RECOMMENDED PRACTICE
DNV-RP-C208
Determination of Structural Capacity by
Non-linear FE analysis Methods
JUNE 2013
The electronic pdf version of this document found through http://www.dnv.com is the officially binding version
DET NORSKE VERITAS AS
FOREWORD
DNV is a global provider of knowledge for managing risk. Today, safe and responsible business conduct is both a license
to operate and a competitive advantage. Our core competence is to identify, assess, and advise on risk management. From
our leading position in certification, classification, verification, and training, we develop and apply standards and best
practices. This helps our customers safely and responsibly improve their business performance. DNV is an independent
organisation with dedicated risk professionals in more than 100 countries, with the purpose of safeguarding life, property
and the environment.
DNV service documents consist of among others the following types of documents:
— Service Specifications. Procedural requirements.
— Standards. Technical requirements.
— Recommended Practices. Guidance.
The Standards and Recommended Practices are offered within the following areas:
A) Qualification, Quality and Safety Methodology
B) Materials Technology
C) Structures
D) Systems
E) Special Facilities
F) Pipelines and Risers
G) Asset Operation
H) Marine Operations
J) Cleaner Energy
O) Subsea Systems
U) Unconventional Oil & Gas
© Det Norske Veritas AS June 2013
Any comments may be sent by e-mail to rules@dnv.com
This service document has been prepared based on available knowledge, technology and/or information at the time of issuance of this document, and is believed to reflect the best of
contemporary technology. The use of this document by others than DNV is at the user's sole risk. DNV does not accept any liability or responsibility for loss or damages resulting from
any use of this document.
Recommended Practice DNV-RP-C208, June 2013
Changes – Page 3
CHANGES – CURRENT
General
This is a new document.
Acknowledgement
This Recommended Practice is developed in a Joint Industry Project that was initiated through a pre-project
supported by Statoil and DNV. The Joint Industry Project was sponsored by the following companies and
institutions (in alphabetic order):
ConocoPhillips Skandinavia AS
Det Norske Veritas AS
Maersk Olie og Gas AS
Petroleum Safety Authority Norway
Statoil ASA
Total E&P Norge AS
In addition to their financial support the above companies are also acknowledged for their technical
contributions through their participation in the project.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Contents – Page 4
CONTENTS
1.
Introduction.......................................................................................................................................... 5
1.1 Objective of the document .....................................................................................................................5
1.2 Validity ...................................................................................................................................................5
1.3 Definitions ..............................................................................................................................................5
1.4 Symbols...................................................................................................................................................6
2.
Basic Considerations............................................................................................................................ 8
2.1 Limit state safety format .........................................................................................................................8
2.2 Characteristic resistance..........................................................................................................................9
2.3 Type of failure modes .............................................................................................................................9
2.4 Use of linear and non-linear analysis methods .......................................................................................9
2.5 Empirical basis for the resistance ...........................................................................................................9
2.6 Ductility ..................................................................................................................................................9
2.7 Serviceability Limit State .....................................................................................................................10
2.8 Permanent deformations .......................................................................................................................10
3.
General requirements ....................................................................................................................... 11
3.1 Definition of failure ..............................................................................................................................11
3.2 Modelling .............................................................................................................................................11
3.3 Determination of characteristic resistance taking into account statistical variation .............................11
3.4 Requirement to the software .................................................................................................................12
3.5 Requirement to the user ........................................................................................................................12
4.
Requirements to The FEM Analysis ............................................................................................... 13
4.1 General..................................................................................................................................................13
4.2 Selection of FE software.......................................................................................................................13
4.3 Selection of analysis method ...............................................................................................................13
4.4 Selection of elements ............................................................................................................................14
4.5 Mesh density .........................................................................................................................................15
4.6 Geometry modelling .............................................................................................................................15
4.7 Material modelling................................................................................................................................16
4.8 Boundary conditions .............................................................................................................................22
4.9 Load application....................................................................................................................................22
4.10 Application of safety factors.................................................................................................................23
4.11 Execution of non-linear FE analyses, quality control...........................................................................23
4.12 Requirements to documentation of the FE analysis..............................................................................23
5.
Representation of different failure modes ....................................................................................... 24
5.1 Design against tensile failure ................................................................................................................24
5.2 Failure due to repeated yielding (low cycle fatigue) ............................................................................26
5.3 Accumulated strain (“Ratcheting”).......................................................................................................30
5.4 Buckling................................................................................................................................................31
5.5 Repeated buckling.................................................................................................................................36
6.
Bibliography ....................................................................................................................................... 37
Appendix A. Commentary.......................................................................................................................... 39
Appendix B. Examples................................................................................................................................ 42
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.1 Introduction – Page 5
1 Introduction
1.1 Objective of the document
This document is intended to give guidance on how to establish the structural resistance by use of non-linear
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 offshore standards.
Non-linear effects that may be included in the analyses are e.g., material and geometrical non-linearity, 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 the characteristic resistance is similar to
what is required by many other structural standards using the limit state safety format and these
recommendations are expected to be valid for determination of capacities to be used with such standards.
The recommendations are foreseen to be used for cases where the characteristic resistance is not directly
covered by codes or standards. The objective 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
standard.
This document is not intended to replace formulas for resistance in codes and standards for the cases where
they are applicable and accurate, but to present methods that allows for using non-linear FE-methods to
determine resistance for cases that is not covered by codes and standards or where accurate recommendations
are lacking.
1.2 Validity
The document is valid for marine structures made from structural steels meeting requirements to offshore
structures with a yield strength of up to 500 MPa.
The recommendations presented herein are adapted to typical offshore steels that fulfil the requirements
specified in DNV-OS-C101 /14/ 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, and failure due to
repeated loading from moderate loads (fatigue) needs to be checked separately. See DNV-RP-C203 /17/.
1.3 Definitions
This Recommended Practise uses terms as defined in DNV-OS-C101. Additional terms are defined below:
characteristic
resistance
conservative load
ductility
engineering shear
strain
equivalent strain
follower load
gross yielding
low cycle fatigue
the resistance that for a particular failure mode is meeting the requirement of having a
prescribed probability that the resistance falls below a specified value (usually 5% fractile)
load that maintains its orientation when the structure deforms (e.g., gravity loads)
the ability to deform beyond the proportionality limit without significant reduction in the
capacity due to fracture or local buckling (originally ductility refers to the behaviour of
the material, but is here also used for the behaviour of structures and structural details)
=2∙
=
1
1+
1
2
=2∙
−
2
+
−
=2∙
2
+
−
2
+3
2+
2+
2
load that changes direction with the structure (e.g., hydrostatic pressure)
yielding across the entire structural component e.g., a flange
the progressive and localised damage caused by repeated plastic strain in the material.
Low cycle fatigue assessments are carried out by considering the cyclic strain level.
net area
area of a cross-section or part of a cross-section where the area of holes and openings are
subtracted
net section ratio
the ratio between the net area and the gross area of the tension part of a cross-section
redundant structure a structure may be characterized as redundant if loss of capacity in one of its structural
elements will lead to little or no reduction in the overall load-carrying capacity due to load
redistribution
shake down
shake down is a state in which a structure after being loaded into the elasto-plastic range
will behave essentially linear for all subsequent cycles
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.1 Introduction – Page 6
1.4 Symbols
b
span of plate
c
flange outstand
C
damping matrix
CFEM
resistance knock down factor
D
outer diameter of tubular sections
E
modulus of elasticity
1
2
stress-strain curve parameter
stress-strain curve parameter
external forces
internal forces
yield stress/yield strength
K
Ramberg-Osgood parameter
!"
M
eigenvalue for governing buckling mode
mass matrix
N
#$
number of cycles to failure
#
design resistance
%
t
design action effect
u
displacement vector
%$
characteristic element size of smallest element
characteristic resistance
characteristic action effect
time, thickness
strain
&'
′
critical strain
engineering (nominal) strain
equivalent strain
fatigue ductility coefficient
εp_ult
εp_y1
stress-strain curve parameter
Δ ℎ"
true (logarithmic) strain
')
Δ ,
stress-strain curve parameter
fully reversible maximum principal hot spot strain range
Δ
fully reversible local maximum principal strain range
.̅
partial factor for actions
-
0
11 ,12
1#
time step
material factor
reduced slenderness
Poisson’s ratio = 0.5 for plastic strain
density
principal stresses
representative stress
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.1 Introduction – Page 7
1
1′
engineering (nominal) stress
1 ,
linearized buckling stress disregarding local buckling modes
1),
true (Cauchy) stress
1
1
fatigue strength coefficient
1 ')
stress-strain curve parameter
critical buckling stress
1 '2
linearized local buckling stress
1
stress-strain curve parameter
1
,$
,$ 2
stress-strain curve parameter
stress-strain curve parameter
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.2 Basic Considerations – Page 8
2 Basic Considerations
2.1 Limit state safety format
A limit state can be defined as: “A state beyond which the structure no longer satisfies the design 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 codes 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 (low cycle fatigue).
The safety format that is used in limit state codes is schematically illustrated in Figure 2-1.
Sd< Rd
Figure 2-1
Illustration of the limit state safety format
The requirement can be written as:
Sd ≤ Rd
(1)
Sd = Sk γf Design action effect
Rd = Rk / γM Design resistance
Sk = Characteristic action effect
γf = Partial factor for actions
Rk = Characteristic resistance
γM = Material factor.
It can be seen from this figure that it is important that the uncertainty in the resistance is adequately addressed
when the characteristic resistance is determined.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.2 Basic Considerations – Page 9
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 prevents an adequate statistical
evaluation so the 5% must be seen as a goal for the engineering judgments that in such cases are needed.
The characteristic resistance given in design codes is determined also on the basis of consideration of other
aspects than the maximum load carrying resistance. Aspects like post-peak 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 the 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 Type of failure modes
When steel structures are loaded to their extreme limits they will either fail by some sort of instability (e.g.,
buckling) that prevents further loading or by tension failure or a combination of the two. For practical cases 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 limiting the plastic strain to avoid cyclic failure
for dynamically loaded structures or deformation limit for structural details failing by plastic strain in
compression. See Section [3.1].
The following types of failure modes are dealt with in this Recommended Practice:
— tensile failure
— failure due to repeated yielding (low cycle fatigue)
— accumulated plastic strain
— buckling
— repeated buckling.
2.4 Use of linear and non-linear analysis methods
Traditionally, the ultimate strength of offshore structures are analysed by linear methods to determine the
internal distribution of forces and moments, and the resistances of the cross-sections are checked according to
design resistances found in design codes. These design resistance formulas often require deformations well into
the inelastic range in order to mobilise the code 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.
2.5 Empirical basis for the resistance
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 codes, 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 non-linear FE methods build on
knowledge that is empirically based. That can be achieved by calibration of the analysis methods to
experimental data, to established practise as found in design codes or in full scale experience.
2.6 Ductility
The integrity of a structure is also influenced by other factors than the value of the characteristic resistance. 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 load-bearing resistance. Such loss of resistance can be due to tensile failure,
instability of cross-sectional parts or member buckling. The design codes give little explicit guidance on this
issue, with exception for stability of cross-sectional parts in yield hinges, which normally are covered by
requirement to cross-sectional class 1. See e.g. DNV OS-C101 /14/.
Steel structures generally behave ductile when loaded to their limits. The established design practise is 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 for use of linear analyses for ULS checks even for structures which behave significantly nonlinear when approaching their ultimate limit states.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.2 Basic Considerations – Page 10
2.7 Serviceability Limit State
Use of non-linear 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. E.g., when plate elements are used beyond their critical load, out of 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 non-linear when loaded to their ultimate limit. The formulas for design
resistance in DNV Offshore Standard /14/ or similar codes and standards are therefore developed on the basis
that permanent deformation may take place before the characteristic resistance is reached.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.3 General requirements – Page 11
3 General requirements
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 loose is load-carrying 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 loadcase, will not lead to a progressive or cyclic failure.
Alternatively a specific check for these failure modes can be carried out. See also [5.2]. Another example is 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
It should be checked that the analysis tool and the modelling adopted represent the non-linear 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 that are excluded from the
analysis and that are assumed to be checked by other methods.
3.3 Determination of characteristic resistance taking into account statistical variation
When FE methods are used to determine the structural resistance it is necessary to take due account of 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 code.
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.
It is therefore necessary to validate the analysis procedure according to one of the following methods:
a) Selection of all governing parameters to be characteristic or conservative values.
In this method all parameters that influence the result (key parameters) are selected to give results to 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.
In some cases values are given in the codes for analysis of specific problems see e.g., [5.4.3].
b) Validation against code values
In this method a selected code case is used for calibration (denoted code 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, imperfections, residual stresses etc. should be selected so the analysis provide the resistance
predicted by the code for the code calibration case. The same parameters are then used when the resistance
of the actual problem is determined.
If the analysis is calibrated against ordinary code values that meet the requirements to characteristic
resistance then the resistance of the analysed structure also will meet the requirement.
c) 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
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.3 General requirements – Page 12
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 /3/ or in
ISO 19902 /10/.
3.4 Requirement to the software
The software used shall be documented and tested for the purpose.
3.5 Requirement to the user
The user should be familiar with FE methods in general and non-linear methods in particular.
The analyst needs to understand the structural behaviour of the problem in question.
The user shall know the theory behind the methods applied as well as the features of the selected software.
When documenting structures to meet a code described reliability level with use of non-linear methods for
determination of the resistance it is necessary the engineers understand the inherent safety requirements of the
governing code.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.4 Requirements to The FEM Analysis – Page 13
4 Requirements to The FEM Analysis
4.1 General
The term non-linear FE analysis covers a large number of analysis types for different purposes and objects. 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 Offshore Standards and other similar standards, such as the Norsok Nseries, /11/ to /13/, and the ISO 19900 suite of standards /9/.
4.2 Selection of FE software
The software must be tested and documented suited for analysing the actual type of non-linear behaviour. This
includes:
— non-linear material behaviour (yielding, plasticity)
— non-linear geometry (Stress stiffening, 2nd order load effects).
Other types of non-linearity that may need to be included are:
— contact
— temperature effects (e.g. material degradation, thermal expansion)
— non-linear load effects (e.g. follower loads).
4.3 Selection of analysis method
4.3.1 Implicit versus explicit solver
Both implicit and explicit equation solvers may be used to solve the general equation system:
-)3
+ 4)5
+
∆ =
!"
0
= !" 7
4
=
(2)
Where
M is the mass matrix
C is the damping matrix
u is the displacement vector
Fint is the internal forces
Fext is the external forces.
In dynamic analyses, explicit solvers are attractive for large equation systems, as the solution scheme does 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.
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 non-linear problems, implicit Newton Raphson
methods are well suited, gradually incrementing the time and iterate to convergence for each time step.
4.3.2 Solution control for explicit analysis
Most explicit FE codes calculate the governing size of the time step based on equations similar to Equation (3).
For problems of longer duration, one often wants to save analysis time by reducing the number of time steps.
This can be done by accelerating the event or mass scaling. Accelerating the event reduces the simulation time
and thus computational time, the mass scaling increases the time step reducing the computational time, see
Equation (3).
The time saving methods are only useful if the inertia forces are small. Thus, it must be documented that the
kinetic energy is small compared to the deformation energy (typically less than 1%) when explicit analyses are
used to find quasi-static response.
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.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.4 Requirements to The FEM Analysis – Page 14
4.3.3 Solution control for dynamic implicit analysis
A large number of time integration procedures exists (e.g. The Newmark family of methods and the α-Method).
For non-linear analyses they should be used in combination with Newton iterations. As a rule of thumb the 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 α-method (HHT method) ref. /27/ the parameters α, β and γ can be selected by the user. The method is
unconditionally stable if:
8=
1
1
1
1 − : 2 , = − : and − ≤ : ≤ 0
4
2
3
(4)
Selecting α less than 0 gives some numerical damping. In order to avoid “noise” from high frequency modes,
parameters that give some numerical damping can be useful. Table 4-1 presents some combinations of
parameters that give unconditional stability.
Table 4-1 Combinations of α, β and γ for unconditional stability
α
β
γ
Comment
0
0.25
0.5
Trapezoidal rule, no numerical damping
-0.05
0.2756
0.55
Numerical damping
-0.1
0.3025
0.6
Numerical damping
4.3.4 Solution control for static implicit analysis
In case the dynamic effects are not important, the equation system to solve may be reduced to:
=
(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 peak-load 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 snap-back problems (passing turning point), or limit/bifurcation point problems that cannot be analysed
using displacement control, an “arc length” method is needed.
Figure 4-1
Limit, Bifurcation and Turning points
4.4 Selection of elements
The selection of element type and formulation is strongly problem dependent. Points to consider are:
— shell elements or solid elements
— elements based on constant, linear or higher order shape functions
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.4 Requirements to The FEM Analysis – Page 15
— full vs. reduced, vs. hybrid integration formulations
— number of through thickness integration points(shell)
— volumetric locking, membrane locking and transverse shear locking
— hourglass control/artificial strain energy (for reduced integration elements).
In general higher order elements are preferred for accurate stress estimates; elements with simple shape
functions (constant or linear) will require more elements to give the same stress accuracy as higher order
elements. Constant stress elements are not recommended used in the area of interest.
Some types of elements are intended as transition elements in order to make the generation of the element mesh
easier and are known to perform poorly. Typically 3-noded plates/shells and 4-noded tetrahedrons are often
used as transition elements. This type of elements should if possible be avoided in the area of interest.
Proper continuity should be ensured between adjacent elements if elements of different orders are used in the
same model.
For large displacements and large rotation analyses, simple element formulations give a more robust numerical
model and analysis than higher order elements.
Care should be taken when selecting formulations and integration rules. Formulations with (selective) reduced
integration rules are less prone to locking effects than full integrated simple elements; however the reduced
integration elements may produce zero energy modes (“hourglassing”) and 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%).
4.5 Mesh density
4.5.1 General
The element mesh should be sufficiently detailed to capture the relevant failure modes:
— For ductility evaluations, preferably several elements should be present in the yield zone in order to have
good strain estimates.
— 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.
The element aspect ratio should be according to requirements for the selected element formulation in the areas
of interest.
Care is required in transitioning of mesh density. Abrupt transitioning introduces errors of a numerical nature.
Load distribution and load type also have an influence on the mesh density. Nodes at which loads are applied
need to be correctly located, and in this situation can drive the mesh design, at least locally.
4.5.2 Mesh refinement study
Often it will be necessary to run mesh sensitivity studies in order to verify that the results from the analyses 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 Appendix [B.2].
4.6 Geometry modelling
Geometry models for FE analyses often need to be simplified compared to drawings of the real structure.
Typically small details need to be omitted as they interfere with the goal of having a good regular element mesh.
The effect the simplifications may have on the result should be evaluated. Typical simplifications include:
— Cut-outs or local reinforcements are not included
— Eccentricities are not included for beam elements or in thickness transitions in shell models
— Weld material is not included
— Welded parts are modelled as two parts and joined using contact surfaces.
For buckling analyses it is necessary to introduce equivalent geometric imperfections in order to predict the
buckling capacity correctly, see Section [5.4]. A common way to include the imperfections is to use one or
more of the structures eigenmodes and scale these such that the buckling capacity is predicted correctly for the
calibration model.
For problems where the geometry of the real structure deviates from the theoretical one, the analysis needs to
reflect that possible geometrical tolerances may have impacts on the result. Example is fabrication tolerances
of surfaces transferring loads by contact pressure.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.4 Requirements to The FEM Analysis – Page 16
4.7 Material modelling
4.7.1 General
The selected material model should at least be able to represent the non-linear behaviour of the material both
for increasing and decreasing loads (unloading). In some cases the material model also needs to be able to
account for reversed loading.
The material model selected needs to be calibrated against empirical data (see [3.3]). 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.
4.7.2 Material models for metallic materials
For metallic materials time independent elasto-plastic 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 metals. 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 plastic strain increment a change in stress gives.
The yield surface function is often used as a flow potential (associated flow).
The von Mises yield function is considered suitable for most capacity analyses of steel structures.
The hardening rule is important for analyses with reversed loading due to the Bauschinger effect. A material
model with kinematic (or combined kinematic/isotropic) hardening rule should be used in such analyses.
Figure 4-2
The von Mises yield surface shown in the σ1-σ2 plane with isotropic (left) and kinematic (right) hardening models
Figure 4-3
Isotropic vs Kinematic hardening
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.4 Requirements to The FEM Analysis – Page 17
4.7.3 Stress strain measures
Stress and strain can be measured in several ways:
— From material testing the results are often given as “Engineering” stress-strain curves (calculated based on
the initial cross section of the test specimen).
— FE software input is often given as “True” stress-strain (calculated based on updated geometry).
— Other definitions of strains are also used in FE formulations, e.g., the Green-Lagrange strain, and the EulerAlmansi strain.
For small deformations/strains, all strain measures give similar results. For larger deformations/strains the
strain measure is important, e.g. the Green-Lagrange measure is limited to “small strains” only. Figure 4-4
shows a comparison of some strain measures. Limitations in the formulations on the use of the selected element
type should always be noted and evaluated for the intended analysis.
Figure 4-4
Comparison of some strain measures
The relationship between Engineering (Nominal) stress and True (Cauchy) stress (up to the point of necking) is:
σtrue = σeng 1 + εeng
(6)
εtrue = ln 1 + εeng
(7)
The relationship between Engineering (Nominal) strain and True (Logarithmic) strain is:
The stress-strain curve should always be given using the same measure as expected by the software/ element
formulation.
4.7.4 Evaluation of strain results
As element strain in FE- analyses is an averaged value dependent on the element type and element size, the
reported strain will always depend on the computer model. It is often necessary to re-mesh 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 ductility limits, the integration point strains or
extrapolated strains from integration points should be used.
4.7.5 Stress - strain curves for ultimate capacity analyses
When defining the material curve for the analysis, the following points should be considered:
— Characteristic material data should normally be used, see Section [3.3].
— The predicted buckling capacity will depend on the curve shape selected, thus equivalent imperfection
calibration analyses and final analyses should be performed using the same material curves.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.4 Requirements to The FEM Analysis – Page 18
— 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 EN-10025 /38/ and EN 10225 /39/ are
proposed in Table 4-2 to Table 4-5. These properties are assumed to be used with the acceptance criteria given
in this RP. Idealized material curves for steel materials delivered according to other codes e.g. DNV Offshore
Standards can be established by comparison with these curves. Figure 4-5 shows the parameters. Figure 4-6
shows the resulting curves for thicknesses less than 16 mm. The stress-strain values are given using the
engineering stress-strain measure. Table 4-6 to Table 4-9 and Figure 4-7 show the corresponding true stressstrain values.
Alternative bi-linear curves may be used for buckling problems e.g. as shown in Figure 5-6.
The curves should also be adjusted for temperature effects as appropriate.
σ
Ep2
Ep1
σult
σyield
σprop
σyield2
E
εp_y1 εp_y2
εp_ult
εp=0
Figure 4-5
Parameters to define stress – strain curves
DET NORSKE VERITAS AS
ε,εp
Recommended Practice DNV-RP-C208, June 2013
Sec.4 Requirements to The FEM Analysis – Page 19
Table 4-2 Proposed non-linear properties for S235 steels (Engineering stress-strain)
S235
Thickness [mm]
t ≤ 16
16 < t ≤ 40
E [MPa]
210000
σprop/σyield
0.9
Ep1/E
σprop [MPa]
σyield [MPa]
σyield2 [MPa]
σult [MPa]
0.001
211.5
202.5
193.5
235
225
215
238.4
228.4
218.4
360
360
360
εp_y1
0.004
εp_y2
0.02
εp_ult
0.2
Ep2/E
40 < t ≤ 63
0.0032
0.0035
0.0037
Table 4-3 Proposed non-linear properties for S355 steels (Engineering stress-strain)
S355
Thickness [mm]
t ≤ 16
16< t ≤ 40
E [MPa]
210000
σprop/σyield
0.9
Ep1/E
0.001
σprop [MPa]
σyield [MPa]
σyield2 [MPa]
σult [MPa]
εp_y1
εp_y2
εp_ult
Ep2/E
40 < t ≤ 63
319.5
310.5
301.5
355
345
335
358.4
348.4
338.4
470
470
450
0.004
0.02
0.15
0.0041
0.0045
0.0041
Table 4-4 Proposed non-linear properties for S420 steels (Engineering stress-strain)
S420
Thickness [mm]
t ≤ 16
16< t ≤ 40
E [MPa]
210000
σprop/σyield
0.9
Ep1/E
0.001
σprop [MPa]
σyield [MPa]
σyield2 [MPa]
σult [MPa]
378.0
360.0
351.0
420
400
390
421.3
401.3
391.3
500
500
480
εp_y1
0.004
εp_y2
0.01
εp_ult
0.12
Ep2/E
40 < t ≤ 63
0.0034
0.0043
DET NORSKE VERITAS AS
0.0038
Recommended Practice DNV-RP-C208, June 2013
Sec.4 Requirements to The FEM Analysis – Page 20
Table 4-5 Proposed non-linear properties for S460 steels (Engineering stress-strain)
S460
Thickness [mm]
E [MPa]
t ≤ 16
16< t ≤ 25
210000
σprop/σyield
0.9
Ep1/E
σprop [MPa]
σyield [MPa]
σyield2 [MPa]
σult [MPa]
0.001
414.0
396.0
373.5
460
440
415
461.3
441.3
416.3
540
530
515
εp_y1
0.004
εp_y2
0.01
εp_ult
0.1
Ep2/E
25< t ≤ 63
0.0042
0.0047
0.0052
Table 4-6 Proposed non-linear properties for S235 steels (True stress strain)
S235
t ≤ 16
16< t ≤ 40
40 < t ≤ 63
E [MPa]
210000
210000
210000
σprop [MPa]
σyield [MPa]
σyield2 [MPa]
σult [MPa]
εp_y1
εp_y2
εp_ult
211.7
202.7
193.7
236.2
226.1
216.1
243.4
233.2
223.0
432.6
432.6
432.6
0.0040
0.0040
0.0040
0.0198
0.0198
0.0198
0.1817
0.1817
0.1817
Thickness [mm]
Table 4-7 Proposed non-linear properties for S355 steels (True stress strain)
S355
Thickness [mm]
t ≤ 16
16< t ≤ 40
40 < t ≤ 63
E [MPa]
210000
210000
210000
σprop [MPa]
σyield [MPa]
σyield2 [MPa]
σult [MPa]
εp_y1
εp_y2
εp_ult
320.0
311.0
301.9
357.0
346.9
336.9
366.1
355.9
345.7
541.6
541.6
518.5
0.0040
0.0040
0.0040
0.0197
0.0197
0.0197
0.1391
0.1391
0.1392
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.4 Requirements to The FEM Analysis – Page 21
Table 4-8 Proposed non-linear 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]
σyield [MPa]
σyield2 [MPa]
σult [MPa]
εp_y1
εp_y2
εp_ult
378.7
360.6
351.6
422.5
402.4
392.3
426.3
406.0
395.9
561.2
561.2
538.7
0.0040
0.0040
0.0040
0.0099
0.0099
0.0099
0.1128
0.1128
0.1128
Table 4-9 Proposed non-linear properties for S460 steels (True stress strain)
S460
Thickness
t ≤ 16
16< t ≤ 25
25< t ≤ 63
E [MPa]
210000
210000
210000
σprop [MPa]
σyield [MPa]
σyield2 [MPa]
σult [MPa]
414.8
396.7
374.2
462.8
442.7
417.5
466.9
446.6
421.2
595.4
584.3
567.8
εp_y1
0.0040
0.0040
0.0040
εp_y2
0.0099
0.0099
0.0099
εp_ult
0.0948
0.0948
0.0948
Figure 4-6
Material curves according to Table 4-2 to Table 4-5 (Engineering stress-strain) (t< 16 mm)
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.4 Requirements to The FEM Analysis – Page 22
Figure 4-7
Material curves according to Table 4-6 to Table 4-9 (True stress-strain) (t< 16 mm)
4.7.6 Strain rate effects
The proposed material curves in Section [4.7.5] can be used for strain rates up to 0.1 s-1. For impact loads higher
strain rates may be experienced, and the increased strength and reduced ductility may be considered.
The Cowper-Symonds (CS) model is one model often used to simulate strain rate effects:
1$
1
5
IJ & = 1" I & K1 + L M N
4
(8)
As seen, the relative effect will be the same for all static stress (strain) levels. Thus the model must be calibrated
for the expected maximum stress (strain), otherwise the effect may be overestimated for large strains.
The constants C and p should be based on experiments and the maximum strain level expected. In lack of data,
C = 4000 [s-1] and p = 5 is proposed for common offshore steel materials.
4.8 Boundary conditions
The selected model boundary condition needs to represent the real condition in a way that will lead to results
that are accurate or to the safe side.
Often it is difficult to decide what the most “correct” or a conservative boundary condition is. In such cases
sensitivity studies should be performed.
4.9 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 non-linear 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. E.g. for an offshore structure subjected to both permanent loads (such as gravity and buoyancy) and
environmental loads (such as wind, waves and current); 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 nonconservative (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.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.4 Requirements to The FEM Analysis – Page 23
4.10 Application of safety factors
Applying load and resistance safety factors in a non-linear 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:
# >% ∙
!2I$ ∙ -I ' I,
(9)
where Rk is the characteristic resistance found from the analysis, and Sk is the characteristic load effect.
4.11 Execution of non-linear FE analyses, quality control
The following points should be considered in a quality control of non-linear FE analyses:
— boundary conditions
— calibration against known values.
— inertia effects in dynamic analyses
— element formulation/ integration rule suited for the purpose
— material model suited for the purpose
— mesh quality suited for the purpose, mesh convergence studies performed for stress strain results.
— equivalent imperfections calibrated for stability analyses
— time/load increments sufficient small, convergence studies performed
— numerical stability
— reaction corresponds to input
— convergence obtained for equilibrium iterations
— hourglass control for reduced integration, hourglass energy remains small
— sensitivity analysis both from idealisation and numerical points of views could be provided in particular
around singularities, for boundary conditions, etc.
— reference recommendations in Standards, Codes or Rules that are applicable directly to the studied system,
or to a similar system with different dimensions
— reference similar analyses for systems or subsystems that are validated from analytical or experimental
sources
— evaluation of analysis accuracy based on performed sensitivity studies.
4.12 Requirements to documentation of the FE analysis
The analysis should be documented sufficiently detailed to allow for independent verification by a third party,
either based on review of the documentation, or using independent analyses. The documentation should 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.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.5 Representation of different failure modes – Page 24
5 Representation of different failure modes
5.1 Design against tensile failure
5.1.1 General
An accurate analysis of tensile failure is demanding as numerous factors affect the problem and determination
of the results from the analysis is highly influenced on how the analysis is carried out.
The recommendations given in this document are not valid for failure that is related to unstable fracture due 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].
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 code methods
based on the forces carried by the welds. See [5.1.4].
5.1.2 Tensile failure resistance from non-linear analysis calibrated against a known solution
The most accurate method to check a structure against tensile failure is by calibrating the non-linear FE 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 code) 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. 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 some selected situations if a calibrated
solution as given in 5.1.2 is not attainable. The simplified check of tensile failure is a two-step check:
i) tensile failure due to gross yielding along the failure line (see [5.1.3.2])
ii) tensile failure due to cracks starting from local strain concentrations (see [5.1.3.3]).
Tensile failure in structures modelled by beam elements is best checked on the basis of the total deflection e.g.
as given in DNV-RP-C204 /18/.
This method is valid for structures made with typical offshore steel that will meet requirements to ductility and
toughness. The structural details need to meet fabrication requirements for offshore steel structures.
The safety factors that should be used for tensile fracture according to this procedure should apply an additional
safety factor γtf = 1.2 compared with standard material factor of the code.
The analyses should apply stress-strain relationship according to [4.7.2].
As the calculated plastic strain in the non-linear FE analysis will always involve inaccuracies depending upon
mesh density, element type, material model etc., the analyst must assure that any deviation from a correct result
is to the safe side. A mesh convergence study can be used to check that satisfactory accuracy is obtained. If it
is found impractical to perform the analysis with the required mesh density, larger elements can be used as long
as the calculated capacity is reduced by e.g. applying a resistance knock down factor CFEM or by reducing the
material key parameters. Such factors should be determined for the actual problem by means of a mesh
convergence study.
5.1.3.2 Strain limits for tensile failure due to gross yielding of plates
This method assumes that a failure line across the plate being analysed can be identified. Along this failure line
the plastic strain can be calculated as an averaged linear strain (gross yield strain) that can be compared with
critical values given in Table 5-1. With the gross yield strain is understood a plastic strain that is averaged over
a length in the direction of the largest principal strain and is linearized (axial and a bending components) in the
transverse directions.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.5 Representation of different failure modes – Page 25
The strain shall be calculated as the maximum principal plastic strain along the likely failure line. The failure
line should be assumed as a straight line.
Local cut outs or holes need to be included in the model if the ratio of the net area to the gross area of the tension
part is less than specified in Table 5-1. Local strains caused by non-loaded attachments like doubler plates or
brackets do not need to be considered.
The strains caused by tension stresses both due to axial tension, in-plane and out-of-plane bending and in-plane
shear should be extrapolated based on the linearized distributed maximum principal stress to the plate edge
corners both transverse to the plate as well as through thickness. The strain can be taken as the average value
over a length lavg given in Equation (10) in the direction of the maximum principal strain.
,IP =
Where:
16
3
−
R
3
S) ,IP ≥ 2
(10)
tp = plate thickness
wp = plate width
Shear strain due to out-of-plane bending should be checked separately as averaged engineering shear across the
failure line. The critical engineering shear strains can be taken as equal to the maximum principal strains given
in Table 5-1.
The linearized averaged plastic strain can be found by curve fitting a straight line using the method of least
squares.
In cases where the direction of the maximum principal plastic strain significantly varies along the failure line
it is recommended to consider different failure line directions.
Table 5-1 Critical strain and net area ratio for uniaxial stress state 1), 2)
Maximum principal plastic linearized strain 1)
S235
S355
S420
Critical gross yield strain
0.05
0.04
0.03
Net section ratio
0.94
0.95
0.96
1)
2)
S460
0.03
0.97
The strain can be calculated as average values over a length (in the direction of the principal strain) equal to the thickness for
in-plane bending and up to 5 times the thickness for pure membrane strains.
Any strain due to cold-forming should be added to the calculated plastic strain considering the direction of the plastic strain due
to cold forming
5.1.3.3 Strain limits for tensile failure due to local yielding in plane plates
The danger of cracks developing in ductile materials due to local concentrated yielding can be checked by the
following simplified procedure.
The local gross yielding can be checked by averaging the strain over a rectangular prismatic volume at the
location with the largest strain. The volume should be taken through the thickness (t) of the plate and should
extend from t to 5*t in the other directions. Where strain gradients are present due to changes in the cross
section or holes the length of the averaging volume should not be larger than a 25% of the length or width of a
notch or 20% of the diameter of a hole for problems dominated by in-plain strains. In case of out-of-plane
bending the length and width of the averaging volume should be taken equal to the thickness. See Figure 5-1.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.5 Representation of different failure modes – Page 26
t
Figure 5-1
Example of rectangular prism for check of local strains
The strain can be calculated as the average value in the direction of the maximum principal strain and should
be linearized (axial and bending component) in the other two directions. The corner with the largest strain
should be compared with the critical strains given in Table 5-2.
For cases with out-of-plane bending the engineering shear strain in the direction normal to the plate should be
checked against the critical values in Table 5-2. The engineering shear strain can be taken as the average
engineering shear strain across the thickness for all cross-sections within the prismatic volume.
Table 5-2 Critical local maximum principal plastic strain for uniaxial stress states 1)
Maximum principal plastic critical strain
S235
S355
S420
Critical local yield strain
0.15
0.12
0.10
1)
S460
0.09
Any strain due to cold-forming should be added to the calculated plastic strain considering the direction of the plastic strain due
to cold forming
5.1.4 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 code requirements e.g. EN 1993-1-8 /7/ or the relevant
code 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 code
requirements e.g. EN 1993-1-8 /7/ or the relevant code for the problem at hand.
Normally it is required that in welded connections the welds are stronger than the base material (overmatch).
See also Section [2.6].
5.2 Failure due to repeated yielding (low cycle fatigue)
5.2.1 General
Non-linear FE-analyses may imply that the structure is assumed to be loaded beyond proportionality limits.
This means that the structure may be weakened against subsequent load cycles by repeated yielding leading to
a possible cyclic failure. This is called low cycle fatigue and need to be treated different 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 underestimated if conventional linear elastic methods, such as those presented in DNV-RP-C203 ref. /17/, 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 low cycle fatigue. See
[5.2.2].
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.5 Representation of different failure modes – Page 27
5.2.2 Fatigue damage accumulation
The fatigue life may be calculated under the assumption of linear cumulative damage, i.e.
U=V
=1
W
,
(11)
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 ∆ε.
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 non-linear 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. 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].
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 /26/.
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 stress-strain curve of the material is applied. The use of monotonic stress-strain
curve must be avoided since it may provide non-conservative 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 the true cyclic stress strain curves presented in
Figure 5-2 can be applied. Kinematic hardening, as illustrated in Figure 4-3 should be assumed. The curves are
described according to the Ramberg-Osgood relation:
=
1
1 10
+X Z .
Y
(12)
The value of the coefficient K is given in Table 5-3.
Table 5-3 Ramberg-Osgood parameters for base material
Grade
K (MPa)
S235
410
S355
600
S420
690
S460
750
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.5 Representation of different failure modes – Page 28
Figure 5-2
The true cyclic stress-strain curve for common offshore steel grades
5.2.5 Low cycle fatigue of welded joints
5.2.5.1 Accumulated damage criterion
The number of cycles to failure, N, for welded joints due to repeated yielding is estimated by solving the
following equation
∆ ℎ" 1 ′
=
2W −0.1 +
2
′
2W −0.5
(13)
Where:
∆εhs is the fully reversible maximum principal hot spot strain range
E
is the modulus of elasticity (material constant)
σf'
is the fatigue strength coefficient (material constant)
εf'
is the fatigue ductility coefficient (material constant)
The parameters in Equation (13) are given in Table 5-4 for air and seawater with cathodic protection.
Table 5-4 Data for low cycle fatigue analysis of welded joints
Environment
σf ' (MPa)
ε f'
Air
175
0.095
Seawater with cathodic protection
160
0.060
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.5 Representation of different failure modes – Page 29
Figure 5-3
ε-N curves for welded tubular joints in seawater with cathodic protection and in air.
5.2.5.2 Derivation of hot spot strain for plated structures
It is recommended to derive the hot spot strain by applying the principles of the procedure given in Section 4
of DNV-RP-C203 ref. /17/. The procedure in ref. /17/ 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 8-noded shell elements or 20-noded brick elements.
For modelling with shell elements without any weld included in the model a linear extrapolation of the 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 three-dimensional 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 DNV-RP-C203 ref. /17/.
5.2.6 Low cycle fatigue of base material
5.2.6.1 Accumulated damage criterion
Despite the fact that the fatigue capacity of structures very often is governed by welded joints there 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 stress-strain 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
∆ , 1′
=
2W −0.1 +
2
′
2W −0.43
(14)
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.5 Representation of different failure modes – Page 30
where
∆εl
E
σf'
εf'
is the fully reversible maximum principal hot spot strain range
is the modulus of elasticity (material constant)
is the fatigue strength coefficient (material constant)
is the fatigue ductility coefficient (material constant)
Values of the parameters in Equation (14) are given in Table 5-5 for air and seawater with cathodic protection.
Table 5-5 Data for low cycle fatigue analysis of base material
Environment
σf' (MPa)
ε f'
Air
175
0.091
Seawater with cathodic protection
160
0.057
Figure 5-4
ε-N curve for low cycle fatigue of base material for tubular joint in seawater with cathodic protection and in air.
5.2.6.2 Derivation of local maximum principal strain
The maximum principal strain is obtained from the local maxima of the considered detail. The local strain 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 [4.5.2] regarding mesh
refinement.
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 be said to achieve 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.
5.3 Accumulated strain (“Ratcheting”)
For cases where the structure will be loaded by 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 Section [5.1].
The criteria for excessive deformations may alternatively be determined on a case by case basis due to
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.5 Representation of different failure modes – Page 31
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 position
or angle of attack. Examples of the first are protection structures that are hit by swinging loads and the latter
may be wheel loads on stiffened plate decks.
5.4 Buckling
5.4.1 General
The buckling resistance of a structure or structural part is a function of the structural geometry, the material
properties, the imperfections and the residual stresses present. When the buckling resistance is determined by
use of non-linear 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) Linearized approach: Apply the FE method for assessing the buckling eigenvalues (linear bifurcation
analysis) and determine the ultimate capacity using empirical formulas,
b) Full non-linear analysis using code defined equivalent tolerances and/or residual stresses and
c) Non-linear analysis that is calibrated against code formulations or tests.
Either of these methods can be used to determine the resistance of a structure or part of a structure 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, non-conservative loads, interaction of local and global stability problems etc.
5.4.2 Determination of buckling resistance by use of linearized buckling values
5.4.2.1 General
In order to establish the buckling resistance of a structure or part of the structure using linearized 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].
ii) Perform a linear analysis for the selected representative load case SRep showing maximum compressive and
von-Mises stresses.
iii) Determine the buckling eigenvalues and the eigenmodes (buckling modes) by the FE analysis
iv) Select the governing buckling mode (usually the lowest buckling mode) and the point for determining 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 von-Mises stress at the point for the representative stress σRep from step ii).
vi) Determine the critical buckling stress as the eigenvalue for the governing buckling mode times the
representative stress:
σki = kg σRep
(15)
Determine the reduced slenderness as:
.=
1
(16)
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.5 Representation of different failure modes – Page 32
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 codes, but if not available the following may be used:
Table 5-6 Buckling curves
Type of buckling
κ
1
] + 7] 2 − .
Column and stiffened plate and
plate without redistribution
possibilities
≤ 1.0
= 1.0 for . ≤ 0.673
. − 0.22
Plate with redistribution
possibilities
.
2
for . > 0.673
Curves to be selected from specific shell buckling codes such as
DNV-RP-C202 /16/ or Eurocode EN-1993-1-6 /6/
Shell buckling
] = 0.5 1 + : . − 0.2 + .
2
2
(17)
α = 0.15 for strict tolerances and low residual stresses
0.3 for strict tolerances and moderate residual stresses
0.5 for moderate tolerances and moderate residual stresses
0.75 for large tolerances and severe residual stresses
viii)Determine the buckling resistance Rd as:
#$ =
a %#
- 1#
(18)
1,4
1,2
buckling factor κ
1
Critical stress (Euler)
Plate
0,8
Column
Shell
0,6
0,4
0,2
0
reduced slenderness
Figure 5-5
Examples of buckling curves showing sensitivity for imperfections etc. for different buckling forms
Empirical buckling curves are needed to account for the buckling resistance reduction effects from
imperfections, residual stresses and material non-linearity. The effect is illustrated in Figure 5-5. 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
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.5 Representation of different failure modes – Page 33
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 cross-sectional classes see DNV-OS-C101 Appendix A /14/.
5.4.2.2 Correction for local buckling effects
There may be cases where a reliable FE representation of local buckling phenomena is not feasible. This 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 (19).
1
σ ki
=
1
σ kig
+
1
(19)
σ kil
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 non-linear analysis using code defined equivalent tolerances
The buckling resistance of a structure or part of a structure can be determined by performing non-linear
analyses where the effects of imperfections, residual stresses and material non-linearity is accounted for by use
of a defined material stress-strain relationship and the use of empirically determined equivalent imperfections.
The defined equivalent imperfections will include effects from real life imperfections, but will in general 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 5-7. For other cases see [5.4.4].
The material model to be used with the equivalent imperfections is shown in Figure 5-6 or with the models
proposed in [4.7.5].
E/100
Stress
1
E
1
Strain
Figure 5-6
Material model for analysis with prescribed equivalent imperfections
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.5 Representation of different failure modes – Page 34
Table 5-7 Equivalent imperfections
Component
Shape
Magnitude
Member
bow
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
Longitudinal
bow
L/400
stiffener girder
webs
buckling
Plane plate
eigenmode
between
stiffeners
bow twist
Longitudinal
stiffener or
flange outstand
s/200
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 5-7. The
values in Table 5-7 apply to the total imperfection from local and global imperfection patterns. Sensitivity
analyses may be required for cases that are particularly imperfection sensitive.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.5 Representation of different failure modes – Page 35
Figure 5-7
Example of local (left) and global (right) imperfections for stiffened panel
5.4.4 Buckling resistance from non-linear analysis that are calibrated against code formulations or
tests
Buckling resistance can be found by non-linear methods where the effect of imperfections, residual stresses
and material non-linearity 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:
i) Prepare a model that is intended to be used for the analysis.
ii) Perform an eigenvalue analysis to determine relevant buckling modes.
iii) Select the object for calibration and prepare a model using the same element type and mesh density as
intended for the model to be analysed.
iv) Perform eigenvalue analysis of the calibration object and determine the appropriate buckling mode for the
calibration object.
v) Determine the magnitude of the equivalent imperfection that will give the correct resistance for the
calibration object.
vi) 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 differ 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 Appendix [B.3].
5.4.5 Strain limits to avoid accurate check of local stability for plates and tubular sections yielding in
compression.
5.4.5.1 General
For cases where compressed parts of the cross-section (as a flange) are experiencing plastic strain 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 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.
Plates supported on both longitudinal edges:
&' ≤ L2.7 L
S
M 2 − 0.0016M
355
S) 0 <
&' < 0.10
(20)
Where b is distance between longitudinal supports and t is plate thickness
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.5 Representation of different failure modes – Page 36
Plates supported on one edge (flange outstand)
2
&' = c0.29 L M − 0.0016e
&
355
S) 0 < &' < 0.10 (21)
Where c is the plate outstand and t is plate thickness
Tubular sections without hydrostatic pressure:
&' = c8.5 L
U
2
M − 0.0016e
355
S) 0 <
&' < 0.10
(22)
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 [4.7].
For structural parts meeting requirements to cross-sectional class 3 or 4 no plastic strain due to compressive
stresses can be allowed without an accurate buckling analysis.
For definition of cross-sectional classes see DNV-OS-C101 /14/.
5.5 Repeated buckling
For cases where buckling of parts of the structure may occur before the total capacity of the entire structure 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 cross-sectional Class 1 and the
reduced member slenderness as a column is above 0.5.
Failure due to low cycle fatigue according to [5.2] needs to be checked also for these cases.
It should be noted that structural parts that are yielding in tension may buckle when unloaded. If cyclic loads
leads to yielding in tension one must check against buckling through the entire dimensioning load cycle.
In certain cases sufficient capacity may be proved by disregarding the structural part that suffer buckling in the
cyclic capacity checks.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.6 Bibliography – Page 37
6 Bibliography
/1/
ISO 2394, General principles on reliability for structures, Second edition 1998-06-01
/2/
/3/
API RP 2A Recommended Practice for Planning, Designing and Constructing Fixed Offshore
Platforms – Working Stress Design, Errata and supplement 3 October 2007
EN 1990, Eurocode - Basis of structural design, April 2002
/4/
EN 1993-1-1 Eurocode 3 Design of steel structures. Part 1-1 General rules and rules for buildings
/5/
/7/
EN 1993-1-5, Eurocode 3 - Design of steel structures - Part 1-5: Plated structural elements, October
2006
EN 1993-1-6, Eurocode 3 - Design of steel structures - Part 1-6: Strength and Stability of Shell
Structures, February 2007
EN 1993-1-8, Eurocode 3 - Design of steel structures - Part 1-8: Design of Joints, 2005/AC:2009
/8/
AISC 360-05, Specification for Structural Steel Buildings, March 9 2005
/9/
/11/
ISO 19900 Petroleum and natural gas industries – General requirements for offshore structures. First
edition 2002-12-01
ISO 19902 Petroleum and natural gas industries – Fixed steel offshore structures, First edition
2007-12-01
Norsok Standard N-001, Integrity of offshore structures, Edition 7, June 2010
/12/
Norsok Standard N-004, Design of steel structures, Revision 4, February 2004
/13/
/14/
Norsok Standard N-006, Assessment of structural integrity for existing offshore load-bearing
structures, Edition 1, March 2009
DNV-OS-C101, Design of Offshore Steel Structures, General (LRFD Method), April 2011
/15/
DNV-RP-C201 Buckling Strength of Plated Structures, October 2010
/16/
DNV-RP-C202 Buckling Strength of Shells
/17/
DNV-RP-C203 Fatigue Design of Offshore Steel Structures October 2012
/18/
DNV-RP-C204 Design against Accidental Loads, October 2010
/19/
/24/
ECCS publication No. 125, Buckling of Steel Shells. European Design Recommendations, 5th Edition,
J.M. Rotter and H. Smith Editors.
DNV-RP-F110 Global Buckling of Submarine Pipelines Structural Design due to High Temperature/
High Pressure, October 2007
DNV-SINTEF-BOMEL: Ultiguide, Best practice for use of non-linear analysis methods in
documentation of ultimate limit state for jacket type offshore structures, April 1999.
Skallerud, Amdahl: Nonlinear analyses of offshore structures, Research studies press ltd., 2002 (ISBN
0-86380-258—3)
Corrocean ASA: Design of offshore facilities to resists gas explosion hazards. Engineering handbook.
Oslo 2001.
ASME Boiler & Pressure Vessel Code 2013 Edition July 1, 2010 VIII Division 2, Alternative Rules
/25/
EN 13445-3:2012 Unfired pressure vessels Part 3
/26/
Hagen, Ø, Solland, G. Mathisen, J. Extreme storm wave histories for cyclic check of offshore structures
OMAE 2010-20941
H. M. Hilber, T. J. R. Hughes and R. L. Taylor: Improved numerical dissipation for time integration
algorithms in structural dynamics, Earthquake engineering and structural dynamics, 5 (1977), page
283-292.
Skallerud, Eide, Amdahl, Johansen: On the capacity of tubular T-joints subjected to severe cyclic
loading. Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering OMAE, v 1, n Part B, p 133-142, 1995.
Weignad, Berman: Behaviour of butt-welds and treatments using low-carbon steel under cyclic
inelastic strains, Journal of Constructional Steel Research, v 75, p 45-54, August 2012.
Boge, Helland, Berge: Proceedings of the International Conference on Offshore Mechanics and Arctic
Engineering - OMAE, v 4, p 107-115, 2007.
/6/
/10/
/20/
/21/
/22/
/23/
/27/
/28/
/29/
/30/
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
Sec.6 Bibliography – Page 38
/31/
/32/
/33/
/34/
/35/
/36/
/37/
Scavuzzo, Srivatsan, Lam: Fatigue of butt welded steel pipes. American Society of Mechanical
Engineers, Pressure Vessels and Piping Division (Publication) PVP, v 374, p 113-143, 1998, Fatigue,
Environmental Factors, and New Materials.
Belytschko, Liu, Moran, Nonlinear Finite Elements and Continua and Structures, John Wiley&Sons,
Ltd., November 2009
Maresca, Milella, Pino: A critical review of triaxiality based failure criteria, IGF 13 Cassino 27 e 28
Maggio, 1997
Kuhlmann: Definition of Flange Slenderness Limits on the Basis of Rotation Capacity Values, Journal
of Constructional Steel Research, 14 (1989) 21-40
Gardner, Wang, Liew: Influence of strain hardening on the behavior and design of steel structures,
International Journal of Structural Stability and Dynamics Vol. 11. No. 5 (2011) 855-875
DNV-OS-F101 Submarine Pipeline Systems, August 2012
/38/
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.
EN-10025 Hot rolled products of structural steels. Part 2, 3, 4 and 6
/39/
EN-10225 Weldable structural steels for fixed offshore structures - Technical delivery conditions
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.A Commentary – Page 39
APPENDIX A COMMENTARY
A.1 Comments to [4.1] General
The element model selected for analysis needs to represent the structure so that any simplifications are 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 codes for offshore structures like ISO 19902 /10/ are
similar and the analysis carried out according to the recommendations in this RP is expected to fulfil the
requirements also in this code.
A.2 Comments to [4.4] Selection of elements
Guidance on selection of suitable elements for non-linear analysis can be found in text books e.g. /32/.
A.3 Comments to [4.7.5] Stress - strain curves for ultimate capacity analyses
The proposed stress-strain curves are based on steel according to /38/ and /39/. The curve is also applicable to
materials according to DNV Offshore Standard /14/.
A.4 Comments to [5.1.1] General
There is not a universal method available that can be used for predicting tensile failure for practical engineering
applications by FE methods.
The value of the acceptable strain will be governed by:
— stress triaxiality
— load history
— cold deformation.
— material properties
— material inhomogeneity
— different material properties of materials being joined. (Even material with the same strength 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. This RP proposes to either use a simplified tensile failure criterion or to
calibrate a problem specific criterion according to a specified procedure.
There are several models describing the local phenomenon of tensile failure. Common for most of these is that
the strain and stress state during the entire loading sequence until failure is considered important for describing
the damage process properly. Unfortunately, a high degree of complexity is a common feature of many of the
models, and the theoretical and practical knowledge required to perform a FE analysis based on these criteria
is judged not to be suited for engineering purposes.
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.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.A Commentary – Page 40
A.5 Comments to [5.1.3] Tensile failure in base material. Simplified approach for plane plates
Dominant structural steel design codes like Eurocode 3 /7/ and AISC /8/ apply a larger material factor for
tensile failure when the capacity is based on the tensile strength. In order to determine a resistance by use of
non-linear FE methods a similar increase in the material factor should be included since the material curve used
is including the strain hardening.
A.6 Comment to [5.2.3] Determination of cyclic loads
The check against cyclic failure should be carried out with the use of a dimensioning load history that has 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. /26/.
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.
A.7 Comment to [5.2.4] Cyclic stress strain curves
The cyclic stress-strain curves are only intended for low cycle fatigue analysis. The use of monotonic stress
strain curve in low cycle fatigue analysis may provide non-conservative results and must therefore be avoided.
The cyclic stress strain curves presented in Table 5-3 are based on cyclic behaviour of similar steels reported
in reference /37/. 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 /37/. Here, the same exponent of
10 in the Ramberg-Osgood 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.
A.8 Comment to [5.2.5.1] Accumulated damage criterion
Laboratory test results presented in references /28/ to /31/ make up the basis for the established ε-N curve for
welded joints. The proposed mean and design curve for air along with the laboratory test data is presented in
Figure A-1. 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 ref. /29/ are
obtained from testing of dog-bone specimens cut out from a butt welded plate. These results have therefore
been weighted less than results from ref. /28/ and ref. /30/ which is based on full scale testing of tubular joints.
The fatigue test results presented in ref. /31/ are from pipes with wall thickness of less than 10 mm. The 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.
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
(DNV-RP-C203 ref. /17/). 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 DNV-RP-C203 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 three
standard deviations. Three standard deviations on log N corresponds to a factor of 103·0.2 ≈ 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
of 2.5. This is identical to the reduction used in DNV-RP-C203 for fatigue lives less than 106.
Due to limited test data the proposed model does not take into account that the fatigue strength decreases with
increasing thickness. It is however believed that this effect is less pronounced in low cycle fatigue compared
to high cycle fatigue, ref. DNV-RP-C203. In order to avoid non-conservative results it is recommended not to
apply the proposed curves for thicknesses above 60 mm. For larger thicknesses it is recommended to multiply
the strain amplitude (∆ε/2) with the thickness correction factor used in DNV-RP-C203. The reference thickness
is set equal to 60 mm. In case that low-cycle fatigue is to be considered together with high cycle fatigue it may
be more practical to use the same reference thickness in both checks.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.A Commentary – Page 41
Figure A-1
Mean and design curve for welded joints along with laboratory test results.
A.9 Comments to [5.2.7] Shake down check
When a structure is loaded beyond linear limits the response for subsequent cycles will be changed. It is
therefore necessary to investigate the behaviour through the full cycles also for the next cycles. See e.g. /22/
for more guidance.
A.10 Comments to [5.4.1] General
The modelling of geometrical imperfections, out-of-straightness etc. is crucial for achieving a credible and 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. For redundant structures the sensitivity of the ultimate
load bearing capacity to the size of the geometrical imperfections will be negligible. In such cases the triggering
of the governing modes rather than accounting for actual tolerance size will be most important for the analyses.
Guidance on analysis of stability problems may be found in e.g. /19/.
A.11 Comments to [5.4.5] Strain limits to avoid accurate check of local stability for plates and
tubular sections yielding in compression.
The strain limits for plates are established from analysis of flanges meeting rotational capacities according to
cross-section class 1 and 2 and by comparison with tests. See /34/ and /35/. Strain limits are also compared with
recommendations given in the DNV Offshore Standard for submarine pipelines /36/.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 42
APPENDIX B EXAMPLES
B.1 Example: Strain limits for tensile failure due to gross yielding of plane plates (uniaxial
stress state)
B.1.1 T-section cantilever beam
Gross yielding check of a T-section cantilever beam, subjected to axial and shear force and moment loading,
is presented in this example. The finite element software ABAQUS is used to perform the analyses.
The geometry and boundary conditions of the beam are shown in Figure B-1. Loading is applied to a reference
point coinciding with the neutral axis of the beam cross section, using kinematic coupling between cross section
and reference point. The beam is modelled using 4-noded shell elements with reduced integration (S4R) with
mesh size of 16 mm x 16 mm. Material grade is S355, modelled according to Section [4.7.5].
The magnitude of the applied forces and moments are given by axial force Nx, shear force Py = − 0.15Nx and
bending moment Mz = − 0.45Nx.
The loading and boundary conditions result in a stress state dominated by uniaxial stress. Hence, the criterion
presented in Section [5.1.3.2] is applied for assessing the beam.
Section A
A
8
3000
M
500
460
N
A
40
P
300
Figure B-1
Geometry and boundary conditions for cantilever beam
According to the criterion presented in Section [5.1.3.2], the strain should be calculated as the linearized
maximum principal plastic strain along the likely failure line and checked against the limit for the critical strain.
The limit is a critical gross yield strain of 0.04 for this example.
Figure B-2 shows a contour plot of the maximum principal plastic strain and the chosen failure line, the 3rd
element column from the clamped end. For the chosen failure line the maximum principal plastic strain is
obtained from integration points and linearized using the method of least squares.
Based on the finite element analysis results and the linearization the critical load is determined to be between
Nx= 489 kN and Nx= 500 kN. The maximum principal plastic strain distribution and corresponding linearized
distribution for load level Nx=489 kN are shown in Figure B-3 and analysis results are shown in Table B-1.
Table B-1 Analysis results
Nx
Maximum linearized
[kN]
principal plastic strain
489
3.9·10-2
500
4.2·10-2
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 43
Chosen failure line
Figure B-2
Maximum principal plastic strain contour plot, with chosen failure line highlighted
Figure B-3
Maximum principal plastic strain and linearized maximum principal plastic strain distributions for web
The design resistance will be found as the characteristic resistance divided by the appropriate material factors.
The selected element mesh is tested to be accurate meaning that a FEM knock down factor CFEM can be taken
as 1.0 in this case.
B.1.2 T-section cantilever beam with notch
Check for tensile failure of a T-section cantilever beam with a notch in the free edge of the web is presented in
this example. The geometry and boundary conditions are shown in Figure B-4. The model, loading and analysis
setup and procedure are the same as in [B.1.1], except the size of the mesh which in this case is 25% of the
notch height, i.e. 25 mm x 25 mm.
In addition to the gross yielding criterion presented in [5.1.3.2], the local tensile failure criterion presented in
[5.1.3.3] must be applied when assessing the beam.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 44
Section A
A
8
3000
100
500
200
M
500
460
N
40
A
P
300
Figure B-4
Geometry and boundary conditions for cantilever beam with notch
For the gross yielding two likely failure lines were chosen; one at mid-notch and one at the notch corner
displaying the highest strain values, see Figure B-5. The maximum principal plastic strain is obtained from
integration points and linearized using the method of least squares. Both failure lines must comply with the
criterion of an allowable maximum principal plastic linearized strain of 0.04. In addition, the local strain,
according to [5.1.3.3], must not exceed the critical strain value of 0.12. In this case the mesh size falls within
the defined volume criteria. Hence, the local strain value is taken as the maximum principal plastic strain in the
element with the largest strain.
Figure B-5
Maximum principal plastic strain contour plot, with chosen failure lines highlighted
Based on the finite element analysis results the linearization line 1 is found to be the critical failure line, with
critical load determined to be between Nx = 310 kN and Nx = 315 kN. The maximum principal plastic strain
distributions and corresponding linearized distributions for both failure lines at load level Nx = 310 kN are
shown in Figure B-6 and analysis results are shown in Table B-2.
Table B-2 Analysis results
Nx[kN]
310
315
Line 1
3.8·10-2
4.1·10-2
Maximum linearized principal plastic strain
Line 2
Largest element strain
3.0·10-2
7.6·10-2
3.3·10-2
8.1·10-2
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 45
Figure B-6
Maximum principal plastic strain and linearized maximum principal plastic strain distributions for web
A convergence study is used to ensure that satisfactory accuracy is obtained. When convergence is reached the
critical load is determined between Nx = 305 kN and Nx = 310kN, resulting in a FEM knock down factor
4
- =
305kN
= 0.968
315kN
The calculations in the convergence study are performed for a fixed volume, i.e. the volume used in the
presented mesh. For the local strain criteria this is illustrated in Figure B-7 and Figure B-8.
The design resistance will be found as the characteristic resistance divided with the appropriate material
factors.
Figure B-7
Element mesh used for convergence study. Elements used in local strain check circled in red
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 46
Figure B-8
Maximum principal plastic strain for three mesh densities, the left mesh is presented in this example
B.2 Example: Convergence test of linearized buckling of frame corner
A symmetric frame of beams with I-section is analysed. The frame with boundary conditions is shown in Figure
B-9 and Figure B-10. The loading is applied as a displacement of the web at one end of the frame, u2,applied =
− 0.01 m. Three different mesh densities and two element types are included in a convergence study, to ensure
a sufficiently refined mesh. See Figure B-11. The element types used are 4 node rectangular shell elements and
8 node rectangular shell elements.
The analyses are performed using the FEM-software ABAQUS.
A
4000
b = 500
a
2000
A
4000
R = 1500
2000
Section A
15
500
20
Figure B-9
Geometry of test example
DET NORSKE VERITAS AS
480
20
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 47
Figure B-10
Displacement/boundary conditions
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 48
Figure B-11
Top: coarse mesh. Middle: fine mesh. Bottom: very fine mesh
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 49
For the eigenvalue analyses and the linear analyses elastic material properties were used and for the buckling
capacity analyses non-linear material properties were used. Details are shown in Table B-3.
Table B-3 Material properties
Density, ρ
7850 kg/m3
Young’s modulus, E
210 GPa
Poisson’s ratio, ν
0.3
The loading is applied as displacement on the web at one end of the frame, as shown in Figure B-10. Hence,
the eigenvalue defines the displacement corresponding to linearized buckling.
A convergence study is performed by analysing 6 cases and the resulting buckling displacements are listed in
Table B-4. From these results all combinations of mesh size and element type except the coarse 4 node
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.
Table B-4 Convergence study of frame
Case
Mesh size
Element type
number
1
Coarse
4-node
2
Fine
4-node
3
Very fine
4-node
4
Coarse
8-node
5
Fine
8-node
6
Very fine
8-node
Linearized buckling displacement [m]
0.0653
0.0624
0.0618
0.0616
0.0615
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.
B.3 Example: Determination of buckling resistance by use of linearized buckling values
B.3.1 Step i) Build model
The same problem as shown in Figure B-9 will be used in this example and the boundary conditions are as in
Figure B-10. The material properties are shown in Table B-5.
Table B-5 Material properties
Density, ρ
7850 kg/m3
Young’s modulus, E
210 GPa
Poisson’s ratio, ν
0.3
Yield strength, σY
355 MPa
The analysis follows the steps as given in [5.4.2]. Step i) is completed as the model from the example in [B.2].
B.3.2 Step ii) Linear analysis of the frame
The results from a linear analysis are shown in Figure B-12 and Figure B-13 for the von-Mises and membrane
compression stresses respectively.
The linear analysis is performed with the same applied displacement as in the eigenvalue analysis
u2,applied = − 0.01m, equivalent to an applied load i y-direction SRep= 75.7 kN.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 50
Figure B-12
Distribution of von-Mises stress from linear analysis
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 51
Figure B-13
Distribution of compressive stress from linear analysis (minimum in-plane principal stress)
B.3.3 Step iii) Determine the buckling eigenvalues
Eigenvalue analysis is performed to find the buckling modes and eigenvalues of the frame. The first eigenvalue
is kg= 6.24, and the corresponding buckling mode shape is shown in Figure B-14.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 52
Figure B-14
First buckling mode
B.3.4 Step iv) Select the governing buckling mode and the point for reading the representative stress
The lowest buckling mode is judged to be a realistic buckling shape for this case and is selected.
The reference stress is taken as the maximum von-Mises stress in the structural part subjected to buckling.
B.3.5 Step v) Determine the von-Mises stress at the point for the representative stress σRep from step ii)
Stress from linear analysis:
σ Rep = 97.4 MPa
B.3.6 Step vi) Determine the critical buckling stress
The critical buckling stress for the governing buckling mode is determined as:
1 = 1# = 608 MPa
The reduced slenderness is determined as:
.̅ =
1
i
= 0.76
B.3.7 Step vii) Select empirically based buckling curve
The buckling curve used here is taken from Table 5-6. The curve selected is the one for column and 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.
a=
1
] + j]2 − .̅2
≤ 1.0
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 53
] = 0.5 [1 + : .̅ − 0.2 + .̅2 ]
α is set to 0.3 for the following calculations.
B.3.8 Step viii) Determine the buckling resistance Rd
With .̅ = 0.76 then the buckling factor is a = 0.767
#$ =
Assuming a material factor
a %#
- 1#
- = 1.15 , the buckling resistance is
#$ =
0.767 ∙ 355 ∙ 0.0757
= 0.184 MN
1.15 ∙ 97.4
B.4 Example: Determination of buckling resistance from non-linear analysis using code
defined equivalent tolerances
B.4.1 Description of model
The same problem as shown in Figure B-9 will be used in this example and the boundary conditions are as in
Figure B-10. The material properties are shown in Table B-5 and the material model is shown in Figure B-15.
Figure B-15
Material model for analysis with material non-linearity
A non-linear analysis (using the arc-length method) is performed, where the effects of imperfections, residual
stresses and material non-linearity is accounted for by use of a defined material stress-strain 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 B-14, and is used as the pattern for the equivalent imperfection.
The magnitude of the equivalent imperfection δ is calculated using the tolerances given in Table 5-7.
The analysed frame can be considered equivalent to a component of longitudinal stiffener or flange outstand,
hence the magnitude is given as
n = 0.02 rad = 0.02&
where c is half the width of the flange. Two values of c are analysed, the largest c; &I = I , where
I = 0.975 J is the distance between where the webs cross in the corner of the frame and the midpoint of the
S
I+2
flange curvature, and an average c; &IP =
, where S = 0.5 J is the width of the flange outside the
2
curved area. See Figure B-9
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 54
nI = 0.0195
nIP = 0.0122
Figure B-16
Stress distribution for non-linear analysis with initial imperfection op at maximum applied force
B.4.2 Results
The stress distribution for the non-linear analysis with initial imperfection is shown in Figure B-16. Figure B17 displays the force-displacement 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 Section [B.3], and from the non-linear analyses.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 55
Figure B-17
Force-displacement from non-linear analyses, linear analysis and the calculated critical value
B.5 Example: Determination of buckling resistance from non-linear analysis that are
calibrated against code formulations or tests
B.5.1 Step i): Prepare model
A conical transition subjected to external hydrostatic pressure and axial tension is chosen for this analysis. The
geometry of the conical transition and the calibration object is shown in Figure B-18. The applied loading is
defined as a hydrostatic pressure = 1.01MPa and an axial tension W = 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 z-direction). Load and boundary conditions and element mesh are shown in Figure B-19.
The conical transition is modelled using 4-noded shell elements (S4R). Material properties are listed in Table B-6.
Figure B-18
Geometry of conical transition (on top) and calibration object (bottom), dimensions in mm
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 56
Figure B-19
Left: Load and boundary conditions. Right: Element mesh
Table B-6 Material properties
Density, ρ
7850 kg/m3
Young’s modulus, E
210 GPa
Poisson’s ratio, ν
0.3
Yield strength, σY
420 MPa
Density water, ρw
1030kg/m3
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 57
B.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 B-20.
Figure B-20
Buckling mode shape for conical transition
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 58
B.5.3 Step iii): Select object for calibration and prepare model
The calibration object is selected as a cylinder. The diameter and wall thickness are equal to the 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 B-21.
Figure B-21
Left: Load and boundary conditions. Right: Element mesh
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 59
B.5.4 Step iv): Determine the appropriate buckling mode for the calibration object
Eigenvalue analysis is performed to find the buckling modes for the calibration object. These buckling modes
are compared to the buckling modes found for the conical transition and a mode with similar pattern is selected.
Figure B-22 shows the first cylinder buckling mode. This shows a similar pattern to the buckling mode of the
conical transition Figure B-20, hence this is determined to be an appropriate buckling mode.
Figure B-22
Buckling mode shape for cylinder
B.5.5 Step v): Determine magnitude of the equivalent imperfection
To determine the magnitude of the equivalent imperfection a non-linear analysis of the cylinder with
imperfections is performed. The imperfection shape from the chosen buckling mode was transferred to the nonlinear analysis, and the same load and boundary conditions as for the eigenvalue analysis were applied. The
material model shown in Figure 5-6 is used for the non-linear analysis.
The imperfection is scaled so the buckling capacity of the cylinder is equal to the buckling capacity for cylinders
given in N-004 /12/. To obtain this capacity the magnitude of the imperfection was found to be 40 mm.
B.5.6 Step vi): Perform non-linear analysis of the model with imperfections
A non-linear analysis of the conical transition with imperfections is performed. The load and boundary
conditions remain the same, and the material model and magnitude of the calibrated imperfection from Step v
is used. The load proportionality factor for this case is shown in Figure B-23. The maximum load proportionality
factor is !r JI = 0.936 . Thus the buckling capacity of the conical transition subjected to the given load
combination is; hydrostatic pressure = 0.95MPa and an axial tension W = 54.7MN . Figure B-24 shows the
von Mises stress at maximum load on the deformed conical transition.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 60
Figure B-23
Load proportionality factor for conical transition with initial imperfection
Figure B-24
Deflected shape showing von Mises stress at maximum load deformations scaled with a factor of 10
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 61
B.6 Example: Low cycle fatigue analysis of tubular joint subjected to out of plane loading.
This example presents a low cycle fatigue analysis of a tubular T-joint subjected to an out-of-plane fully
reversible load of ± 60 kN. The objective of the analysis is to estimate the design life based on the
recommendations in Section [5.2.5]. The assumed geometry and dimensions are given in Table B-7 and Figure
B-25.
Table B-7 Dimensions
Chord diameter
Chord thickness
Chord length
Brace diameter
Brace thickness
Brace length
[mm]
D = 300
T = 15.9
L = 1800
d = 160
t = 11.5
l = 500
Figure B-25
Geometry of test example, dimensions in mm
It is assumed that the cyclic stress-strain behaviour is well described by the Ramberg-Osgood relationship:
=
1
1
1 ′
+ X ′Z
Y
The values for the Ramberg-Osgood parameters are presented in Table B-8 for the chord and the brace.
Table B-8 Ramberg-Osgood parameters
K’
n’
[MPa]
Chord
731.7
0.096
Brace
699.5
0.108
In order to obtain the cyclic strains a finite element analysis was carried out using the FEM-software ABAQUS.
An 8-noded shell element (S8R) model was established with load and support conditions as shown in Figure
B-26.
The chord was constrained at each end for all translational and rotational degrees of freedom. The out-of-plane
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 B-27.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 62
Figure B-26
Boundary and loading conditions for tubular joint
Figure B-27
Load steps
Figure B-28 (a) shows an overview of the finite element mesh. Figure B-28 (b) shows a close-up of the bracechord intersection area. The finite element mesh in the hot spot region is in accordance with the recommended
practice DNV-RP-C203 for tubular joints. In this example the element nodes coincide with the specified
extrapolation points (a and b). Hence, nodal values are applied in the extrapolation procedure for calculating
the hot spot strain range. The distance from the hot spot to the first extrapolation point, a is obtained by means
of Equation (23). The distance to the second extrapolation point, b is obtained by Equation (24).
I = 0.2√' = 6.1mm
S=
t#
= 13.1mm
36
(23)
(24)
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 63
Figure B-28
Left: (a) Meshed model, Right: (b) Close-up of brace-chord intersection area
Figure B-29 shows the principal strain range due to the out-of-plane cyclic loading.
Figure B-29
Maximum principal strain range
The hotspot strain range is obtained according to the following procedure:
i) Establish the total strain ranges ( Δ , Δ , Δ etc.) 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.
ii) Extract the principal strain ranges at the two extrapolation points, i.e, at distances a and b.
iii) The hotspot strain range Δ ℎ" is calculated by means of the following equation:
I
Z Δ S−Δ I
S−I
Δ ℎ" = Δ I − X
(25)
The hot spot strain range along with values at distances a and b are presented in Table B-9.
Table B-9 Hotspot strain range
Nodal value saddle
vwp
0.0036
vwx
0.0021
vwyz
0.0048
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 64
Air environment is assumed. Hence, the characteristic design life due to the cyclic loading is obtained by
solving the following equation, ref. Section [5.2.5]:
∆εhs σ′f
=
2N −0.1 + ε′f 2N −0.5
2
E
→ W = 1270 cycles
B.7 Example: Low cycle fatigue analysis of plate with circular hole.
In this example a low cycle fatigue analysis of a plate with a circular hole subjected to cyclic displacement of
1.0mm is presented. The objective of the analysis is to estimate the design life based on recommendations in
Section [5.2.6]. The dimensions of the plate are presented in Figure B-30. The plate material is of grade S355
and the cyclic stress-strain curve is obtained from Table 5-3.
Figure B-30
Geometry of considered specimen
The maximum principal strain range is obtained by performing a finite element analysis with the FEM-software
ABAQUS. The finite element analysis was performed with 8-noded shell elements with reduced integration
(S8R). The material modelling is according to specifications in Section [4.7.5]. The boundary conditions and
cyclic displacement is applied as illustrated in Figure B-32. 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 B-31
Geometry and loading
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 65
Figure B-32
Load steps
The maximum principal strain range is obtained according to the following procedure:
i) Perform a strain convergence study. The mesh around the hole is refined until the strain value in the
relevant nodal point converges. Based on the convergence study it was found sufficient to use 48 elements
around the hole.
ii) Establish the strain component ranges ( Δ , Δ , Δ , 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.
iii) Calculate the maximum principal strain range based on the strain component ranges.
iv) Calculate the design fatigue life based on the seawater with cathodic protection curve.
Figure B-33 shows the maximum principal strain range due to the specified cyclic loading of the plate. The
hysteresis loop in the location adjacent to the hole with the highest cyclic strain is plotted in Figure B-34. The
maximum principal strain range of ∆εl = 0.011 obtained from the last cycle in the finite element analysis is used
as basis for calculating the design fatigue life. Hence, by solving Equation (13) in Section 5.2.6 a design fatigue
life of N = 139 is obtained.
Figure B-33
Equivalent strain range
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-C208, June 2013
App.B Examples – Page 66
Figure B-34
Stress versus strain in y – direction (parallel to the 1st principal strain direction).
DET NORSKE VERITAS AS