 63
63< t ≤ 100
E [MPa]
210000
210000
210000
210000
σprop [MPa]
211.7
202.7
193.7
193.7
σyield [MPa]
236.2
226.1
216.1
216.1
σyield2 [MPa]
243.4
233.2
223.8
223.8
εp_y1
0.004
0.004
0.004
0.004
εp_y2
0.02
0.02
0.02
0.02
K[MPa]
520
520
520
520
n
0.166
0.166
0.166
0.166
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Figure 4­5 Definition of stress­strain curve
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Table 4­3 Proposed properties for S275 steels (true stress strain)
S275
Thickness [mm]
t≤ 16
16< t ≤ 40
40< t ≤ 63
E [MPa]
210000
210000
210000
σprop [MPa]
248.0
238.0
228.0
σyield [MPa]
276.5
266.4
256.3
σyield2 [MPa]
283.9
273.6
263.4
εp_y1
0.004
0.004
0.004
εp_y2
0.017
0.017
0.017
K[MPa]
620
620
620
n
0.166
0.166
0.166
Table 4­4 Proposed properties for S355 steels (true stress strain)
S355
Thickness [mm]
t≤ 16
16< t ≤ 40
40< t ≤ 63
63< t ≤ 100
E [MPa]
210000
210000
210000
210000
σprop [MPa]
320.0
311.0
301.9
284
σyield [MPa]
357.0
346.9
336.9
316.7
σyield2 [MPa]
366.1
355.9
345.7
323.8
εp_y1
0.004
0.004
0.004
0.004
εp_y2
0.015
0.015
0.015
0.015
K[MPa]
740
740
725
725
n
0.166
0.166
0.166
0.166
Table 4­5 Proposed properties for S420 steels (true stress strain)
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
395.9
εp_y1
0.004
0.004
0.004
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Thickness [mm]
t≤ 16
16< t ≤ 40
40< t ≤ 63
εp_y2
0.012
0.012
0.012
K[MPa]
738
703
686
n
0.14
0.14
0.14
Table 4­6 Proposed properties for S460 steels (true stress strain)
S460
Thickness [mm]
t≤ 16
16< t ≤ 40
40< 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
εp_y1
0.004
0.004
0.004
εp_y2
0.01
0.01
0.01
K[MPa]
772
745
703
n
0.12
0.12
0.12
4.6.7 Recommendations for parameters for steel material qualities to obtain
mean capacity
The recommended material curve to be used for analyses when the expected resistance of a structure should
be calculated is given in the commentary; see Table 7­1 to Table 7­5. These material parameters are only
intended to be used when a low capacity can be unfavourable. The typical application is to determine the
forces imposed to a structure from a ship colliding with the structure.
4.6.8 Strain rate effects
­1
For strain rates above 0.1 s increased strength and reduced ductility will be experienced. In most cases it
will be safe to exclude the effect.
Strain rate hardening is sensitive to the strain magnitude, and this must be accounted for when selecting the
models and model parameters to simulate strain rate effects. Generally the relative increase in flow stress is
less for large strains than for small strains, i.e. at the yield point. See [7.10].
If strain­rate hardening effects are included in a simulation, it should be documented that the selected strain­
rate hardening model and corresponding parameters result in the expected response.
4.7 Boundary conditions
The selected model boundary condition needs to represent the real condition in a way that will lead to results
that are accurate or to the safe side.
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S420
4.8 Load application
Unlike linear elastic analyses, where results from basic load cases can be scaled and added together, the
sequence of load application is important in non­linear analyses. Changing the sequence of load application
may change the end response.
The loads should be applied in the same sequence as they are expected to occur in the condition/event to
be simulated. For an offshore structure subjected to both permanent loads (such as gravity and buoyancy)
and environmental loads (such as wind, waves and current), for example, the permanent loads should be
incrementally applied first 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 carrying
capacity for the final load case, in such cases a sensitivity study on the effect of reduced initial load should be
performed.
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 predicted
response. Increment sensitivity studies should be performed to ensure that all failure modes are captured.
4.9 Contact modelling
4.9.1 Contact pair definitions
Several options for contact pair definition are available in most programs for non­linear FE analysis. Typical
options are:
—
—
—
—
—
surface to surface
single surface (self­contact)
edge (line) to surface
edge to edge
node to surface.
Both meshed regions and analytical surfaces may be used in the contact pair definitions.
General contact or automatic contact is available in some programs. These options automatically assign
contact pairs and ease the modelling work.
The analyst should verify that all possible contact pairs that may get into contact are included in the contact
definitions.
4.9.2 Symmetric and asymmetric contact
The contact pair can be symmetric or asymmetric. For asymmetric contact, one of the contact surfaces is
defined as the slave and the other surface is defined as the master. The slave surface nodes or integration
points are not allowed to penetrate the master surface elements. For symmetric contact each surface is both
slave and master.
For asymmetric contacts, it is normally recommended that the surface with the fine mesh is defined as the
slave. It is also beneficial if the slave is the softer part as this may improve the convergence.
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Often it is difficult to decide what the most “correct” or a conservative boundary condition is. In such cases
sensitivity studies should be performed.
Several different approaches are used to enforce the contact constraint:
— penalty­based methods
— augmented Lagrange methods
— pure Lagrange methods (direct methods).
Penalty­based methods and Augmented Lagrange methods introduce linear or non­linear contact stiffness
and will thus always allow for some contact penetration. Augmented Lagrange methods add a term not
dependent on the penetration, and can thus to some extent control the penetration.
Pure Lagrange methods introduce extra degrees of freedoms to solve (contact pressure) instead of contact
stiffness and can thus give solutions with zero penetrations.
Pure Lagrange methods give the highest accuracy and are often used for small­sliding problems. However,
due to computational expense as well as possible convergence issues, penalty­based methods are often used
for finite­sliding problems.
4.9.4 Controlling the accuracy of contact analyses
The typical problems that need to be monitored and dealt with in contact analyses are:
—
—
—
—
convergence problems
excessive penetrations
sticking (when sliding is expected)
start­up problems for analyses with parts kept in place only by contact.
The default contact stiffness assigned to the contact surfaces may be adjusted in order to improve
convergence (reduce stiffness) or to reduce the penetration (increase stiffness). Non­default values should
however be used with caution as the solution accuracy will be influenced.
Sticking problems may arise from inaccurate models, e.g. faceted surface representing a cylinder, or
small radii that are omitted. Refining the mesh or adding geometrical smoothing and initial over­closure
adjustments may improve the solution.
Start­up problems due to parts initially with no constraint may be solved by adding stabilization algorithms,
or by manually adding springs or boundary conditions. It is recommended to remove the stabilization
measures as soon as contact is established, before the full load is applied.
4.10 Application of safety factors
Applying load and resistance safety factors in a non­linear analysis can be challenging as application of safety
factors 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
for all failure modes to be analysed for, and then apply all the safety on the load side, defining a target load
level that accounts for both load and resistance safety. Using this approach, the same model may be used for
both ULS and ALS type of analysis without recalibration of the model:
(9)
where Rk is the characteristic resistance found from the analysis, and Sk is the characteristic load effect.
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4.9.3 Contact constraint enforcement methods
The following items should be considered in a quality control of non­linear 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 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 finite element 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
include description 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 also
be documented
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4.11 Execution of non­linear finite element analyses, quality control
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 the results
from the analysis is highly influenced on how the analysis is carried out.
In ordinary engineering situations tensile failure is seldom decisive as it is associated with large permanent
deformations and other failure modes will in such cases govern. Tensile failure is mostly relevant for checking
of structures against accidental loads like explosions or collisions.
The recommendations given in this document are not valid for failure that is related to unstable fracture due
to either insufficient material toughness, defects outside fabrication specifications or cracks. In such cases
fracture mechanics methods need to be used.
In general accurate prediction of tensile failure needs to be made by analyses that are calibrated against
tests or a known solution where the conditions for tensile failure are similar as in the structural detail being
investigated. This method is described in [5.1.2] below.
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 eventual
failure takes place in the base material. Welds should therefore be checked according to ordinary standard
methods based on the forces carried by the welds. See [5.1.5].
Tensile failure in structures modelled by beam elements is best checked on the basis of the total deflection
e.g. as given in DNVGL­RP­C204 /12/.
5.1.2 Tensile failure resistance calibrated against a known solution
The most accurate method to check a structure against tensile failure is by calibrating the non­linear FE
analysis against 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 standard) as the reference object.
The reference object should have the similar conditions for tensile failure as the actual problem such as
the type of 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. maximum
principal strain) at failure 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
level when 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 selected situations if a calibrated
solution is not attainable. The simplified check is intended for shell models of plated structures with element
sizes from t × t to 5t × 5t. The requirements to the element size are only relevant for the areas subjected to
plastic straining in tension. Other parts of the structure may be modelled by larger elements if found suitable,
but results from analyses using element larger than 5t should not be relied upon if the maximum principal
strain is larger than 2%.
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SECTION 5 REPRESENTATION OF DIFFERENT FAILURE MODES
The safety factors that should be used for tensile fracture according to this procedure should include an
additional safety factor
= 1.2 when determining the maximum failure load compared with standard
material factor of the standard in question. The resulting material factor should be the product of the
ordinary material factor
and the
making a resulting
.
These tensile failure criteria are valid for monotonic loading. In case of cyclic loads, see [5.2].
This method is valid for structures made with typical offshore steel that will meet requirements to ductility
and toughness. The structural details need to meet fabrication requirements for offshore steel structures.
The analyses should be carried out using the defined material curves given in [4.6.5]. Other materials will
need to be calibrated according to the general procedure given in [5.1.2].
The analyses should be made using von Mises yield function.
The structure should be checked for a general requirement for all areas subjected to plastic strains called
gross yielding given in [5.1.3.2]. For concentrated yield zones larger strain can be allowed as given in
[5.1.3.3].
5.1.3.2 Gross yielding check
The strain limit for gross yielding reflects that real structures will include elements of inhomogeneity that will
not be accurately modelled in the analyses. This will mean that the strain measured over a long length of a
real structure will in average not reach the values that can be found in standardized tensile tests.
With gross yielding is meant that plastic deformations with strain above 2% are taking place over a zone
lyz > 20t in the direction of the maximum plastic strain.
The maximum gross yielding strain in any integration point, in any element within the yield zone, should be
limited to the gross yielding critical strain εcrg. The gross yielding critical strain should be found by making a
calibration analysis with the actual element type and with an element size relative to the thickness t between
t × t and 5t × 5t by use of calibration case CC01 as shown in Figure 5­1. The gross yielding strain limit εcrg
should be determined from the deformation limits given in Figure 5­1.
Figure 5­1 Calibration case CC01, steel plate under uniaxial load plane strain conditions
The critical strain should be determined by the maximum strain found by analysing the calibration case CC01
using the element type and size for the considered area.
The analyst may select a preferred failure parameter for the calibration, but it is recommended to establish a
critical maximum principal strain value that should not be exceeded in the analysis.
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The criteria are determined under the assumption that the resulting structural capacity should represent a
5% low fractile. In case when a high capacity is unfavourable, the recommendations in [5.1.6] should be
followed.
The effect of holes that are not modelled and located within the yield zone, will need to be accounted for
by reducing the critical strain if the diameter of the hole is larger than 5% of the plate width measured
perpendicular to the direction of the maximum principal strain.
Table 5­1 Deformation limits for gross yielding check
δx (mm)
CC01
S235
S275
S355
S420
S460
25
24
21
18
18
5.1.3.3 Local yielding check
General
When yielding takes place in a limited area it will be due either to strain gradients or out of­plane bending or
a combination of these two effects.
No strain concentration caused by attachment holes etc., that is not modelled, should be within the yield
zone for this check.
Care should be exercised when representing welds or other elements that will impact the plastic deformations
of the structure. Normally tensile failure will take place outside the welds and check of failure in the welds
itself should be made by checking the forces imposed on the weld see [5.1.5]. However the modelling of
the welds may influence the strain in the base material. It is recommended to increase the strength and/or
thickness of the elements representing the welds so plastic deformations take place outside the elements
representing the weld. However, the weld strength should not be increased more than avoiding plasticity to
take place in the weld in order not to impose artificial strength into the detail.
In case the structure is made with use of cold­forming (e.g. rolling of plates to tubular sections) then one
either need to work with a reduced critical local strain value or include the forming­process in the analysis.
When establishing a reduced local criterion, the maximum tensile plastic strain from the forming process at
the actual position should be used to modify the local critical strain. When evaluating the reduction effects
from the forming process one can account for the direction and through thickness variation of the plastic
tensile strain imposed from the forming.
Strain gradients
For problems dominated by membrane strains, but where the extent of the plastic zone as defined in
[5.1.3.2] is less than 20 × t then the maximum principal strain in any integration point in any element should
be less than obtain from the analysis of CC01 modified as follows:
The maximum principal strain should be less than
(10)
where l is element length in the direction of the maximum principal strain.
The analyst may select a preferred failure parameter for the calibration, but it is recommended to establish a
critical maximum principal strain value that should not be exceeded in the analysis.
Out­of­plane bending
For problems dominated by out­of­plane bending, the local strain at the surface should be limited to what is
obtained as the maximum surface strain found from analysing calibration case CC02 as defined in Figure 5­2.
The elements in the calibration analysis should have the same relative size relative to the thickness as for
the elements in the area of interest. The mid­point strain should be limited to εcrl as defined from CC01 and /
(10).
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The strain values obtained by analysing the calibration case CC01 will be valid for structures made with not
more than one weld or stiffener transverse to the maximum principal strain within the yield zone. In case the
structure within the yield zone is less homogeneous the critical strains need to be reduced.
This procedure is valid, when the plane plate part of structural details from support or from load to point
of counter­flexure, should be longer than 0.5 t and when out of plane dynamic effects can be neglected.
Problems not meeting this requirement need to be checked for shear failure (locking failure or punching
shear failure) by calibrating to a known case.
Figure 5­2 Calibration case CC02, steel plate under out­of­plane bending and membrane tension.
Plane strain conditions
The deflections due to out­of­plane bending and membrane tension should be simultaneously and
proportionally introduced in the model.
Table 5­2 Deformation limits for CC02
δx (mm)
CC02
δz (mm)
S235
S275
S355
S420
S460
S235
S275
S355
S420
S460
55
53
50
45
40
75
73
70
65
60
5.1.4 Representation of tensile failure applying element erosion
In analyses of accidental loads such as dropped object, explosion, and ship impact, it can be useful to
represent the tensile fracture using an element erosion approach. For shell elements, the trough­thickness
layers may be deactivated individually.
It is proposed to initially use the local criteria derived from section [5.1.3.3] as the strain limits where
element layers are deactivated. The calibration cases CC01 and CC02 should be rerun to confirm that
element deletion occurs at or before the defined deformation limits in case a low­fractile characteristic
capacity is sought.
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The analyst may select a preferred failure parameter for the calibration, but it is recommended to establish a
critical maximum principal strain value that should not be exceeded in the analysis.
5.1.5 Failure of welds
The welds may or may not be represented with separate elements. For cases where the welds are not
modelled the check of the strength of welds should be based on stress resultants determined by integration
of stresses from the closest elements and checked against ordinary standard requirements e.g. EN
1993­1­8 /4/ or the relevant standard for the problem at hand.
If welds are modelled the linearized stress components (axial, bending, shear) should be determined from
integration of the stresses in the elements representing the welds and checked against ordinary standard
requirements e.g. EN 1993­1­8 /4/ or the relevant standard for the problem at hand.
Normally it is required that in welded connections the welds are stronger than the base material (overmatch).
See also [2.6].
In the representation of the welds in a shell model it is recommended that the welds are given a certain
overstrength to represent weld overmatch (e.g. by increasing stress level by 25% in the stress­strain
curves for the material representing the weld material compared with the base material and or by increased
thickness). The failure will normally be located close to the welds so a node should be located at the weld
toe. It should be checked that the weld is not experiencing significant plastic strain before critical strain is
reached in the base material. At the same time it is necessary that the weld strength is not given too high
strength leading to an artificial too high capacity. Generally, it will be necessary to check the sensitivity of the
assumptions.
5.1.6 Simplified tensile failure criteria in case low capacity is unfavourable
5.1.6.1 General
There are cases when it is unsafe to assume a too low capacity. When performing a collision analysis
assuming energy dissipation on both objects, for example, it will be unfavourable to assume low fractile
failure criteria for the striking object when the task is to evaluate the structural integrity of the struck object.
For such cases, the procedure given below is recommended for simulation of behaviour and tensile failure of
the striking object.
5.1.6.2 Tensile failure for estimation of mean capacity
In order to analyse a structure without underestimating how tensile failure will impact the structural capacity
the following recommendations are proposed.
Tensile failure may be assumed to take place if the critical strain values exceed the strain found from the
calibration cases CC01 as shown in Figure 5­1, using the deformation limits given in Table 5­3 Furthermore
mean material curves should be used as given in the commentary, see Table 7­1 to Table 7­5.
If the elements used will be unstable due to thinning, the strain levels at start of instability can be taken as
the critical strain.
The principal strain implying failure may be assumed when the maximum principal strain exceeds:
(11)
The global strain εcrg can be calculated from CC01 as described in [5.1.3.3]. When the mean capacity is
sought, the gross yielding limits as given in [5.1.3.2] should not be considered.
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The gross yielding check still applies, and the applied erosion criteria may have to be adjusted if large zones
of plastic deformation are present.
δx (mm)
CC01
δz (mm)
S235
S275
S355
S420
S460
S235
S275
S355
S420
S460
80
78
75
70
70
0
0
0
0
0
5.2 Failure due to repeated yielding (low cycle fatigue)
5.2.1 General
Non­linear FE­analyses 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
to a possible cyclic failure. This is called low cycle fatigue and need to be treated differently from how high
cycle fatigue 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 DNVGL­RP­C203 /11/, are
applied. 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.
For such cases the damage from high cycle fatigue should be added to the damage from low­cycle fatigue.
See [5.2.2]
Welded joints ([5.2.5]) and base material ([5.2.6]) are covered. Note that the procedure for assessing the
strain amplitude is somewhat different in these two cases. Reference is made to [5.2.5.2] and [5.2.5.3] for
welded joints and [5.2.6.2] for base material.
5.2.2 Fatigue damage accumulation
The fatigue life may be calculated under the assumption of linear cumulative damage, i.e.
(12)
where D is the accumulated fatigue damage. ni is the number of cycles in block i and Ni the number of cycles
to failure at constant strain range Δε.
In cases where the fatigue damage from high cycle fatigue (HCF) is considerable the total damage is
obtained by 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 to partial
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
to repeated plastic straining. This check is necessary as non­linear analysis allows parts of the structure
to undergo significant 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
and seismic action. When cyclic loads are present it is necessary to define a load history that will imply a
probability of failure that is similar or less than intended for static loads. See also [3.1].
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Table 5­3 Deformation limits for cases when a large capacity is unfavourable (mean values)
All the remaining cycles in the storm of the maximum wave action may be assumed to come from the same
direction as the dimensioning wave.
5.2.4 Cyclic stress strain curves
It is required that the cyclic stress­strain curve of the material is applied. The use of monotonic stress­
strain curve must be avoided since it may provide non­conservative fatigue life estimates, especially for high
strength steels. It is required that the welds are produced with overmatching material. Consequently the
cyclic stress­strain properties of the base material should be used when assessing welded joints.
Unless the actual cyclic behaviour of the material is known from tests according to a recognized testing
standard, the true cyclic stress strain curves presented in Figure 5­3 can be applied. Kinematic hardening,
as illustrated in Figure 4­3 should be assumed. The curves are described according to the Ramberg­Osgood
relation:
(13)
The value of the coefficient K is given in Figure 5­4.
Table 5­4 Ramberg­Osgood parameters for base material
Grade
K (MPa)
S235
410
S355
600
S420
690
S460
750
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The load­history for the remaining waves in a 10 000 year dimensioning storm investigated for southern
North Sea conditions have been found to have a maximum value equal to 0.93 of the dimensioning wave,
a duration of 6 h and a Weibull shape parameter of 2.0. This applies for check of failure modes where
the entire storm will be 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 /15/.
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 the
following equation
(14)
Where:
Δεhs
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)
The parameters in Equation (14) are given in Table 5­5 for air and seawater with cathodic protection.
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Figure 5­3 The true cyclic stress­strain curve for common offshore steel grades
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Table 5­5 Data for low cycle fatigue analysis of welded joints
σf' (MPa)
εf'
Air
175
0.095
Seawater with cathodic protection
160
0.060
Environment
Figure 5­4 ε­N 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 DNVGL­
RP­C203 Sec.4 /11/. The procedure in /11/ is originally developed for assessing the hot spot stress of a linear
elastic material in relation to high cycle fatigue assessments. However, by substituting maximum principal
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
localised in a limited area near the hot spot. In order to reflect steep strain gradient in a good manner it is
recommended to use finite elements with mid side nodes, such as 8­noded shell elements or 20­noded brick
elements.
For modelling with shell elements without any weld included in the model a linear extrapolation of the strains
to the intersection line from the read out points at 0.5t and 1.5t from the intersection line can be performed
to derive hot spot strain. For modelling with three­dimensional elements with the weld included in the model
a linear extrapolation of the strains to the weld toe from the read out points at 0.5t and 1.5t from the weld
toe can be 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 DNVGL­RP­C203 /11/.
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(15)
The thickness model is identical to that of DNVGL­RP­C203 /11/ and values for k and tref are determined
according to [2.4] in this document.
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 are
situations 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 stress­strain level. A low cycle fatigue check of the base
material may therefore be necessary.
As opposed to assessments of welded joints where the fatigue damage is determined by means of the cyclic
hot spot strain, low cycle fatigue analysis of base material is based on the maximum principal strain range.
The strain 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 the
following equation
(16)
where
Δεl
is the fully reversible local maximum principal 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)
Values of the parameters in Equation (16) are given in Table 5­6 for air and seawater with cathodic
protection.
Table 5­6 Data for low cycle fatigue analysis of base material
σf' (MPa)
εf'
Air
175
0.091
Seawater with cathodic protection
160
0.057
Environment
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5.2.5.4 Thickness effect
The low cycle fatigue strength is to some extent dependent on plate thickness /29/. The thickness effect is
accounted for by multiplying the strain amplitude obtained from the FE analysis by the following factor
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
state will be underestimated if the finite element mesh is too coarse. A mesh sensitivity study should
therefore be carried out to ensure that the applied strain is not underestimated. Reference is made to [4.5.4]
regarding mesh refinement. Modelling of sharp corners must be avoided as the assessed stain amplitude will
approach infinity with decreasing mesh size.
5.2.7 Shake down check
Structures loaded beyond the elastic range may alter their response behaviour for later cycles. However, if a
structure is behaving essentially linear for all cyclic loads after the first few cycles following the dimensioning
load, it will achieve a stable state called shake down, and further checks of failure due to repeated yielding or
buckling is not necessary.
In the general case it is necessary to define a characteristic cyclic load and to use this load with appropriate
partial safety factors. It should be checked that yielding only takes place in the first few loading cycles and
that later load repetitions only cause responses in the linear range. This may then serve as an alternative to
a low cycle 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 load
cycles. The checks may be carried out using a linear stress­strain relationship up to the yield stress specified
for the material.
5.3 Accumulated strain (ratcheting)
For cases where the structure is subjected to cyclic loads in a way that incremental plasticity may accumulate
and in the end lead to tensile failure or excessive deformations the maximum accumulated strain needs to be
checked against the strain values in [5.1].
The criteria for excessive deformations may alternatively be determined on a case by case basis due to
requirements to the structural use or performance. Cases where accumulated strain may need to be checked
can be structures that are repeatedly loaded by impacts in the same direction or functional loads that change
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Figure 5­5 ε­N curve for low cycle fatigue of base material in seawater with cathodic protection
and in air
5.4 Buckling
5.4.1 General
The buckling resistance of a structure or structural part is a function of the structural geometry, the material
properties, the imperfections and the residual stresses present. When the buckling resistance is determined
by use of non­linear methods it is important that all these factors are accounted for in a way so that the
resulting resistance 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)
b)
c)
linearized approach: apply the FE method for assessing the buckling eigenvalues (linear bifurcation
analysis) and determine the ultimate capacity using empirical formulas
full non­linear analysis using standard defined equivalent tolerances and/or residual stresses and
non­linear analysis that is calibrated against standard formulations or tests.
Either of these methods can be used to determine the resistance of a structure or part of a structure and
recommendations for their use are given in the following sections.
The proposed methods are valid for ordinary buckling problems that are realistically described by the FE
analysis. Care should be exercised when analysing complex buckling cases or cases that involve phenomena
like snap through, non­conservative 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 buckling
values (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 any
simplifications are leading to results to the safe side. If certain buckling failure modes are not seen as
appropriate to be represented by the model their influence on the resulting resistance can be established
according to [5.4.2.2] below.
ii) Perform a linear analysis for the selected representative load case SRep showing maximum compressive
and von­Mises 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 the
buckling representative stress. The point for reading the representative stress is the point in the model that
will first reach yield stress when the structure is loaded to its buckling resistance.
v) Determine the von­Mises stress at the point for the representative stress σRep from step ii).
vi) Determine the critical buckling stress as the eigenvalue (kg) for the governing buckling mode times the
representative stress:
(17)
Determine the reduced slenderness as:
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position or direction. Examples of the first are protection structures that are hit by swinging loads and the
latter may be wheel loads on stiffened plate decks.
vii) Select empirically based buckling curve to be used based on the sensitivity of the problem with respect to
imperfections, residual stresses and post buckling behaviour. Relevant buckling curves can be selected from
standards, but if not available the following may be used:
Table 5­7 Buckling curves
Type of buckling
κ
Column and stiffened plate and
plate without redistribution
possibilities
Plate with redistribution
possibilities
Shell buckling
1)
Curves to be selected from specific shell buckling standards such as
1)
DNVGL­RP­C202 /10/ or Eurocode EN­1993­1­6 /3/
Please note that DNVGL­RP­C202 defines the reduced slenderness differently
(19)
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
viii) Determine the buckling resistance Rd as:
(20)
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(18)
Empirical buckling curves are needed to account for the buckling resistance reduction effects from
imperfections, residual stresses and material non­linearity. The effect is illustrated in Figure 5­6. For all
buckling forms the usable buckling resistance is less than the critical stress for reduced slenderness less
than 1.2. Above this value, plates with possibility of redistributing stresses to longitudinal edges may reach
buckling capacities above the critical value, column buckling problems will be less than the critical value, but
approach the critical value for large slenderness. Shell buckling is more sensitive to imperfections and the
difference between the buckling capacities that may be exploited in real shell structures are considerably less
than the critical value also for large slenderness.
Members will buckle as columns for cross section classes 1,2 and 3 with exception of tubular sections
exposed to external hydrostatic pressure. For definition of cross­sectional classes see DNVGL­OS­C101 App.A
/9/.
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 may
for instance be torsional buckling of stiffener or local stability of stiffener flange and web. For such cases the
eigenvalue analysis should be carried out without the local buckling modes represented and the interaction
of local and global buckling may be accounted for in a conservative manner by linear interaction as shown in
Equation (21).
(21)
σkig is the linearized buckling stress when local buckling modes are disregarded and σkil is the linearized local
buckling stress.
5.4.3 Buckling resistance from non­linear analysis using standard defined
equivalent tolerances
The buckling resistance of a structure or part of a structure can be determined by performing non­linear
analyses where the effects of imperfections, residual stresses and material non­linearity is accounted for
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Figure 5­6 Examples of buckling curves showing sensitivity for imperfections etc. for different
buckling forms
The material model to be used with the equivalent imperfections is shown in Figure 5­7 or with the models
proposed in [4.6.5].
Figure 5­7 Material model for analysis with prescribed equivalent imperfections
Table 5­8 Equivalent imperfections
Component
Member
Shape
bow
Magnitude
L/300 for strict tolerances
and low residual stresses
L/250 for strict tolerances
and moderate residual
stresses
L/200 for moderate
tolerances and moderate
residual stresses
L/150 for large tolerances
and severe residual stresses
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by use of a defined material stress­strain relationship and the use of empirically determined equivalent
imperfections. The defined equivalent imperfections will include effects from real life imperfections, but will
in general be different in shape and size. This method is only valid for buckling problem similar to the cases
where the equivalent imperfections are given in Table 5­8. For other cases see [5.4.4].
Shape
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Component
Magnitude
Longitudinal
bow
stiffener girder
webs
L/400
Plane plate
between
stiffeners
buckling
eigenmode
s/200
Longitudinal
stiffener
or flange
outstand
bow twist
0.02 rad
It is required that an eigenvalue analysis is carried out to determine the relevant buckling modes. Usually
the pattern from the buckling can be used as the selected pattern for the imperfections, but in certain cases
e.g. when the shape of the buckling load differ from the deflected shape from the actual loads it may be
necessary to investigate also other imperfection patterns.
It may be useful to divide the imperfections into local and global imperfections as shown in Figure 5­8. The
values in Table 5­8 apply to the total imperfection from local and global imperfection patterns. Sensitivity
analyses may be required for cases that are particularly imperfection sensitive.
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For member systems the equivalent imperfections can be taken from EN­1993­1­1. Care should be taken in
order to assure that possible sway modes are adequately covered.
5.4.4 Buckling resistance from non­linear analysis that are calibrated
against standard formulations or tests
Buckling resistance can be found by non­linear methods where the effect of imperfections, residual stresses
and material non­linearity is accounted for by use of equivalent imperfections and/or residual stresses by
calibrating the magnitude of the imperfections (and, or the residual stresses) to the resistance of a known
case that 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 to
obtain realistic capacities:
Prepare a model that is intended to be used for the analysis.
Perform an eigenvalue analysis to determine relevant buckling modes.
Select the object for calibration and prepare a model using the same element type and mesh density as
intended for the model to be analysed.
Perform eigenvalue analysis of the calibration object and determine the appropriate buckling mode for the
calibration object
Determine the magnitude of the equivalent imperfection that will give the correct resistance for the
calibration object
Define an equivalent imperfection for the most relevant failure mode for the problem under investigation
based 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 required
to 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 differs from the shape of
the buckling eigenmodes. In cases of doubt several patterns may be needed.
Example of the use of this procedure is included in the [8.3].
5.4.5 Strain limits to avoid accurate check of local stability for plates and
tubular sections yielding in compression.
For cases where compressed parts of the cross section (as a flange) are experiencing plastic strain in
compression, but one wants to avoid an accurate stability analysis of the local buckling effects the stability
can be assumed to be satisfactory if the plastic strain are limited to the values given below. The requirements
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Figure 5­8 Example of local (left) and global (right) imperfections for stiffened panel
Plates supported on both longitudinal edges:
(22)
Where b is distance between longitudinal supports and t is plate thickness
Plates supported on one edge (flange outstand)
(23)
Where c is the plate outstand and t is plate thickness
Tubular sections without hydrostatic pressure:
(24)
The strain shall be calculated as plastic strain and may be taken as the average value through a cross section
of the compressed plated for element length no less than 2 times the plate thickness. Material properties
should be according to [4.6].
For structural parts meeting requirements to sectional class 3 or 4 no plastic strain due to compressive
stresses can be allowed without an accurate buckling analysis.
For definition of sectional classes, see DNVGL­OS­C101 /9/.
5.4.6 Buckling strength in case low capacity is unfavourable
There are cases when it is unsafe to assume a too low capacity. When analysing a collision assuming energy
dissipation on both objects, for example, it will be unfavourable to assume low fractile failure criteria for
the striking object when the task is to evaluate the structural integrity of the struck object. For such cases
using the procedure to establish buckling strength in [5.4.1], [5.4.5] should not be used. Instead it is
recommended to neglect imperfections when it is unsafe to calculate too low capacity and to use mean
material properties as given in [7.3].
5.5 Repeated buckling
For cases where buckling of parts of the structure may occur before the total capacity of the entire structure
is reached, it is necessary to investigate if the buckling may cause reduced capacity against cyclic loads.
When significant cyclic loads are present one should limit the capacity to the load level that corresponds to
the first incident 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 due
to 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
than 120.
— Member buckling if all parts of the cross section meet requirements to cross­sectional class 1 and the
reduced member slenderness as a column is above 0.5.
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are valid for plates that are loaded in the longitudinal direction and supported on one or both of their
longitudinal edges, and for tubular sections.
It should be noted that structural parts that are yielding in tension may buckle when unloaded. If cyclic loads
lead 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 suffers buckling in
the cyclic capacity checks.
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Failure due to low cycle fatigue according to [5.2] needs to be checked also for these cases.
6.1 Bibliography
The latest valid edition of each of the DNV GL reference documents applies.
For other standards and recommended practices, the edition valid at the time of publishing this document
applies, unless dated references are given.
/1/
ISO2394, General principles on reliability for structures
/2/
EN 1990, Eurocode ­ Basis of structural design
/3/
EN 1993­1­6, Eurocode 3 ­ Design of steel structures ­ Part 1­6: Strength and Stability of Shell
Structures
/4/
EN 1993­1­8, Eurocode 3 ­ Design of steel structures ­ Part 1­8: Design of Joints, 2005/AC:2009
/5/
AISC 360­05, Specification for Structural Steel Buildings
/6/
ISO 19902 Petroleum and natural gas industries – Fixed steel offshore structures
/7/
Norsok Standard N­004, Design of steel structures
/8/
DNVGL­OS­B101 Metallic materials
/9/
DNVGL­OS­C101 Design of offshore steel structures, general ­ LRFD method
/10/
DNVGL­RP­C202 Buckling strength of shells
/11/
DNVGL­RP­C203 Fatigue design of offshore steel structures
/12/
DNVGL­RP­C204 Design against accidental loads
/13/
ECCS publication No. 125, Buckling of Steel Shells. European Design Recommendations, 5
Edition 2008, J.M. Rotter and H. Smith Editors
/14/
Skallerud, Amdahl: Nonlinear analyses of offshore structures, Research studies press ltd., 2002
(ISBN 0­86380­258—3)
/15/
Hagen, Ø, Solland, G. Mathisen, J. Extreme storm wave histories for cyclic check of offshore
structures OMAE 2010­20941
/16/
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 283­292
/17/
Skallerud, Eide, Amdahl, Johansen: On the capacity of tubular T­joints subjected to severe
cyclic loading. Proceedings of the International Conference on Offshore Mechanics and Arctic
Engineering ­ OMAE, v 1, n Part B, p 133­142, 1995
/18/
Weignad, Berman: Behaviour of butt­welds and treatments using low­carbon steel under cyclic
inelastic strains, Journal of Constructional Steel Research, v 75, p 45­54, August 2012
/19/
Boge, Helland, Berge: Proceedings of the International Conference on Offshore Mechanics
and Arctic Engineering ­ OMAE, v 4, p 107­115, 2007, Proceedings of the 26th International
Conference on Offshore Mechanics and Arctic Engineering 2007, OMAE2007
/20/
Scavuzzo, Srivatsan, Lam: Fatigue of butt welded steel pipes. American Society of Mechanical
Engineers, Pressure Vessels and Piping Division (Publication) PVP, v 374, p 113­143, 1998,
Fatigue, Environmental Factors, and New Materials
/21/
Belytschko, Liu, Moran, Nonlinear Finite Elements and Continua and Structures, John Wiley&Sons,
Ltd., November 2009
th
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SECTION 6 BIBLIOGRAPHY
Kuhlmann: Definition of Flange Slenderness Limits on the Basis of Rotation Capacity Values,
Journal of Constructional Steel Research, 14 (1989) 21­40
/23/
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)
855­875
/24/
DNVGL­ST­F101 Submarine pipeline systems
/25/
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.
/26/
EN­10025 Hot rolled products of structural steels. Part 2,3 4, and 6
/27/
EN­10225 Weldable structural steels for fixed offshore structures ­ Technical delivery conditions
/28/
DNV GL AS. Follow­up of Project on Non­Linear FE analysis, Tensile Failure Criteria, Background
Report. Høvik; Norway: DNV GL; 2016­01­11. Report No. 2015­0955, Rev. 1.
/29/
Tateishi, Hanibuchi. Effect of plate thickness on extremely low cycle fatigue strength of welded
joints. International Institute of Welding Commission XIII. Doc XIII­2335­10.
/30/
Simulia 2012, ABAQUS/CAE 6.12 User’s Manual
/31/
Ansys. Mechanical APDL v. 16.0
/32/
EN 1090­2:2008 Execution of steel structures and aluminium structures, Part 2 Technical
requirements for steel structures
/33/
EN 1993­1­2 Eurocode 3: Design of steel structures, Part 1­2: General rules Structural fire
design.
/34/
Martin Storheim & Jørgen Amdahl (2015): On the sensitivity to work hardening and strain­rate
effects in nonlinear FEM analysis of ship collisions, Ships and Offshore Structures,
DOI: 10.1080/17445302.2015.1115181
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/22/
7.1 Comments to [4.1] General
The element model selected for analysis needs to represent the structure so that any simplifications are
leading to results to the safe side. This is especially important for the selection of boundary conditions and
the representation of the load. The analyst needs to assess the possibility that simplification may lead to
an overrepresentation of the resistance. An example may be the representation of neighbouring elements
that also are subjected to buckling. In the case that the stiffness of the adjoining structure is uncertain it is
recommended to use boundary condition corresponding to simple support. If there are uncertainties with
respect to simplification in load it is recommended to vary the load pattern and perform alternative analyses
to check the effect.
The requirements to characteristic resistance in other standards for offshore structures like ISO 19902 /6/
are similar and the analysis carried out according to the recommendations in this RP is expected to fulfil the
requirements also in this standard.
7.2 Comments to [4.5.2] Selection of element
Guidance on selection of suitable elements for non­linear analysis can be found in text books e.g. /21/.
7.3 Comments to [4.6.6] Recommendations for steel material
qualities (low fractile)
The proposed stress­strain curves are based on steel according to /26/ and /27/. The curve is also applicable
to materials according to /8/. The curves apply the nominal yield stress for the respective steel grade and
thickness. The tensile strength is determined based on the expected yield to tensile strength ratio. The stress
strain relationship is aimed to be suited for determination of characteristic resistance and should be regarded
as a nominal stress strain relationship and may deviate from the actual stress strain relationship.
The material curves in engineering stress and strain is given in Figure 7­1.
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SECTION 7 COMMENTARY
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Figure 7­1 Proposed material curves in engineering stress and strain terms
7.4 Comment to [4.6.8] Strain rate effects
Often the constitutive models available in the NLFE software do not include this dependency. The model
should then be calibrated for the expected maximum stress (strain); otherwise the strain rate hardening will
be overestimated for large strains.
Examples of much used strain rate hardening models are Johnson­Cook (JC) and the Cowper­Symonds (CS).
Johnson­Cook (JC):
(25)
Cowper­Symonds (CS):
(26)
As seen, for both these models the relative effect will be the same for all static stress (strain) levels.
The constants D, C and p should be based on experiments corresponding to the material quality used and the
maximum strain level expected.
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7.5 Comments to [5.1.1] General
In most ordinary design situations, variable or cyclic loads will limit the plastic strain before tensile failure will
occur. Other limit states as accumulated plastic strain, low cycle fatigue, deformations or other failure modes
will govern. However, tensile failure is important for limiting the energy that can be resisted by structures
exposed to accidental loads, especially impact loading.
This recommended practice presents two methods for predicting tensile failure. One is based on calibration
to a known case while the other is a simplified method. It should be acknowledged that the tensile failure
phenomenon is complex and that both methods should be used with caution. The following aspects should be
considered when tensile failure is checked:
—
—
—
—
—
—
—
stress triaxiality
load history
loading rate
cold deformation
material properties
Material inhomogeneity
different material properties of materials being joined. (Even material with the same strength specification
may 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 the
conditions of the actual problem.
There are several failure models describing the local phenomenon of tensile failure. Such models may be
used if they are calibrated against known cases where the above aspects are considered.
7.6 Comments to [5.1.3] Tensile failure in base material ­ simplified
approach for plane plates
The simplified criteria are based on studies reported in /28/.
Traditionally, design standards have based the capacity against tensile failure on the value of the nominal
yield stress. In the proposed procedure, a realistic stress­strain curve is assumed with considerable
hardening. This will mean that the resulting capacity for simple tensile failure modes will be considerably
increased. In order to maintain the same safety margins for tensile failure compared with other failure
modes, as used in most structural design standards, an additional material factor is introduced. Eurocode
3 /4/ and AISC /5/, for example, use increased material factor when the capacity is using the tensile strength
and not the yield strength as the material parameter in the capacity formulations. Recommended values for
the material factor in Eurocode 3 are 1.25 and 1.0 for tensile strength and yield strength as the reference
strength, respectively.
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The published model parameters for various steel types are often calibrated to give accurate result at small
plastic strain levels, care should be taken to not overestimate the effect for large deformation analyses.
Strain rate effects for ship collision analyses are discussed in /34/, some guidance in selecting safe model
parameters can be found there.
Base material has in general better toughness properties than weld material. It is therefore regarded as
good design practice to ensure that large plastic deformation occurs in the parent material and not in the
weld. This is normally the case for full penetration welds where the overmatching material ensures limited
plastic deformation in the weld. Weld material may, however, contain defects of considerable size. In such
cases, a fracture mechanics assessment is necessary in order to determine if fracture in the weld may be the
governing failure mode.
7.8 Comment to [5.1.6] Simplified tensile failure criteria in case low
capacity is unfavourable
For cases where it will be unfavourable to calculate too low capacity like when the problem is to determine
the forces imposed on a structure when struck by a ship. Then the ship should not be analysed using
characteristic material properties intended to result in 5% low fractile resistances. For this reason values
aiming to represent mean values of common offshore steels are given in Table 7­1 to Table 7­5.
Table 7­1 Proposed mean properties for S235 steels (true stress strain)
S235
Thickness [mm]
t≤ 16
16< t ≤ 40
40< t ≤ 63
63< t ≤ 100
E [MPa]
210000
210000
210000
210000
σprop [MPa]
285.8
273.6
251.8
242.1
σyield [MPa]
318.9
305.2
280.9
270.1
σyield2 [MPa]
328.6
314.8
289.9
278.8
εp_y1
0.004
0.004
0.004
0.004
εp_y2
0.02
0.02
0.02
0.02
K[MPa]
700
700
675
650
n
0.166
0.166
0.166
0.166
Table 7­2 Proposed mean properties for S355 steels (true stress strain)
S275
Thickness [mm]
t≤ 16
16< t ≤ 40
40< t ≤ 63
E [MPa]
210000
210000
210000
σprop [MPa]
297.6
273.7
250.8
σyield [MPa]
331.8
306.4
282.0
σyield2 [MPa]
340.6
314.7
289.7
εp_y1
0.004
0.004
0.004
εp_y2
0.017
0.017
0.017
K[MPa]
740
700
685
n
0.166
0.166
0.166
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7.7 Comments to [5.1.5] Failure of welds
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Table 7­3 Proposed mean properties for S355 steels (true stress strain)
S355
Thickness [mm]
t≤ 16
16< t ≤ 40
40< t ≤ 63
63< t ≤ 100
E [MPa]
210000
210000
210000
210000
σprop [MPa]
384.0
357.7
332.1
312.4
σyield [MPa]
428.4
398.9
370.6
348.4
σyield2 [MPa]
439.3
409.3
380.3
350.6
εp_y1
0.004
0.004
0.004
0.004
εp_y2
0.015
0.015
0.015
0.015
K[MPa]
900
850
800
800
n
0.166
0.166
0.166
0.166
Table 7­4 Proposed mean properties for S420 steels (true stress strain)
S420
Thickness [mm]
t≤ 16
16< t ≤ 40
40< t ≤ 63
E [MPa]
210000
210000
210000
σprop [MPa]
435.5
432.7
421.9
σyield [MPa]
485.9
482.9
470.8
σyield2 [MPa]
490.2
487.2
475.1
εp_y1
0.004
0.004
0.004
εp_y2
0.011928571
0.011928571
0.011928571
K[MPa]
738
703
686
n
0.14
0.14
0.14
Table 7­5 Proposed mean properties for S460 steels (true stress strain)
S460
Thickness [mm]
t≤ 16
16< t ≤ 40
40< t ≤ 63
E [MPa]
210000
210000
210000
σprop [MPa]
485.3
484.0
460.3
σyield [MPa]
541.5
540.1
513.5
546.3
544.9
518.1
εp_y1
0.004
0.004
0.004
εp_y2
0.01
0.01
0.01
K[MPa]
772
745
703
yield2
σ
[MPa]
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Thickness [mm]
t≤ 16
16< t ≤ 40
40< t ≤ 63
n
0.12
0.12
0.12
7.9 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 the
prescribed probability of occurrence as required for a single extreme load. For environmental loads like wave
and wind it should be established a dimensioning storm that the structure is required to survive. It would be
in line with check for other failure modes to check the structure for one single storm from each of the critical
directions, 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
North Sea conditions have been found to have a maximum value equal to 0.93 of the dimensioning wave,
a duration of 6 h and a Weibull shape parameter of 2.0. This applies for check of failure modes where the
entire storm will be 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 /15/.
The load history for the remaining waves in a 100 year dimensioning storm investigated for southern North
Sea conditions have been found to have a maximum value equal to 0.95 of the dimensioning wave, a
duration of 6 h and a Weibull shape parameter of 2.0. The largest remaining waves after the dimensioning
wave (e.g. for cases like buckling) the largest wave is found as 0.92 of the dimensioning wave.
7.10 Comment to [5.2.4] Cyclic stress strain curves
The cyclic stress­strain curves are only intended for low cycle fatigue analysis. The use of monotonic stress
strain curve in low cycle fatigue analysis may provide non­conservative results and must therefore be
avoided.
The cyclic stress strain curves presented in Table 5­4 are based on cyclic behaviour of similar steels
reported in reference /25/. In order to account for uncertainties in material behaviour the curves are based
on conservative assumptions. A steel grade similar to S235 was not reported in /25/. Here, the same
exponent of 10 in the Ramberg­Osgood relation was assumed. K was assessed by assuming a strain value of
approximately 0.005 when the stress has approached the monotonic stress level of 235MPa.
7.11 Comment to [5.2.6] Low cycle fatigue of base material
The ε­N curve is based on laboratory test results reported in the literature. As for welded joints the design
curve is established by subtracting three standard deviations from the mean curve. A standard deviation of
0.2 in log N scale is assumed.
The influence of cold forming during production does not need to be included in the assessment as this effect
is considered to be implicitly accounted for in the design curve.
7.12 Comment to [5.2.5.1] Accumulated damage criterion
Laboratory test results presented in references /17/­/20/ make up the basis for the established e­N curve for
welded joints. The proposed mean and design curve for air along with the laboratory test data is presented in
Figure 7­2. Note that some of the results presented in the figure are not obtained directly from the referred
articles. 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 /18/ are
obtained from testing of dog­bone specimens cut out from a butt welded plate. These results have therefore
been weighted less than results from /17/ and /19/ which is based on full scale testing of tubular joints.
The fatigue test results presented in /20/ are from pipes with wall thicknesses of less than 10 mm. The
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S460
Because the fatigue test data come from several different sources it was not found reasonable to establish
the standard deviation from a regression analysis. Instead, a standard deviation of 0.2 in log N scale is
assumed for constructing the design curve in air. A standard deviation of 0.2 is identical to what is used in
high cycle fatigue (DNVGL­RP­C203 /11/). It is a general opinion within the body of fatigue expertise that the
statistical deviation in fatigue test results, decreases with decreasing fatigue life. Hence, assuming a standard
deviation value of 0.2 should be conservative.
The high cycle fatigue design curve in DNVGL­RP­C203 is defined as the mean curve minus two standard
deviations. In order to account for limited test data, the design curve has been established by subtracting
3·0.2
three standard deviations. Three standard deviations on log N corresponds to a factor of 10
≈ 4, i.e the
design curve 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 factor
6
of 2.5. This is identical to the reduction used in DNVGL­RP­C203 for fatigue lives less than 10 .
Figure 7­2 Mean and design curve for welded joints along with laboratory test results
7.13 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 is
therefore necessary to investigate the behaviour through the full cycles also for the next cycles. See e.g. /14/
for more guidance.
7.14 Comments to [5.4.1] General
The modelling of geometrical imperfections, out­of­straightness etc. is crucial for achieving a credible and
safe estimate of the buckling and ultimate strength limits. The less redundant the structure is the more
important it will be to model the geometrical deviations from perfect shape in a consistent way using the
eigenmode, post­buckling shapes, combinations thereof or similar. In such cases triggering the governing
modes rather than accounting for actual tolerance size will be most important in the analyses. However,
in all but the simplest of cases, if a structure is believed to be insensitive to geometric imperfections (or
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fatigue strength of welded joints is to some extent dependent on the wall thickness and since the thickness
of structural elements normally is significantly larger than this the results have been weighted less.
Guidance on analysis of stability problems may be found in e.g. /13/.
7.15 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 to
cross section class 1 and 2 and by comparison with tests. See /22/ and /23/. Strain limits are also compared
with recommendations given in /24/.
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imperfection size) it may be prudent to confirm this by undertaking sensitivity analyses on the imperfection
size. It should be noted that it is not always easy to identify in advance which form of imperfection may be
the most critical.
8.1 Example: Strain limits for tensile failure due to gross yielding of
plane plates (uniaxial stress state)
8.1.1 T­section cantilever beam
Gross yielding check of a T­section cantilever beam, subjected to axial and shear force and moment loading,
is presented in this example. The finite element software ABAQUS /30/ is used to perform the analyses.
The geometry and boundary conditions of the beam are shown in Figure 8­1. Loading is applied to a
reference point coinciding with the neutral axis of the beam cross section, using kinematic coupling between
the cross section and the reference point. The beam is modelled using 4­noded shell elements with reduced
integration (S4R). Material grade is S355, modelled according to Table 4­4.
The magnitude of the applied forces and moments are given by the axial force Nx, the shear force
Py = −0.15Nx and the bending moment Mz =−0.45Nx · a, a = 1 m.
Check against tensile failure is made according to [5.1.3].
Figure 8­1 Geometry and boundary conditions for cantilever beam, dimensions in mm
First, the critical gross yielding strain is determined by running calibration case CC01, see [5.1.3]. As seen
from Figure 8­2, the critical gross yielding strain εcrg is found to be 0.044. The element length selected in the
model is 2t which means 30 mm for the 15 mm thick CC01.
The load limit when the gross yielding strain is reached at a load of 421 kN is shown in Figure 8­3.
As the extent of the yielding zone with principal plastic strain above 0.02 is less than 20t it can be
documented a larger capacity by increasing the load until either the length of the yield zone reaches 20t or
the maximum principal plastic strain reaches the critical local strain:
The element length l = 2t = 16 mm.
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SECTION 8 EXAMPLES
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Figure 8­2 Principal plastic strain obtained with S4R elements for calibration case CC01
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In Figure 8­4 the direction of the maximum principal strain is plotted. The length of 20t in this direction is
indicated with the red line in Figure 8­5.
Figure 8­4 Plot showing direction of maximum plastic strain
Figure 8­5 shows the principal plastic strain when the extent of the yielding zone with principal plastic strain
above 0.02 reach 20t which corresponds to a load level of N = 459 kN. It should be noted that from the
strain plot it can be seen an even larger plastic zone away from the support, but as the plastic strain values
within this zone are all below the critical gross yielding value, the length restriction of 20t does not apply.
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Figure 8­3 Principal plastic strain at load level corresponding to critical gross yielding strain, N =
421 kN
From Figure 8­5 it can also be seen that the principal plastic strain is below the critical local strain so this
criterion is fulfilled. Consequently the calculated capacity of the beam can be set to:
Where γtf is the additional tensile failure material factor of 1.2 as given in [5.1.3], and
material factor according to the actual design standard.
γm is the ordinary
8.1.2 T­section cantilever beam with notch
Check for tensile failure of a T­section cantilever beam with a notch in the free edge of the web is presented
in this example. The geometry and boundary conditions are shown in Figure 8­6. The model, loading and
analysis setup and procedure are the same as in [8.1.1], except the size of the mesh which in this case is
25% of the notch height, i.e. 25 mm × 25 mm.
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Figure 8­5 Principal plastic strain at load level corresponding to the length of the yielding zone
reaching 20t, N = 459 kN
Plot of principal plastic strain at a load level corresponding to gross yielding strain of 0.044 is shown in Figure
8­7. The load is 287 kN. From the figure it can also be seen that the extent of the yield zone with plastic
strain above 0.02 is less than 20t meaning that a larger load can be documented.
Figure 8­7 Principal plastic strain at load level corresponding to critical gross yielding strain, N =
287 kN
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Figure 8­6 Geometry and boundary conditions for cantilever beam with notch
The direction of the maximum plastic strain is shown in Figure 8­8 and a red line with length of 20t is shown
in Figure 8­9.
In Figure 8­9 the plastic strain at the load level where the yield zone is extending 20t is plotted. The load is
306 kN. It can also be seen that the principal plastic strain in all areas is below the critical local strain:
Consequently, the calculated capacity of the beam can be set to
kN
where γtf is the additional tensile failure material factor of 1.2 as given in [5.1.3] and
material factor according to the actual design standard.
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γm is the ordinary
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Figure 8­8 Direction of maximum principal plastic strain at the load level corresponding to a zone
with principal plastic strain above 0.02, N = 306 kN
8.2 Example: Convergence test of linearized buckling of frame
corner
A symmetric frame of beams with I­section is analysed. The frame with boundary conditions is shown in
Figure 8­10 and Figure 8­11. The loading is applied as a displacement of the web at one end of the frame,
m. Three different mesh densities and two element types are included in a convergence
study, to ensure a sufficiently refined mesh. See Figure 8­12. The element types used are 4 node rectangular
shell elements and 8 node rectangular shell elements.
The analyses are performed using the finite element software ABAQUS /30/.
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Figure 8­9 Principal plastic strain at the load level corresponding to a zone with principal plastic
strain above 0.02
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Figure 8­10 Geometry of test example
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Figure 8­11 Displacement/boundary conditions
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Figure 8­12 Top: coarse mesh. Middle: fine mesh. Bottom: very fine mesh
For the eigenvalue analyses and the linear analyses elastic material properties were used and for the buckling
capacity analyses non­linear material properties were used. Details are shown in Table 8­1.
Table 8­1 Material properties
Density, ρ
Young’s modulus, E
Poisson’s ratio, ν
3
7850 kg/m
210 GPa
0.3
The loading is applied as displacement on the web at one end of the frame, as shown in Figure 8­11. 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
in Table 8­2. From these results all combinations of mesh size and element type except the coarse 4 node
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Table 8­2 Convergence study of frame
Case
number
Mesh size
Element type
Linearized buckling
displacement [m]
1
Coarse
4­node
0.0653
2
Fine
4­node
0.0624
3
Very fine
4­node
0.0618
4
Coarse
8­node
0.0616
5
Fine
8­node
0.0615
6
Very fine
8­node
0.0615
In summary the convergence test has shown that case number 2 and case 4 will produce sufficiently accurate
results of the linearized buckling value. Case 2 is preferred as the analysis is more efficient compared to case
4. The increased mesh refinement of case 3, 5 and 6 will not significantly improve the accuracy for the actual
problem solution.
8.3 Example: Determination of buckling resistance by use of
linearized buckling values
8.3.1 Step i) Build model
The same problem as shown in Figure 8­10 will be used in this example and the boundary conditions are as
in Figure 8­11. The material properties are shown in Table 8­3.
Table 8­3 Material properties
Density, ρ
7850 kg/m
Young’s modulus, E
210 GPa
Poisson’s ratio, ν
0.3
Yield strength, σY
355 MPa
3
The analysis follows the steps as given in [5.4.2]. Step 1 is completed as the model from the example in
[8.2].
8.3.2 Step ii) Linear analysis of the frame
The results from a linear analysis are shown in Figure 8­13 and Figure 8­14 for the von­Mises and membrane
compression stresses respectively.
The linear analysis is performed with the same applied displacement as in the eigenvalue analysis
m, equivalent to an applied load i y­direction
.
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combination, seems to be sufficiently refined. However, the stress results wanted are also highly dependent
on the mesh refinement, and a fine mesh in the area where high stress values are reached is preferable. An
analysis using the very fine mesh is time consuming, hence the mesh size and element type combination
chosen is the 4 node elements with fine meshing.
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Figure 8­13 Distribution of von­Mises stress from linear analysis
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8.3.3 Step iii) Determine the buckling eigenvalues
Eigenvalue analysis is performed to find the buckling modes and eigenvalues of the frame. The first
eigenvalue is
and the corresponding buckling mode shape is shown in Figure 8­15.
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Figure 8­14 Distribution of compressive stress from linear analysis (minimum in­plane principal
stress)
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Figure 8­15 First buckling mode
8.3.4 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 von­Mises stress in the structural part subjected to buckling.
8.3.5 Step v) Determine the von­Mises stress at the point for the
representative stress σRep from step ii).
Stress from linear analysis:
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The critical buckling stress for the governing buckling mode is determined as:
The reduced slenderness is determined as:
8.3.7 Step vii) Select empirically based buckling curve
The buckling curve used here is taken from Table 5­7. The curve selected is the one for column and stiffened
plate and plate without redistribution possibilities as it is judged that the corner plate cannot redistribute
stresses in a way so the plate curve could be used.
α is set to 0.3 for the following calculations.
8.3.8 Step viii) Determine the buckling resistance Rd
With
then the buckling factor is
Assuming a material factor
, the buckling resistance is
8.4 Example: Determination of buckling resistance from non­linear
analysis using standard defined equivalent tolerances
8.4.1 Description of model
The same problem as shown in Figure 8­10 will be used in this example and the boundary conditions are as
in Figure 8­11. The material properties are shown in Table 8­3 and the material model is shown in Figure
8­16.
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8.3.6 Step vi) Determine the critical buckling stress
A non­linear analysis (using the arc­length method) is performed, where the effects of imperfections, residual
stresses and material non­linearity is accounted for by use of a defined material stress­strain relationship
and the use of empirically determined equivalent imperfections. The shape of the governing buckling mode
is taken as the lowest buckling mode as shown in Figure 8­15, 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
5­8.
The analysed frame can be considered equivalent to a component of longitudinal stiffener or flange outstand,
hence the magnitude is given as
δ = 0.02 rad =0.02c
where c is half the width of the flange. Two values of c are analysed, the largest c; ca = a, where a =
0.975 m 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 b = 0.5 m is the width of the flange outside the curved
area. See Figure 8­10.
8.4.2 Results
The stress distribution for the non­linear analysis with initial imperfection is shown in Figure 8­17. Figure
8­18 displays the force­displacement curves for the displaced end of the frame for the linear analysis and
the force­displacement corresponding to the critical buckling stress where imperfections are taken into
consideration as calculated in [8.3], and from the non­linear analyses.
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Figure 8­16 Material model for analysis with material non­linearity
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Figure 8­17 Stress distribution for non­linear analysis with initial imperfection δ# at maximum
applied force
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Figure 8­18 Force­displacement from non­linear analyses, linear analysis and the calculated
critical value
8.5 Example: Determination of buckling resistance from non­linear
analysis that are calibrated against standard formulations or tests
8.5.1 Step i: Prepare model
A conical transition subjected to external hydrostatic pressure and axial tension is chosen for this analysis.
The geometry of the conical transition and the calibration object is shown in Figure 8­19. The applied loading
is defined as a hydrostatic pressure p =1.01MPa and an axial tension Ny = 58.4MN.
The boundary conditions are modelled using constraints with kinematic coupling between a reference point
in the cross section centre and the nodes on the circumference of the conical transition ends. At the bottom
all translations and rotations of the reference point are constrained and the top reference point is constrained
in the horizontal plane (x­ and z­direction). Load and boundary conditions and element mesh are shown in
Figure 8­20. The conical transition is modelled using 4­noded shell elements (S4R). Material properties are
listed in Table 8­4.
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Figure 8­19 Geometry of conical transition (on top) and calibration object (bottom), dimensions
in mm
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Figure 8­20 Left: Load and boundary conditions. Right: Element mesh
Table 8­4 Material properties
Density, ρ
Young’s modulus, E
7850 kg/m
3
210 GPa
Poisson’s ratio, ν
0.3
Yield strength, σY
420 MPa
Density water, ρw
1030kg/m
3
8.5.2 Step ii: Determine relevant buckling modes
Eigenvalue analysis is performed to find the buckling modes for the conical transition. The first relevant
buckling mode (with positive eigenvalue) is mode 3, shown in Figure 8­21.
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Figure 8­21 Buckling mode shape for conical transition
8.5.3 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 lower
cylindrical part of the conical transition, while the length is chosen as 2/3 of the conical transition length
(lower part, conical part and a part of the top part). The load and boundary conditions, element type and
mesh density used is the same as for the model of the conical transition, see Figure 8­22.
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Figure 8­22 Left: Load and boundary conditions. Right: Element mesh
8.5.4 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 8­23 shows the first cylinder buckling mode. This shows a similar pattern to the buckling
mode of the conical transition Figure 8­21, hence this is determined to be an appropriate buckling mode.
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Figure 8­23 Buckling mode shape for cylinder
8.5.5 Step v: Determine magnitude of the equivalent imperfection
To determine the magnitude of the equivalent imperfection a non­linear analysis of the cylinder with
imperfections is performed. The imperfection shape from the chosen buckling mode was transferred to the
non­linear analysis, and the same load and boundary conditions as for the eigenvalue analysis were applied.
The material model shown in Figure 5­7 is used for the non­linear analysis.
The imperfection is scaled so the buckling capacity of the cylinder is equal to the buckling capacity for
cylinders given in N­004 /7/. To obtain this capacity the magnitude of the imperfection was found to be
40 mm.
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A non­linear 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 8­24. The maximum load
proportionality factor is LPFmax = 0.936. Thus the buckling capacity of the conical transition subjected to the
given load combination is; hydrostatic pressure p = 0.95MPa and an axial tension Ny =54.7MN.
Figure 8­25 shows the von Mises stress at maximum load on the deformed conical transition.
Figure 8­24 Load proportionality factor for conical transition with initial imperfection
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8.5.6 Step vi: Perform non­linear analysis of the model with imperfections
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Figure 8­25 Deflected shape showing von Mises stress at maximum load deformations scaled with
a factor of 10
8.6 Example: Buckling check of jacket frame structure during deck
installation
8.6.1 Float­over concept
A float­over installation of a jacket topsides structure is characterised by the topside being floated in
between the jacket leg, by use of a barge or installation vessel, as exemplified in Figure 8­26. The topside is
supported in an elevated position on the vessel deck so that it can be lowered into place on top of the jacket
legs. However, the concept requires two jacket sides/rows to be un­braced to make room for the vessel. The
jacket is thus weak in the direction parallel to the un­braced faces.
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8.6.2 Model description
The example geometry is a four­legged jacket with straight, vertical legs in a 20 m × 20 m square and
diagonal bracing in 3 bays in the elevations shown in Figure 8­27. The upper (fourth) bay, from elevation
­10.0 m to 0.0 m, is only braced on two faces, in order to make room for a topside installation barge.
A typical, unbraced MSF structure reaches from elevation 0.0 m to +18.0 m where a stiff topside frame
connecting the four MSF legs acts as a simplified model of a topside module.
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Figure 8­26 Float­over installation. The topside structure, jacked up on the Black Marlin, is moved
in between the jacket legs and lowered into place. Source: Boskalis
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Figure 8­27 Overview of simplified model for jacket designed for float­over topside installation.
Legs are numbered from 1 to 4.
Only three tubular member sizes are defined, plus the stiff, tubular bracing of the topside frame.
Table 8­5 Tubular member dimensions
Tubular
Dimensions
Legs – upper
Ø1200 mm × 25mm
Legs ­ lower
Ø1400 mm × 60mm
Bracing
Ø600 mm × 20mm
8.6.3 FE program and element types
The general finite element program ANSYS /31/ is used for the FE analyses.
The jacket model is defined using the three­node, second­order plastic BEAM189 element with six degrees
of freedom at each node. The topside mass is implemented as a single mass element in the centre of gravity
(COG) of the topsides structure. See also [8.6.5] for load application.
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The non­linear material model used for the jacket legs and braces is an isotropic hardening, bi­linear curve as
shown in Figure 5­7. The von Mises yield criterion is applied.
Table 8­6 Material Data
Yield Stress
345 MPa
Young’s Modulus
210.000 MPa
Poisson´s ratio
0.3
Density
7850 kg/m
3
Figure 8­28 Non­linear material model
8.6.5 Boundary conditions and loads
The bottom ends of the four legs at mudline are fully fixed.
The primary load scenario is topside stabbing/mating, and the topside mass is applied at a single node near
the centre of the jacket, 5 m above the top of the legs at elevation +23.0 m.
The load is distributed stiffly to the jacket legs using rigid links.
In the present base analyses, the topside COG is offset by 1.0 m in X­direction, i.e. in the weak direction.
The topsides load setup is shown in Figure 8­29.
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8.6.4 Materials
2
Basic gravity is included in the model as vertical acceleration of 9.806 m/s . Load and material safety factors
have been purposefully omitted in the example.
8.6.6 Analyses
The buckling example covering a basic float­over installation load case follows the basic workflow shown
below:
—
—
—
—
—
—
—
—
—
—
—
linear (eigenvalue) buckling analysis
build model
run linear static analysis
run eigenvalue buckling analysis
calculate buckling capacity
non­linear analysis
extract imperfections from eigenvalue buckling analysis, mode shape 1
apply imperfections to non­linear model
run non­linear analysis, ramping topside load until failure (force­control)
test: re­run non­linear analysis, ramping topside load until failure (displacement­control).
post­process non­linear result: P­d curves etc.
Where possible, calculation and post­processing of FE results are done for all four jacket legs, although legs 1
and 4 nearest to the topside COG, see Figure 8­27, are sustaining the largest axial forces.
In addition to the base analyses, a number of sensitivity analyses has been performed, covering mesh
density and size of imperfections.
8.6.7 Linear (eigenvalue) buckling analysis
8.6.7.1 Initial linear­elastic analysis
As input to the eigenvalue buckling analysis a simple linear­elastic analysis is set up. The only loads are
gravity acting on the jacket steel and the single mass element representing the topside mass of 4000 t.
The stress state from the linear analysis is used as a pre­stressed starting point for the eigenvalue buckling
analyses. Most FE­analysis packages have quite streamlined approaches for this setup. Other output
needed from the linear analysis is the maximum representative stress σrep in each of the four legs and the
corresponding representative total force Srep corresponding to the total topside load of 4.000 t ~39.23 MN.
Stresses from linear analyses are shown in Figure 8­30 and axial forces are shown in Figure 8­31.
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Figure 8­29 Application of topside load: Single­node mass with stiff offset from centre of jacket
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Figure 8­30 Stresses (MPa) from linear analysis
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Figure 8­31 Axial forces (N) from linear analysis, legs from ­10.0 m to +18.0 m only
As this is a linear­elastic analysis the stress magnitude relative to the material yield stress is not critical or
relevant.
8.6.7.2 Eigenvalue buckling analysis
Starting with the stress­state from the linear­elastic analysis, the eigenbuckling analysis yields the requested
number of eigenvalues corresponding to the loaded model. Typically, the first eigenmode, and lowest buckling
eigenvalue, is governing.
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Figure 8­32 First buckling mode shape and corresponding eigenvalue, 0.861
It is seen from Figure 8­32 that the first eigenmode is a sway mode of the unbraced parts of the jacket legs,
between elevations ­10.0 m and +18.0 m.
The critical buckling stress σki is calculated as the chosen buckling eigenvalue kg times the representative
stress:
σki = kg σrep = 0.861 ∙ 123.4 MPa = 106.3 MPa
The reduced slenderness is determined as:
The buckling curve is taken from Table 5­7. The selected curve is for column and stiffened plate and plate
without redistribution possibilities:
where
Alpha
α is taken as 0.30 for strict tolerances and moderate residual stresses.
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where Srep is the total topside load and
σrep is the maximum stress found in the four legs.
Table 8­7 shows axial forces and stresses for all four legs.
Table 8­7 Linearized buckling results for all four jacket legs
Axial force (MN)
Max. stress σrep (MPa)
Leg 1
Leg 2
Leg 3
Leg 4
Total load
(Srep)
10.76
8.86
8.79
10.82
39.23
123.44
106.37
105.91
123.18
­
For the purpose of this example, the material factor
buckling resistance:
γM is taken as 1.0, which yields the following system
The system buckling capacity, expressed here in total topside mass, is calculated using the maximum stress
in any of the legs together with the total topside load. This can then be compared to the system capacity
found in the non­linear analyses, described in the following.
8.6.8 Non­linear buckling analysis with standard­defined equivalent
tolerances
8.6.8.1 General
The non­linear analysis is carried out using the same model as for the linear analyses, but with imperfections
and material and geometric non­linearities included.
8.6.8.2 Imperfections, misalignments and residual stresses
The effects of imperfections, misalignments and residual stresses must be taken into account for the non­
linear analysis. This is done by imposing initial, stress­free displacements on all nodes or elements in the
shape of the first eigenmode from the eigenbuckling analysis. These equivalent imperfections are scaled
in accordance with Table 5­8 for a member component with the magnitude taken as for strict tolerances
and moderate residual stresses. The same classification was chosen for the linear buckling analysis alpha
parameter (α = 0.3). This means that the imperfections are scaled so that the points of maximum deflection
in the eigenmode shape at the top of the jacket legs are displaced by L/250, where L is taken as the Euler
buckling length for a single leg based on the maximum axial load for the linearized buckling load:
which leads to the max. equivalent imperfection of 59.5/250 = 0.238 m
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The above yields a buckling factor κ = 0.256, and finally the buckling resistance can be calculated as:
Various methods can be employed to impose the scaled mode shape deflections to the model. All major FEA
packages are able to do this, either by use of built­in, automated procedures or through the use of custom
scripting. Some iteration may be needed to achieve the chosen maximum deflection.
Figure 8­33 Imperfections based on first eigenbuckling mode shape (exaggerated)
8.6.8.3 Material and geometric non­linearities
Material non­linearities are taken into account (see material curve in [8.6.4])
Geometric non­linearities (also often called large deformation effects) must be activated in the FE program.
This setting causes the solver to re­calculate/update the stiffness matrix for every load sub­step, based on
the deflected geometry.
8.6.8.4 Load in non­linear analysis – ‘force control’
For the non­linear analysis, the applied load must be higher than what is needed to reach failure. As the
linear analysis showed a buckling capacity of ~2900 t, it is chosen to apply a total equivalent force of up to
4000 t. The non­linear buckling capacity is then defined by the maximum force applied when the analysis
fails to converge.
For the previously mentioned non­linearities to have any effect, the load must be applied gradually, ramped
over several steps. In the present analyses the load (topside mass under standard gravity) is applied at a
starting rate of 1/100 increments of the total load, but the solver is allowed to apply the load in down to
1/1000 increments if necessary for convergence.
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The imposed imperfections are sketched in Figure 8­33.
Therefore, as a test, the same problem is analysed with the topside load applied as a Z­displacement. The
magnitude of the displacement is taken as twice the resulting Z­displacement seen in the previous, force­
controlled analysis, in order to drive the analysis well past the load limit.
8.6.8.6 Non­linear analysis – results
The stress distribution at maximum load for the non­linear analysis is shown in Figure 8­34.
Figure 8­34 Stress distribution at maximum load, non­linear analysis w. initial imperfections
The force­displacement curves in Figure 8­35 show the axial forces in the four legs plotted against
displacement in X­direction of the topside node. It is seen that the legs are almost equally loaded in leg pairs
1 and 4, and 2 and 3, where pair 1 and 4 is loaded higher, because the topside COG is offset towards these
legs.
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8.6.8.5 Load in non­linear analysis – displacement control
In the present case, the problem is simple enough that the topside load can be applied as a displacement in
the load direction (standard gravity in the vertical direction).
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Figure 8­35 Non­linear force­displacement curves (force­controlled)
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Figure 8­36 show the force displacement curves for the axial force in the four legs when the analysis is
displacement controlled. In Figure 8­37 the force/displacement curves for force­controlled and displacement­
controlled analyses are compared. As expected, the curves are practically coincident, although the
displacement­controlled analysis is able to drive the analysis beyond the maximum capacity.
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Figure 8­36 Non­linear force­displacement curves (displacement­controlled)
Again, the system buckling resistance in the present case is defined as the total topside load at the time of
the limit load. For comparison, the result from the linear buckling analysis is included by a horizontal line.
Table 8­8 Non­linear buckling results for all four jacket legs. Axial forces are compressive.
Max. Axial Force (MN)
System Buckling Resistance
Leg 1
Leg 2
Leg 3
Leg 4
7.52
6.11
6.15
7.48
27.26MN ~ 2780 t (topside mass)
8.6.9 Comparison – linear vs non­linear results
A comparison of the capacities from the linear buckling analysis and the non­linear analysis shows reasonably
good agreement between the two methods, with the non­linear capacities being slightly lower. In general, it
would be expected that the non­linear analysis yields the highest capacity, but the difference depends on the
actual model.
Table 8­9 Result comparison – linear buckling vs non­linear analysis
Force (MN)
Topside mass (t)
Linear buckling resistance
28.18
2873
Non­linear capacity
27.26
2780
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Figure 8­37 Test: comparison between force­controlled and displacement­controlled analysis
Topside mass (t)
­3.2
Difference (%)
8.7 Example: Joint of rectangular hollow section (RHS) and circular
hollow section (CHS) under tension loading
8.7.1 Introduction
The RHS­CHS bolted joint shown in Figure 8­38 is to be designed according to EN­1993­1­8 /4/. The joint
is subjected to an axial tension force. When the joint is subjected to tension, prying forces will influence the
bolt tensile force and flange stress field. In order to properly account for these effects, a non­linear FE model
is developed and analyses carried out in accordance with method b) of [3.4].
The following design checks will be performed:
— tension resistance of the individual bolt
— punching shear resistance at the individual bolt
— failure by plastic collapse of the flange.
The following design procedure is proposed: For an appropriate calibration object, the allowable load is
found from the standard. A non­linear FE model is developed for the calibration object. The maximum first
principal plastic strain value when the characteristic load is applied in the calibration object is noted. This is
used as failure parameter. A non­linear FE model is developed for the RHS­CHS joint using a similar setup
as for the calibration case. The resistance is found as the maximum tensile force that can be applied without
exceeding the first principal strain value found from the calibration. Bolt tension resistance and punching
shear is checked using the bolt (reaction) force found from the FEA.
Using the outlined procedure, the safety level inherent in the proposed standard is ensured for the RHS­CHS
joint.
Figure 8­38 RHS­CHS joint
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Force (MN)
8.7.2.1 Definition of calibration case
For the RHS­CHS joint depicted in Figure 8­38, prying forces will influence the bolt tension force and flange
stress field when the joint is subjected to tension. The proposed calibration object is a simple T­stub in
tension, depicted in Figure 8­39. When subjected to tension, the same failure mode is expected in the T­stub
as in the RHS­CHS joint. The T­stub is considered a section cut out from a long, continuous joint. The flange
dimensions from the RHS­CHS joint are used in the T­stub section as well as the same bolts, M24 8.8.
Figure 8­39 T­stub geometry
8.7.2.2 T­stub design
The T­stub design is performed according to EC3,1­8,Table 6.2 /4/. Calibration is performed for characteristic
resistance, implying that all partial factors are set to γM = 1.0. End row effects are not considered, implying
that the effective length Σleff = 100 mm. Also, it is checked that the punching shear resistance exceeds the
tension resistance of one fastener. Details of the design are given in Table 8­10.
The failure mode giving the resistance is Mode 1, implying that complete yielding of the flange is the failure
mode. The resulting characteristic tension resistance is FT,Rc = 214.6 kN
It is also noted that
The calculated bolt preload is Fp,Cc = 197.7 kN
The tension resistance of one bolt is Ft,Rc = 254.2 kN
The punching shear resistance at one bolt is Bp,Rc = 498.8 kN
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8.7.2 Calibration case
General input
Partial safety factors
γM0
1.0
γM2
1.0
γM7
1.0
Bolt data
fub
Ultimate tensile strength
800
MPa
21.2
mm
Bolt head height
15
mm
Diameter of washer/nut/bolt head
36
mm
Flange thickness
15
mm
Number of bolts
2
Tension diameter
dw
Tab. 3.1
Joint geometry and material data
tf
­
fy
Yield strength
355
MPa
fu
Ultimate strength
490
MPa
0.9
­
353
mm
Tension resistance
k2
factor
As
Tension area
Ft,Rd
Tension resistance, one bolt
0.63 for countersunk bolt; 0.9
k2 fub As / γM2
254152
Tab. 3.4
2
N
Tab. 3.4
Punching shear resistance
dm
Mean of across points/flats
dimensions of the bolt head/nut,
whichever smaller
dw
36
mm
tp
Thickness of plate under bolt/nut
tf
15
mm
Bp,Rd
Punching shear resistance
0.6
π dm tp fu / γM2
498759
N
Tab. 3.4
197674
N
3.6.1
(2)
508305
N
Preload
Fp,Cd
Design preload
0.7 fub As / γM7
Resistance of equivalent T­stub
ΣFt,Rd
Total value of tension resistance
Ft,Rd · number of bolts
Σ leff,1
100
mm
Σ leff,2
100
mm
Mpl,1,Rd
Mpl,2,Rd
2
0.25 Σ leff,1 tf fy / γM0
0.25 Σ
2
leff,2 tf fy
/ γM0
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Tab. 6.2
1996875
Tab. 6.2
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Table 8­10 T­stub design parameters and result according to EN­1993­1­8 /4/
Partial safety factors
ew
9
emin
50
Fig. 6.2
m
44
Fig. 6.2
n
50
Lb
Bolt elongation length
45
Tab. 6.2
8.8 m As / (Σ leff,1 tf )
784
Tab. 6.2
May prying forces develop?
Yes if Lb ≤ Lb*;No
Yes
Tab. 6.2
TR, Mode 1. Method 1
4 Mpl,1,Rd / m
1.815E+05
N
Tab. 6.2
TR, Mode 1. Method 2
(8n­2ew)Mpl,1,Rd / (2 m·n ­ew(m+n)) 2.146E+05
N
Tab. 6.2
2.146E+05
N
Tab. 6.2
3
Lb*
3
FT,1,Rd
Tension resistance, Mode 1
FT,2,Rd
Tension resistance, Mode 2
(2Mpl,2,Rd + n ΣFt,Rd) / (m+n)
3.129E+05
N
Tab. 6.2
FT,3,Rd
Tension resistance, Mode 3
ΣFt,Rd
5.083E+05
N
Tab. 6.2
FT,Rd
Tension resistance
min {FT,1,Rd; FT,2,Rd; FT,3,Rd}
2.146E+05
N
8.7.3 T­stub finite element model
8.7.3.1 Software used
The analyses are performed using ANSYS /31/ within the ANSYS Workbench environment. The geometries
are prepared in ANSYS DesignModeler.
8.7.3.2 Material models
For the bolts, a linear­elastic material model is used. The elastic material properties are given as follows:
Poisson’s ratio: ν = 0.3
Young’s modulus: E = 210 000 MPa
For the base material, the elastic­plastic material S355 described in [4.6.6] Provisional tensile failure criteria
is implemented using an elastic material model together with a multi­linear kinematic hardening material
model. The elastic material properties are given as follows:
Poisson’s ratio:
ν = 0.3
Young’s modulus: E = 210 000 MPa
A multilinear kinematic hardening material model is used to define plastic behaviour. Twenty points are used
to define the stress­strain curve, using an uneven distribution such that the first part of the curve has a finer
resolution. The curve is given as a true stress vs true plastic strain curve, as required by ANSYS Mechanical.
The stress­strain curve used is shown in Figure 8­40. Numerical values are given in Table 8­11.
The weld is modelled with a material similar to the S355 material, but modified to be 25% stronger. The
resulting stress­strain curve is given in Table 8­11.
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General input
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Figure 8­40 Stress­strain curve for S355
Table 8­11 True stress vs true plastic strain
Strain [­]
Stress [MPa]
Strain [­]
Stress [MPa]
0
320
0
400
0.004
357
0.004
446.25
0.015
366.1
0.015
457.63
0.018
377.77
0.018
472.21
0.021
387.87
0.021
484.84
0.025
399.56
0.025
499.45
0.03
412.11
0.03
515.14
0.039
430.79
0.039
538.48
0.051
450.67
0.051
563.34
0.075
480.76
0.075
600.95
0.1
504.44
0.1
630.55
0.15
539.74
0.15
674.67
0.2
566.23
0.2
707.79
0.3
605.75
0.3
757.19
0.4
635.43
0.4
794.29
0.5
659.44
0.5
824.3
0.6
679.73
0.6
849.66
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Stress [MPa]
Strain [­]
Stress [MPa]
0.7
697.36
0.7
871.7
0.8
713
0.8
891.26
0.9
727.09
0.9
908.86
8.7.3.3 Geometry modelling
The T­stub geometry dimensions are shown in Figure 8­39. The bolt shaft is modelled using the tension area.
The bolt heads and nuts are modelled as the cylinder inscribed within the hexagonal prism. Washers are not
modelled.
The material assignment is shown in Figure 8­41.
Figure 8­41 Material assignment
8.7.3.4 Element types and mesh
The geometry is meshed with higher order hexahedral solid elements, SOLID186. Two mesh densities are
investigated. One fine mesh has element size approximately tf/3 × tf/3 × tf/3; one coarse has element size
approximately tf × tf × tf. The bolts are meshed with even smaller elements. For the fine mesh, uniform
reduced integration is used. For the coarse mesh, full integration is used, as this is the recommended setting
for models with only one solid element through the thickness.
The resulting meshes are shown in Figure 8­42 and Figure 8­43.
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Strain [­]
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Figure 8­42 Coarse element mesh
Figure 8­43 Fine element mesh
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The four contacts between the bolt heads/nuts and flanges are defined as frictional contacts with properties
as defined in Table 8­12. The friction coefficient is taken as the lowest value for slip factor, from EN1993­1­8,
Table 3.7 /4/. A typical contact­target designation is shown in Figure 8­44. The coarser meshed side, in this
case the flange, is selected as the target (master) side.
The contact between the flanges is defined as a frictional contact with properties as defined in Table 8­12.
The contact­target designation is shown in Figure 8­45.
Table 8­12 Contact settings
Bolt head/nut to flange
Flange to flange
Friction coefficient
μ = 0.2
μ = 0.2
Behaviour
Asymmetric
Asymmetric
Formulation
Augmented Lagrange
Augmented Lagrange
Interface treatment
Adjust to Touch
Adjust to Touch
Figure 8­44 Contact definition, bolt head/nut to flange
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8.7.3.5 Connections
Non­linear contact is defined between bolt heads/nuts and flanges and at the interface between the two
flanges.
8.7.3.6 Analysis settings
The analyses are run as static structural analyses. Under analysis settings, large deflection is turned on. Auto
time stepping is used in order to enhance convergence as well as for results accuracy.
8.7.3.7 Loads and boundary conditions
Frictionless supports are applied at both capped ends of the assumed continuous T­stub. One side of the T­
stub is fixed in the global X­direction. The applied supports are shown in Figure 8­46.
Preload is applied to the bolts in the first step. The value is taken as the calculated bolt preload Fp,Cc as given
in [8.7.2.2]. For succeeding steps, the bolts are kept in a locked position, implying that the bolt elongation
is determined by the bolt shaft stiffness. In the second load step, one side of the T­stub is subjected to a
tensile force with value given by the allowable force FT,Rc found in [8.7.2.2]. The applied loads are shown in
Figure 8­47.
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Figure 8­45 Contact definition, flange to flange
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Figure 8­46 Applied supports
Figure 8­47 Applied loads
8.7.3.8 Results
The first principal plastic strain plot is given in Figure 8­48 and Figure 8­49. For the coarse mesh model, the
maximum values for plastic strain is found under the bolt heads. However, since punching shear is checked
by code requirements, local strains in this region may safely be disregarded. The relevant plastic strain
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Table 8­13 Results Summary
Maximum EPPL1 [­]
Force reaction in bolts [N]
Coarse mesh model tf × tf × tf
0.00153
221 220
221 460
Fine mesh model tf/3 × tf/3 × tf/3
0.00605
222 420
222 380
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values are found in inner corner of the flanges, as seen in Figure 8­48 and Figure 8­49. The maximum first
principal strain values as well as bolt force reactions are given in Table 8­13.
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Figure 8­48 First principal plastic strain results. Coarse mesh model
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Figure 8­49 First principal plastic strain results. Fine mesh model
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8.7.4.1 FE Model
The model geometry for the RHS­CHS joint is shown in Figure 8­50. The materials used are the same as
described for the T­stub model in [8.7.3.1].
As for the T­stub model, two mesh densities are investigated. The coarse mesh has element size
approximately tf × tf × tf; the fine mesh has element size approximately tf/3 × tf/3 × tf/3. The resulting
meshes are shown in Figure 8­51 and Figure 8­52 respectively.
Figure 8­50 Joint geometry
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8.7.4 RHS­CHS Joint finite element model
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Figure 8­51 Coarse mesh
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Figure 8­52 Fine mesh
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As only a quarter of the actual geometry is modelled, two symmetry planes are defined using frictionless
support. One side of the joint is constrained in the axial direction. The applied supports are shown in Figure
8­53.
Preload is applied to the bolts in the first step. The value is taken as the calculated bolt preload Fp,Cc as given
in [8.7.2.2]. For succeeding steps, the bolts are kept in a ‘locked’ position. One side of the joint is subjected
to a force leading to axial tension in the joint. The applied loads are shown in Figure 8­54.
It is not known in advance at which applied load the allowable value for EPPL1 from [8.7.3.8] is exceeded.
Trial­and­error is required. In the final analysis setup, the load is gradually applied such that the principal
plastic strain is just below the allowable value in analysis step 2 and just above in step 3. The allowed load
will be the load applied in the last substep in which the allowable first principal plastic strain is not exceeded.
Using relatively fine substepping in step 3, the utilization is maximized.
Table 8­14 Analysis settings
Number of steps
Auto time stepping
3
On
On
On
1
2
3
10
10
10
Minimum substeps
5
10
10
Maximum substeps
50
50
50
Settings for step number…
Initial substeps
Large deflection
Solver type
Direct
Figure 8­53 Applied supports
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8.7.4.2 Loads and boundary conditions
The applied analysis settings are shown in Table 8­14.
8.7.4.3 Results
Results are summarized in Table 8­15. All values are given for the substep at which the occurring first
principal plastic strain value is closest to but not exceeding the allowed value found in [8.7.3.8]. Again, the
strain results are scoped to the flanges, close to the weld, to avoid the results around the bolt holes.
The applied axial force is reported in Table 8­15. In order to find the total axial force in the member, the force
applied in the analysis is multiplied by a factor 4, accounting for the double symmetry used. The bolt force
reactions are also reported in the table.
Table 8­15 Results Summary
Coarse mesh model
Fine mesh model
t f × tf × t f
tf/3 × tf/3 × tf/3
Allowable EPPL1 [­] (from Table 8­13)
0.00153
0.00605
Occurring EPPL1 [­]
0.00152
0.00597
Applied force [N]
233 500
233 500
Total axial force [N]
934 000
934 000
226 800
225 350
226 720
225 370
Force reactions in bolts [N]
The first principal plastic strain values are presented in Figure 8­55 and Figure 8­56 for the coarse and the
fine element mesh, respectively.
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Figure 8­54 Applied loads
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Figure 8­55 First principal plastic strain. Coarse mesh model
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Figure 8­56 First principal plastic strain. Fine mesh model
8.7.5 Discussion
8.7.5.1 Obtaining characteristic resistance
The characteristic tension resistance is taken as the force at the time step at which the occurring first
principal plastic strain value is closest to but not exceeding the allowed value found in [8.7.3.8].
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Referring to Table 8­15, the difference in allowable strain for the two mesh densities compared with the
difference in characteristic resistance for the two mesh densities may be noted. This underscores the
importance of using the same setup, including mesh density, for the calibration object as for the designed
object.
8.7.5.2 Obtaining design resistance
Since calibration is performed both for the coarse and fine mesh, the same allowed force is expected. This
is also the case in this specific example. However, the resulting allowed force could differ slightly due to the
uncertainties in FE­modelling. In that case, any of the two results would be considered acceptable.
All calculations and analyses are performed using characteristic values. In order to obtain design resistance,
the resistance found in [8.7.5.1] must be divided by the appropriate partial factor. In this example, the
design mode is mode 1, where partial factor γM0 is relevant. Also, if all design calculations in [8.7.2.2] are
performed with the appropriate values assigned to material factors, mode 1 is still the design mode. For this
reason, we argue that
In the case that mode 2 would give the design value, both γM0 and γM2 influence on the design resistance.
In this case, the calibration process could be performed using design values. Alternatively, the calibration
process could be performed using characteristic values and the material factor with the highest numerical
value could be used to find the design resistance.
If, however, the mode 3 would give the design mode, dividing by
γM2 is appropriate.
8.7.5.3 Notes on modelling tolerances
For both the T­stub and the RHS­CHS joint models, the flanges are modelled perfectly aligned, resulting in
perfect contact between the flanges. As long as the fabrication tolerances of the structure comply with EN
1090­2:2008 /32/ and the same modelling approach is used for both calibration object and structure, this
approach is considered valid. However, if fabrication tolerances exceed the above mentioned tolerances, the
worst expected tolerance should be modelled in both models.
8.7.5.4 Check points for the current analysis
For the pretension of the bolts to function properly, the mesh on the bolt shaft should be mapped meshed
and at least two elements should be used along the bolt shaft.
Since the washers are not included in the model, the unrealistic small contact area may lead to an artificially
large contact pressure, resulting in a large penetration and/or local plasticity. These effects may lead to loss
of the bolt pretension. As a part of the results verification, the resulting penetration as well as local plasticity
is verified. The penetration should be an order of magnitude smaller than the flange compression. Also, the
resulting pretension is checked.
8.8 Example: Check of stiffened plate exposed to blast loads
8.8.1 Description of stiffened plate wall
The wall shown in Figure 8­26 is to be checked for blast pressure. The wall is built as a stiffened plate
spanning between floor and roof girders.
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Furthermore, it is checked that the tension resistance of one fastener and punching shear resistance of one
fastener is not exceeded. This is done by comparing force reaction in bolts in Table 8­15 with the tension
resistance of one bolt Ft,Rc and punching shear resistance Bp,Rc from [8.7.2.2].
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Figure 8­57 Blast wall example
8.8.2 Model description
The example geometry is a strip of the wall shown in Figure 8­57 with only one single longitudinal stiffener.
The width of the wall is taken equal to the distance between stiffeners, with the single stiffener at the centre
of the plate, so that symmetry can be utilised to simulate an infinitely wide wall. For the present example,
it is judged that this is an adequate representation of the weakest spot on the wall, i.e. that the wall will be
stronger in proximity to the corners of the room.
The wall plate is connected to heavy HEM400 girders at the top and bottom, representing the roof and floor
levels of the structure. The girders are also reinforced with a 15 mm plate between the flanges, in line with
the longitudinal stiffener.
The wall plate is welded to the girders using a welded, L­shaped bracket with two lap joints, as seen in Figure
8­58.
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Symmetry is also utilised at the mid­elevation between the top and bottom girders, to further reduce the
model size, see Figure 8­59.
Figure 8­59 Symmetry conditions in FE model
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Figure 8­58 Bracket connecting wall and girder
Plate thicknesses are listed in Table 8­16.
Table 8­16 Plate thicknesses
Component
Dimensions
Wall plate
10 mm
Stiffener
10 mm
L­brackets
10 mm
All steel is S355 material, except the HEM girder, which is defined as linear­elastic steel because it is not
meant to be checked in the present analysis. See also [8.6.4].
Since the analysis will be of a transient dynamic type, any extra mass on the wall such as insulation or
architectural cladding should be accounted for, if deemed relevant. If the cladding does not contribute to the
wall stiffness, this could for example be done by scaling the steel density of the relevant portions of the wall
accordingly.
For the purpose of this example, it is assumed that the wall is bare, without external cladding.
8.8.3 Software, element types and mesh
The general FE program ANSYS Mechanical v16.1 is used for the FE analyses.
The geometry is meshed with eight­node, second­order plastic SHELL281 elements. The mesh density in the
critical areas is equal the plate thickness of 10 mm, while larger elements have been allowed in less critical
regions. The mesh is shown in Figure 8­60.
Figure 8­60 Element mesh
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See also [8.6.5] for boundary conditions.
—
—
—
—
second­order, 8­node shell elements with full integration, SHELL281
element size = 10 × 10 mm, i.e. t × t
five integration points through thickness
non­linear material S355 (t ≤ 16mm) as specified in [8.6.4].
8.8.4 Materials
For the present analysis a low fractile (5%) steel capacity is sought and the applied material model according
to [4.6.6] is presented in Figure 8­61.
Figure 8­61 Non­linear material model
For part 4, the relation between stress and strain is as follows:
For the chosen steel type, S355, the relevant parameters for plate thickness less than 16 mm are shown in
Table 8­17.All relevant plate sections are below 16 mm in thickness, so only one set of non­linear material
properties is used in the model.
Table 8­17 Non­linear properties for S355 steel (true stress­strain)
Thickness [mm]
t ≤ 16
E [MPa]
210000
σprop [MPa]
320.0
σyield [MPa]
357.0
σyield2 [MPa]
366.1
εp_y1
0.004
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In summary the model is set up as follows:
t ≤ 16
εp_y2
0.015
K[MPa]
740.0
n
0.166
3
Density [kg/m ]
7850
Poisson’s ratio
0.3
The material type is defined in ANSYS using the von Mises yield criterion. Note that strain rate hardening
effects as mentioned in [4.6.8] have not been accounted for in the present analyses.
For calibration of tensile failure strain criteria, please see [8.8.8].
8.8.5 Boundary conditions and loads
Symmetry conditions are applied at the centre elevation and along the side of the wall strip, see [8.8.2].
The flange at the far end of the HEM400 girder is fully fixed (see Figure 8­59), as it is assumed that the
girder is part of a roof or floor structure that is supported both horizontally and vertically by intermediate
bracing. Note that the girder and supporting structure is not intended to be checked in the present analysis.
As the explosion load is considered an accidental scenario, the load and material safety factors are 1.0,
except for an additional safety factor for the tensile failure check, as prescribed in [5.1.3]. This factor, γtf,
equals 1.20.
An explosion load is a high­velocity, short­duration load, and is therefore often best described in non­linear
transient analysis as a pressure time history. The present explosion load is shown as a time­pressure curve
in Figure 8­62. For non­linear analyses it is often practical to apply all safety factors on the load side, as
explained in [4.10]. Therefore the tension failure safety factor is implemented by multiplying the entire curve
by 1.2, as is also shown in Figure 8­62.
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Thickness [mm]
It is seen that the pressure peaks rapidly, and the pressure drop is followed by a slight negative pressure
before diminishing completely. The duration of the transient analysis must be at least long enough to capture
the peak response, which may be delayed due to inertia effects.
The time history used, including the additional safety factor of 1.2, is given in Table 8­18.
The pressure is applied in the positive Y­direction, on the entire wall plate at the stiffener side, and to
the exposed, vertical part of the bracket. No pressure has been applied to the girder web, as the girder is
primarily included as a support or boundary condition.
Table 8­18 Tabularised blast pressure time­history, including tensile failure safety factor
Point
Time [s]
Pressure [MPa]
1
0.000
0.000
2
0.022
0.002
3
0.028
0.005
4
0.032
0.010
5
0.036
0.019
6
0.040
0.034
7
0.050
0.082
8
0.054
0.029
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Figure 8­62 Blast pressure time­history graph
Time [s]
Pressure [MPa]
9
0.057
0.000
10
0.060
­0.007
11
0.064
­0.012
12
0.068
­0.012
13
0.074
­0.010
14
0.078
­0.005
15
0.085
0.000
16
0.090
0.001
17
0.095
0.000
18
0.100
0.000
19
0.105
0.000
20
0.110
0.000
8.8.6 Solution parameters for transient implicit analysis in ANSYS
The present transient analysis in ANSYS uses the HHT method (or alpha­method). Recommendations for
solution parameters are given in [4.3.2]. It should be noted that the parameter nomenclature in ANSYS is
different, as shown in Table 8­19.
Table 8­19 Correlation between ANSYS and notation used in [4.3.3] for HHT­method
ANSYS
ANSYS
value used
Table 4­1
γ
0.05
­α
α
0.2756
β
δ
0.55
γ
The time step size should be chosen sufficiently small to obtain an accurate solution. ANSYS recommends
that the maximum integration step size is 1/20 of the system response period, so there should be 20 points
per cycle. The response frequency is calculated by ANSYS during a full transient analysis and the weighted
average of the responses for all frequencies excited by a given load is calculated. This means that steps per
cycle can be tracked in the solution output.
Generally, the defined maximum number of sub­steps per load step is 100 and the minimum number is 10
for the present transient analysis. A load step is defined as the step between two points on the load curve.
The steps­per­cycle value was found to be ~35­130 for the load steps near the peak load on the time­history
curve, and much larger values were seen at other load steps. It is possible to fine­tune the allowable time
step sizes and to use various ANSYS commands such as CUTBACK and SOLCONTROL, to optimize the clock
time of the solution. It is vital to test the sensitivity of the used time control settings to verify that the chosen
parameters are adequate for a converged result.
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Point
The primary failure mode is tensile failure in the wall plate and/or L­bracket. As the explosion pressure ramps
up, the longitudinal stiffener is subject to compression and is expected to eventually buckle.
The effects of imperfections, misalignments and residual stresses must be taken into account in the non­
linear analysis. This is done by imposing initial (stress­free) displacements on all nodes/elements in the
shape of the first relevant eigenmode from an eigenbuckling analysis. These equivalent imperfections
are scaled in accordance with Table 5­8 for longitudinal stiffener or flange outstand. This means that the
imperfections are scaled so that the point of maximum deflection in the “bow twist” eigenmode shape is
displaced by δT0, calculated as Tan(0.02) · (125+10) mm ~ 3.0 mm.
The analysis procedure is as follows:
A linear­elastic analysis is set up with a nominal pressure load applied in the same manner and direction as
for the transient analysis. The magnitude of the pressure load is not important in this linear analysis.
Starting with the stress state from the linear­elastic analysis, a buckling eigenvalue analysis yields the
requested number of eigenvalues corresponding to the loaded model. Typically the first eigenmode, and
lowest buckling eigenvalue, is governing.
The shape of the first relevant eigenbuckling mode is imposed to the FE model used in the transient analysis.
Various methods can be employed to impose the scaled mode shape deflections to the model. All major FEA
packages are able to do this, either by use of built­in automated procedures (some iteration may be needed
to achieve the chosen maximum deflection) or through the use of custom scripting.
In the present case, the imperfections are scaled to 3.0 mm at the stiffener flange close to the mid­span
symmetry boundary condition, see Figure 8­63.
Figure 8­63 First relevant eigenbuckling mode, stiffener buckling
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8.8.7 Stiffener buckling and imperfections
In [5.1.6] the critical strain at tensile failure can be estimated. In order to determine the critical strain
values, two calibration cases CC01 and CC02, see [5.1.3], should be tested for the same element type and
mesh density as is used in the ‘actual’ analysis.
Calibration case CC01 covers both gross yielding (where strain above 2% is seen in a zone larger than
20 times the plate thickness t) and local yielding (zone < 20t). For the local yielding check the maximum
principal plastic strain (used for gross yielding) is modified according to Equation (10) as follows:
for the plate thickness t = 10 mm and the element length l = 10 mm, as used in the actual model.
The calibration examples are set up as follows:
—
—
—
—
Second­order, 8­node shell elements w. full integration, SHELL281
Element size = 15 ×15 mm, i.e. corresponding to t × t like in the actual model
Five integration points through thickness
Non­linear material S355 (t ≤ 16mm) as specified in [8.6.4].
The maximum principal plastic strain (EPPL1) is chosen as the critical result parameter, so basically the
maximum EPPL1 value found in the calibration tests must not be exceeded in the current transient explosion
analysis.
Using the geometry and boundary conditions specified for the two calibration cases the following results are
shown in Table 8­20:
Table 8­20 Result from calibration cases
Applied displacement
Critical/maximum
principal strain
Δx (mm)
Δz (mm)
Surface layer
Middle layer
Failure Type
CC01
21
0
0.045
0.045
Gross
CC01
21
0
0.119
0.119
Local
CC02
50
70
0.250
0.150
Local, Bending
Strain plots can be seen in Figure 8­64 through Figure 8­66.
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8.8.8 Calibration of tensile failure strain
It is seen that the critical principal strain values for the bending cases, CC02, are very high. In the region of
maximum strain in the actual model, see [8.8.9], the extent of the yield zone is only in the order of 1 times
the thickness in the strain direction, normal to the weld.
Therefore, the critical tensile strain is decided as 0.119, corresponding to calibration case CC01 for local
yielding, see Table 8­20.
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Figure 8­64 Principal plastic strain, calibration case CC01
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Figure 8­65 Principal plastic strain, calibration case CC02 (surface layer)
Figure 8­66 Principal plastic strain, calibration case CC02 (middle layer)
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8.8.9.1 Blast response
The deformation of the blast panel is examined to verify that the model is responding to the pressure load as
expected.
The maximum deformation in the blast direction along the Y­axis is shown in Figure 8­67.
Figure 8­67 Wall deformation for explosion load (plot is scaled with a factor of 2.0)
The Y­axis deformation is tracked over time, showing the response to the explosion pressure time­history,
see the red curve in Figure 8­68. It is noted that the maximum deformation of the wall plate ‘lags’ behind
the explosion pressure because of inertia in the dynamic analysis. The peak load is at ~0.05 s, but the peak
displacement occurs at ~0.07 s.
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8.8.9 Results
Also in the plot is the response curve from a static analysis with the same time­history pressure, i.e. an
analysis of the same system, but without time integration (no dynamic effects). The static analysis is using
the same time reference for the loading, but the time value is arbitrary, or not physical. It is seen that the
deformation response in the static analysis follows the first peak of the pressure history exactly (~0.05 s).
Similar curves for transient, dynamic analyses with only one half and one quarter of the steel density are
seen to position themselves between the other two curves, so the transient solution resembles the static
analysis more as the system mass is reduced.
This comparison illustrates the dynamic effects in a transient, dynamic analysis with a short­duration load,
and the importance of modelling the system mass correctly. For a blast wall, this means, as mentioned
previously, that the analyst should take care to include all relevant masses, like insulation and architectural
cladding.
8.8.9.2 Stress results
Looking at the overall stress levels in the wall strip (at the time of maximum stress), it is seen that the
bracket and plate are experiencing relatively high stresses. Stresses in the plate are concentrated near the
top where the stiffener ends and at the mid­span, symmetry boundary. The bracket shows peak stresses
along the bend and along the weld closest to the HEM girder.
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Figure 8­68 Max. deformation curves, transient and static responses
Looking closer at the bracket in Figure 8­70, it is seen that tensile stresses develop at the top of the
horizontal part and on the ‘inside’ of the vertical part, which corresponds well with the deformation pattern.
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Figure 8­69 Equivalent stress in wall strip (at time of maximum stress)
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Figure 8­70 Tensile stress in bracket (displacement scale =3.0)
The stiffener experiences high stresses at the mid­span due to buckling, see Figure 8­71. There is no tensile
failure in the middle of the stiffener plate, and there is no plastic strain in the wall plate near the mid­span
location, so the explosion scenario is not limited by the effects from stiffener buckling.
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Figure 8­71 Stress at stiffener (buckling). Displacement scale =1.0
8.8.10 Tensile failure ­ critical strain
In the tensile failure check the plate and bracket is checked against the critical maximum (plastic) principal
strain determined in [8.8.8].
The plate shows a local, maximum principal plastic strain of 0.055 at the end of the girder, near the bracket
connection, see Figure 8­72. With the strain being as localized as this, the result shall be checked against the
local yielding criterion for calibration case CC01. The local strain criterion is determined to 0.119 in [8.8.8],
which is larger than the calculated strain.
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The bracket shows a uniform line of plastic strain at the border between the first and the second element row
along the first girder weld. The maximum principal plastic strain, located right at the bend in the bracket, is
0.038, which is below the critical strain level of 0.119, found in CC01 for local yielding, and obviously also
well below the criteria for bending, see [8.8.8].
Figure 8­73 Local yielding in bracket along first girder weld and at bend
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Figure 8­72 Local principal strain in plate, max. value of 0.055
The wall plate is connected to the top and bottom girders using L­brackets with lap joints, i.e. 4 welds per
bracket, see Figure 8­74. The welds are assessed according to the recommendations in [5.1.5].
Figure 8­74 Naming of fillet welds (bonded contact pairs)
The welds are not modelled, but are instead represented by line­to­line bonded contact pairs. The maximum
total reaction force (global XYZ components) is extracted in ANSYS for each weld/contact pair, see Table
8­21.
Table 8­21 Weld reaction forces
Fx [kN] *
Fy [kN]
Fz [kN]
Resultant [kN]
Weld A
­0.57
­7.79
129.31
129.55
Weld B
­3.47
­18.12
219.46
220.24
Weld C
0.15
­137.24
­15.24
138.09
Weld D
0.07
187.57
­118.54
221.89
(Loads are reported for different time steps for each individual weld, and do not represent a simultaneous
load balance)
*: (The minimal transverse reactions are likely due to the buckled stiffener)
It is seen that weld B, the first weld between plate and bracket, and weld D, the first weld between girder
and bracket, transfer the largest total forces, see Figure 8­75 and Figure 8­76.
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8.8.11 Weld check
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Figure 8­75 Weld B reactions
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Figure 8­76 Weld D reactions
Weld D is checked using the “simplified method for design resistance of fillet weld”, Section 4.5.3.3 in /4/.
According to this, the design resistance of a fillet weld may be assumed to be adequate if, at every point
along its length, the resultant of all the forces per unit length transmitted by the weld satisfy the following
criterion:
where:
Fw,Ed is the design value of the weld force per unit length;
Fw,Rd is the design weld resistance per unit length.
Since the wall strip is 500 mm wide in the FE model, the weld force per unit length is:
The above calculation expects the forces to be uniformly distributed along the weld, and for the present
symmetry/strip model this is judged to be an adequate assumption. If peaks are experienced along the weld,
these should be taken into consideration.
These weld forces are taken from an analysis with a tensile failure safety factor of 1.2 applied to the pressure
load. This factor is not required for the weld check, but the forces are used as­is for the present example.
However, due to the non­linear, dynamic nature of the analysis it cannot be assumed that the weld forces can
simply be divided by 1.20 to get unfactored forces. The proper way to get unfactored weld forces is to re­run
the analysis without the factor applied to the pressure time­history.
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where:
a is the fillet weld size = 7 mm
fu is the nominal tensile strength of the weaker part joined (S355) = 470 MPa
It is seen that the fillet weld resistance is higher than the maximum acting forces, and the welds are all ok.
8.9 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 T­joint subjected to an out­of­plane
fully reversible load of ±60 kN. The objective of the analysis is to estimate the design life based on the
recommendations in [5.2.5]. The assumed geometry and dimensions are given in Table 8­22 and Figure
8­77.
Table 8­22 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
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With the fillet weld size equal to a = 7 mm, the design resistance per unit length is calculated as:
It is assumed that the cyclic stress­strain behaviour is well described by the Ramberg­Osgood relationship:
The values for the Ramberg­Osgood parameters are presented in Table 8­23 for the chord and the brace.
Table 8­23 Ramberg­Osgood parameters
K' [MPa]
n'
Chord
731.7
0.096
Brace
699.5
0.108
In order to obtain the cyclic strains a finite element analysis is carried out using the FEM­software ABAQUS.
An 8­noded shell element (S8R) model is established with load and support conditions as shown in Figure
8­78.
The chord is constrained at each end for all translational and rotational degrees of freedom. The out­of­plane
load was applied by means of a reference point located at the cross section centre of the brace end. This
reference point is connected to the circumference of the brace end by means of kinematic coupling. The load
was applied using three steps as illustrated in Figure 8­79. FE element analysis of additional load steps gives
similar results.
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Figure 8­77 Geometry of test example, dimensions in mm
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Figure 8­78 Boundary and loading conditions for tubular joint
Figure 8­79 Load steps
Figure 8­80 (a) shows an overview of the finite element mesh. Figure 8­80 (b) shows a close­up of the
brace­chord intersection area. The finite element mesh in the hot spot region is in accordance with the
recommended practice DNVGL­RP­C203 for tubular joints. In this example the element nodes coincide
with the specified extrapolation points (a and b) as given below. Hence, nodal values are applied in the
extrapolation procedure for calculating the hot spot strain range. For chord side failure, the distance from the
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Figure 8­80 Left: (a) Meshed model, Right: (b) Close­up of brace­chord intersection area
Figure 8­81 shows the principal strain range due to the out­of­plane cyclic loading.
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hot spot to the first extrapolation point, a is obtained by means of the equation below. The distance to the
second extrapolation point, b is obtained by the equation below.
The hot spot strain range is obtained according to the following procedure:
1)
Establish the total strain range components (
) by subtracting the minimum strain values
of load step 2 by the maximum values of load step 3. In ABAQUS this is done by using the “Create Field
Output” option.
2)
Calculate the extrapolated strain range for each strain component (
3)
the equation below.
st
Calculate the 1 principal hot spot strain range
) by means
, see equation below.
The resulting hot spot strain range is 0.0048.
Air environment is assumed. The thickness of the chord member is below the reference thickness of 25 mm,
so the thickness effect does not need to be taken into account. Hence, the characteristic design life due to
the cyclic loading is obtained by solving the following equation, see [5.2.5]:
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Figure 8­81 Maximum principal strain range
In this example a low cycle fatigue analysis of a plate with a circular hole subjected to cyclic displacement of
1.0 mm is presented. The objective of the analysis is to estimate the design life based on recommendations
in [5.2.6]. The dimensions of the plate are presented in Figure 8­82. The plate material is of grade S355 and
the cyclic stress­strain curve is obtained from Table 5­4. Elastic modulus of 210 000 MPa and Poisson’s ratio
of 0.3 is assumed.
The finite element analysis is carried out using ABAQUS. The plasticity is specified using the combined
hardening option. The plastic hardening is specified using the half cycle option where the stress/plastic strain
relation is tabulated according to specifications given in [5.2.4]. The number of back stresses is set to 10.
Figure 8­82 Geometry of considered specimen
The maximum principal strain range is obtained by performing a finite element analysis with the FEM­
software ABAQUS. The finite element analysis is performed with 8­noded shell elements with reduced
integration (S8R). The boundary conditions and cyclic displacement is applied as illustrated in Figure 8­84.
Note that a total of 6 complete load cycles is specified in the analysis in order to see how the stress strain
hysteresis curve developed.
Figure 8­83 Geometry and loading
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8.10 Example: Low cycle fatigue analysis of plate with circular hole
The maximum principal strain range is obtained according to the following procedure:
1)
Perform a strain convergence study. The mesh around the hole is refined until the strain value in the
relevant nodal point converges. Based on the convergence study it was found sufficient to use 48
elements around the hole.
2)
Establish the strain component ranges (
, etc.) by subtracting the strain component values
of load step 12 from the values of load step 13. Hence, the stress/strain output from the last cycle is
used as basis for calculating the design fatigue life. In ABAQUS the strain range is obtained by using the
“Create Field Output” option.
Calculate the maximum principal strain range based on the strain component ranges.
Calculate the design fatigue life based on the seawater with cathodic protection curve.
3)
4)
Figure 8­85 shows the maximum principal strain range due to the specified cyclic loading of the plate. The
hysteresis loop in the location adjacent to the hole with the highest cyclic strain is plotted in Figure 8­86. The
maximum principal strain range of Δεl = 0.011 obtained from the last half cycle in the finite element analysis
is used as basis for calculating the design fatigue life. The maximum principal strain range of Δεl = 0.011 is
considered to be a representative value based on the first cycles. By solving Equation (14) in [5.2.6] a design
fatigue life of N = 139 is obtained.
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Figure 8­84 Load steps
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Figure 8­85 Equivalent strain range
Figure 8­86 Stress versus strain in y – direction (parallel to the 1st principal strain direction)
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A.1 Element library of offshore supply vessels
The joint industry project that established this RP prepared structural models for offshore supply vessels
(OSV). The models are described in http://rules.dnvgl.com/docs/pdf/DNVGL/RP/2016­09/DNVGL­2015­0984­
rev1.pdf . The models are prepared in Abaqus and LS­Dyna formats that can be downloaded from http://
rules.dnvgl.com/docs/pdf/DNVGL/RP/2016­09/RP­C208­Ship­model­Library­Abaqus.zip and http://
rules.dnvgl.com/docs/pdf/DNVGL/RP/2016­09/RP­C208­Ship­model­Library­LS­Dyna.zip .
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APPENDIX A STRUCTURAL MODELS FOR SHIP COLLISION
ANALYSES
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Changes – historic
CHANGES – HISTORIC
There are currently no historical changes for this document.
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