RECOMMENDED PRACTICE
DNV-RP-F401
Electrical Power Cables in
Subsea Applications
FEBRUARY 2012
The electronic pdf version of this document found through http://www.dnv.com is the officially binding version
DET NORSKE VERITAS AS
FOREWORD
DET NORSKE VERITAS (DNV) is an autonomous and independent foundation with the objectives of safeguarding life,
property and the environment, at sea and onshore. DNV undertakes classification, certification, and other verification and
consultancy services relating to quality of ships, offshore units and installations, and onshore industries worldwide, and
carries out research in relation to these functions.
DNV service documents consist of among others the following types of documents:
— Service Specifications. Procedual 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
© Det Norske Veritas AS February 2012
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-F401, February 2012
Changes – Page 3
CHANGES
General
This is a new document.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-F401, February 2012
Contents – Page 4
CONTENTS
1.
Introduction............................................................................................................................................ 5
2.
Scope........................................................................................................................................................ 5
2.1 Application................................................................................................................................................5
2.2 Applicable standards.................................................................................................................................5
2.3 Terminology..............................................................................................................................................5
3.
Requirements.......................................................................................................................................... 6
3.1 General construction requirements ...........................................................................................................6
3.2 Conductor..................................................................................................................................................7
3.3 Electrical insulation of core – breakdown strength...................................................................................7
3.4 Screen/sheath for prevention water exposure to the insulation system ....................................................7
3.5 Water blocking..........................................................................................................................................8
3.6 Degassing..................................................................................................................................................8
3.7 Longitudinal gas barrier............................................................................................................................8
3.8 Armour......................................................................................................................................................9
3.9 Anchoring of armour.................................................................................................................................9
3.10 Radial compression – load carrying capacity ...........................................................................................9
3.11 Flexibility/Compliance .............................................................................................................................9
3.12 Bending radius ........................................................................................................................................10
3.13 Coiling.....................................................................................................................................................10
4.
References............................................................................................................................................. 10
Appendix A. Qualification with Respect to Fatigue................................................................................... 11
A.1 Limitations ............................................................................................................................................. 11
A.2 Definitions.............................................................................................................................................. 11
A.3 Input data ............................................................................................................................................... 12
A.4 Pre-test straining .................................................................................................................................... 12
A.5 Qualification principles.......................................................................................................................... 12
A.6 Qualification based on components ....................................................................................................... 13
A.7 Qualification of complete cable cross section ....................................................................................... 14
A.8 Electrical verification tests..................................................................................................................... 15
Appendix B. Test Methods – Fatigue Loading of Complete Cables......................................................... 17
B.1 General................................................................................................................................................... 17
B.2 4-point-bending...................................................................................................................................... 17
B.3 Bending against template....................................................................................................................... 17
Appendix C. Fatigue Testing Detection Techniques .................................................................................. 19
C.1 Metallic materials................................................................................................................................... 19
C.2 Plastic materials ..................................................................................................................................... 20
Appendix D. Estimation of Fatigue Design Curves – Least Squares Method ......................................... 22
Appendix E. Estimation of Fatigue Design Curves Incomplete Observations of Number of Cycles to Failure ......................................................................... 23
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-F401, February 2012
Sec.1. Introduction – Page 5
1. Introduction
This Recommended Practice is to be used as a supplement to ISO 13 628-5 /1/ with regards to electrical power
cables. This ISO standard does not give requirements to such cables on a detailed level. This RP covers
additional requirements for power cables being submerged in seawater at large water depths and/or being
exposed to dynamic excitation, e.g. when suspended from floating production units.
The RP is intended to be used together with /1/. In case of conflict between the ISO standard and this document
the ISO standard shall prevail.
It is a pre-requisite that power cables are designed and fabricated according to existing IEC standards.
2. Scope
2.1 Application
The RP covers electrical power cables, as single cables or integrated in an umbilical in an application covered
by ISO 13 628-5 /1/.
The RP covers cables which comply with IEC 60 502-1 /2/ and IEC 60 502-2 /3/.
The RP applies to cables used for AC power transmission. DC cables are not covered.
Guidance note:
Examples of single cables may e.g. be power supply to direct electrical heating systems for pipelines, main power
supply from shore to floating production units, power supply from floating units to subsea installation etc.
---e-n-d---of---G-u-i-d-a-n-c-e---n-o-t-e---
Guidance note:
The following definition of an umbilical is given in /1/: “group of functional components, such as electric cables,
optical fibre cables, hoses, and tubes, laid up or bundled together or in combination with each other, that generally
provides hydraulics, fluid injection, power and/or communication services”.
---e-n-d---of---G-u-i-d-a-n-c-e---n-o-t-e---
2.2 Applicable standards
Power cables shall comply with the following standards unless otherwise stated:
— IEC 60 502-1. Power cables with extruded insulation and their accessories for rated voltages from 1 kV
(Um = 1,2 kV) up to 30 kV (Um = 36 kV) - Part 1: Cables for rated voltages of 1 kV (Um = 1,2 kV) and 3
kV (Um = 3,6 kV) /2/
— IEC 60 502-2. Power cables with extruded insulation and their accessories for rated voltages from 1 kV
(Um = 1,2 kV) up to 30 kV (Um = 36 kV) - Part 2: Cables for rated voltages from 6 kV (Um = 7,2 kV) up
to 30 kV (Um = 36 kV) /3/
— IEC 60 228. Conductors of insulated cables /4/.
and the following recommendations:
— Electra 189: Recommendations for Testing of Long AC Sub-marine Cables with Extruded Insulation for
System Voltage Above 30(36) to 150(170) kV /5/.
2.3 Terminology
The terminology used in the document follows the terminology specified in /2/ or /3/:
Alternating current.
Covering consisting of a metal tape(s) or wires, generally used to protect the cable from
external mechanical effects.
Note:
In this RP Armour is used for the components providing the longitudinal strength and
stiffness to the cable.
Conductor:
Part of a cable which has the specific function of carrying current.
Conductor screen: Electrical screen of non-metallic and/or metallic material covering the conductor.
Core:
Assembly comprising a conductor with its own insulation (and screens if any)
DC:
Direct current.
Insulation screen: Non-metallic, semi-conducting layer in combination with a metallic layer applied on the
insulation.
Insulation:
Assembly of insulating materials incorporated in a cable with the specific function of
withstanding voltage.
Oversheath:
Non-metallic sheath applied over a covering, generally metallic, ensuring the protection of
the cable from the outside.
AC:
Armour:
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-F401, February 2012
Sec.3. Requirements – Page 6
Conducting layer or assembly of conducting layers having the function of control of the
electric field within the insulation.
Sheath:
Uniform and continuous tubular covering of metallic or non-metallic material, generally
extruded. (North America: jacket)
Strand:
One of the wires in a stranded conductor.
In addition the term Water blocking is used for powder, tape, grease, compound, yarn or glue applied under a
sheath or into the interstices of a conductor to prevent water migrating along the cable.
A Barrier sheath, IEC 60 050-461 /6/, having the function of protecting the insulation and its screen from
outside contamination may be specified by the purchaser.
Screen:
3. Requirements
3.1 General construction requirements
3.1.1 Insulation system
The insulation system shall consist of a fully bonded true triple extruded XLPE system (extrusion of conductor
screen, insulation and insulation screen simultaneously). The insulation screen is not required for cables
according to /2/. The use of other insulation system(s) is the subject of agreement between manufacturer and
purchaser.
3.1.2 Conductor
A joint of the entire conductor cross section of the conductor shall not be located in a dynamic part of a cable,
i.e. parts of the cable not resting on the seabed or otherwise prevented from motion.
3.1.3 Armour
Cables shall be balanced with respect to torsion. Un-balanced designs may be used subject to agreement
between manufacturer and purchaser. Test methods and acceptance criteria may have to be modified for unbalanced designs.
3.1.4 Thermal conditions
Cable routing and installation method (e.g. burial, rock dumping, guide tubes etc.) may reduce the heat
transport from the cable. Ancillary equipment like bend stiffeners may act as thermal insulators on the outside
of the cable reducing the heat transport from the cable. Hence, the cable system shall be designed to meet the
worst case thermal loads. The temperature shall not exceed the thermal limitations for any materials in the
power cable.
3.1.5 Longitudinal static strength of cable
The conductor and sheath(s) or screen(s) shall not be taken into account when assessing the longitudinal
capacity of the cable cross section. The strain in the conductor and sheath(s)/screen(s) shall be limited by the
strain in the load carrying elements in the cable cross section.
For applications where it can be shown that it is acceptable that the conductor contributes to the longitudinal
capacity of the cable, e.g. at smaller water depths, the load carrying capacity of the conductor may be taken into
account. In such a case it shall be demonstrated that failure due to creep or any other failure mechanism will
not occur.
A joint of the entire cable cross section shall not be located in dynamic part(s) of a cable, i.e. parts of the cable
not resting on the seabed or otherwise supported.
3.1.6 Fatigue strength of cable.
Cables exposed to dynamic excitation (e.g. cables suspended between floating installations and the sea bottom,
cables exposed to vortex induced vibrations) shall be qualified with respect to fatigue as specified in this
document. For static applications (e.g. cables resting on the sea bed) such qualification is not mandatory.
However, dynamic effects during installation shall be considered.
3.1.7 Hydrostatic strength
The components in the cable cross section shall, when the complete cross section is subjected to an external
hydrostatic pressure not smaller than the larger of 3.5 MPa or the pressure corresponding to the maximum water
depth multiplied by a factor of 1.25, not exhibit any damage that may impair its capacity with respect to
mechanical loads.
Casings (including seals) for cable joints or terminations shall, when subjected to an external hydrostatic
pressure not smaller than the larger of 3.5 MPa or the pressure corresponding to the maximum water depth
multiplied by a factor of 1.5, not exhibit any damage or leakage.
The effect of external hydrostatic pressure on the electrical properties of the cable is handled elsewhere.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-F401, February 2012
Sec.3. Requirements – Page 7
3.2 Conductor
3.2.1 Static strength of conductor
The conductor shall be supported in the longitudinal direction of the cable such that failure due to creep is
prevented. This shall be confirmed by calculations based on adequate test data or data available in literature.
The evaluation of creep shall consider the effect of service temperature on the rate of creep. Variations in the
service temperature shall be considered or a conservative approach chosen.
Guidance note:
For cables suspended in large water depths the self-weight of the conductor may induce unacceptable creep in the
conductor if the conductor is not supported.
---e-n-d---of---G-u-i-d-a-n-c-e---n-o-t-e---
3.2.2 Fatigue strength of conductor
Conductors in cables exposed to dynamic loading shall be qualified with respect to fatigue. A procedure for the
qualification of power cables with respect to fatigue is given in Appendix A.
A failed strand may have a detrimental effect on the conductor screen. It is recommended that failure of one
strand is used as fatigue failure criterion. Alternative failure criteria may be agreed upon between purchaser
and manufacturer.
The safety factor on fatigue life, determined by calculation or testing, shall not be smaller than 10 unless
otherwise agreed between manufacturer and purchaser.
Guidance note:
Electra 189 includes tests for cable joints for submarine cables. These tests are however considered sufficient for static
applications only with respect to demonstrating sufficient fatigue strength. Depending on the application, further tests
need to be carried out.
---e-n-d---of---G-u-i-d-a-n-c-e---n-o-t-e---
3.3 Electrical insulation of core – breakdown strength
Cable insulation for wet applications shall be qualified in accordance with CENELEC HD 605 S2 /7/ Sec.5.4.15.
It is recognised that these tests are carried out using tap water at a small water head (for practical purposes the
specified tests can be considered carried out at atmospheric pressure) and at a temperature of 40ºC. The cables
will be exposed to sea water, a significantly higher hydrostatic pressure and temperatures within a wide range,
the maximum temperature exceeding 40ºC. An evaluation of the significance of the actual service conditions
with respect to the test conditions shall be carried out and the test conditions modified accordingly.
For dry applications, i.e. where the cable cross section includes a sheath preventing radial water transport to
the insulation and where this sheath is duly qualified in accordance with this document, the qualification
specified above need not be carried out.
Guidance note:
Dynamic response of the cable is assumed not to have any adverse effect on the capacity of the insulation with respect
to breakdown strength. It is, however, advised that research in the matter is consulted when becoming available.
---e-n-d---of---G-u-i-d-a-n-c-e---n-o-t-e---
3.4 Screen/sheath for prevention water exposure to the insulation system
3.4.1 General
Sheaths for prevention or slowing down of radial water transport are not prescribed in cables conforming to
IEC 60 502-1 and 2, /2/ and /3/. The requirement for a metallic sheath is the subject of agreement between
manufacturer and purchaser. Herein such a sheath is referred to as a Barrier sheath /6/.
3.4.2 Static strength/overstraining of screen/sheath
In suspended cables the barrier sheath shall be supported in the longitudinal direction of the cable such that
failure due to creep is prevented. Alternatively it shall be demonstrated by analysis based on adequate data on
creep that creep failure will not occur. The evaluation of creep shall consider the effect of service temperature
on the rate of creep.
Guidance note:
The hydrostatic pressure on the cable may be sufficient to support the barrier sheath if the coefficient of friction to the
adjacent components in the cross section and the load carrying capacity of these components are sufficiently large.
---e-n-d---of---G-u-i-d-a-n-c-e---n-o-t-e---
3.4.3 Buckling
If not properly supported, bending of a barrier sheath may induce local buckling, particularly for combinations
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-F401, February 2012
Sec.3. Requirements – Page 8
of thin materials made from material with a large Young modulus (e.g. metallic tubes or foils). Local buckling
shall be prevented.
3.4.4 Corrosion of sheath/screen
The material in the barrier sheath shall be chosen such that it has the sufficient resistance to corrosion
considering the service environment: exposure to sea water, temperature. There shall be no penetration of the
sheath due to corrosion (holes, pits, cracks etc.) during the service life of the cable.
3.4.5 Fatigue strength of barrier sheaths – global loads
Barrier sheaths in cables exposed to dynamic loading shall be qualified with respect to fatigue. A procedure for
the qualification of power cables with respect to fatigue is given in Appendix A.
Penetration of the sheath, e.g. a crack, hole etc., shall constitute failure. The safety factor on fatigue life,
determined by calculation or testing, shall not be smaller than 10 unless otherwise agreed between
manufacturer and purchaser.
3.4.6 Fatigue strength of barrier sheath – thermal effects/radial expansion
Repeated thermal expansion/contraction of components inside of the sheath may induce fatigue stress in the
circumferential direction. The number of load cycles may be small, but the circumferential strain induced in
the sheath may be relatively large. The number and magnitude of the load cycles shall be specified by the
purchaser as well as the relevant service temperatures.
A satisfactory fatigue life of the sheath shall be demonstrated by recognised methods taking into account
deformation in the non-linear regime of the materials. The design with respect to fatigue shall be based on
fatigue design curves or by direct testing on cable samples. Fatigue design curves shall be determined by testing
at the strain levels that are relevant and be expressed as fatigue life in number of load cycles vs. strain range.
Damage accumulation shall be carried out in accordance with recognised methods.
The fatigue life may alternatively be determined by testing of samples of cable by exposing the cable to heating
cycles while applying the relevant external pressure.
Penetration of the sheath, e.g. a crack, hole etc., shall constitute failure. The safety factor on fatigue life,
determined by calculation or testing, shall not be smaller than 10 unless otherwise agreed between
manufacturer and purchaser.
3.5 Water blocking
3.5.1 Water blocking
Block against water transport along the conductor and the interstice between screen and external sheath shall
be fitted.
The type of water block is subject to agreement between purchaser and manufacturer.
3.5.2 Longitudinal water blocking along conductor
The cable shall be tested in accordance with and comply with the requirements given in /5/ Sec.4.8.3 –
Conductor penetration test. The test shall be carried out with sea water, preferably with artificial sea water
according to recognized standards or according to agreement between purchaser and manufacture. The test
pressure shall not be lower than the maximum hydrostatic pressure in operation.
3.5.3 Longitudinal water blocking between screen and external sheath
The cable shall be tested in accordance with and comply with the requirements given in /5/ Sec.4.8.3 – Outer
sheath penetration test. Alternative requirements may be agreed upon between manufacturer and purchaser.
The simultaneous effect of an external hydrostatic pressure on the external sheath on the water transport along
the cable may be considered and included in the test procedure.
3.6 Degassing
Cables shall be de-gassed as part of the manufacturing process. The de-gassing procedure is subject to
agreement between the purchaser and manufacturer.
Guidance note:
There is at present no generally accepted procedure for degassing.
---e-n-d---of---G-u-i-d-a-n-c-e---n-o-t-e---
3.7 Longitudinal gas barrier
The use of gas blocks, continuous or intermittent, is the subject of agreement between manufacturer and
purchaser.
The design and fabrication of the cable shall aim at minimising the volume of trapped voids inside the cable.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-F401, February 2012
Sec.3. Requirements – Page 9
3.8 Armour
3.8.1 Static strength
The allowable utilization factors for load carrying elements for longitudinal loads are specified in Table 3-1.
The utilisation factor shall be taken as the ratio of the applied load to the lesser of the specified minimum yield
strength and 90% of the specified minimum ultimate tensile strength of the steel material in the armour.
For armour of other material than steel the utilisation factors are subject to agreement between the purchaser
and the manufacturer.
Table 3-1 Armour utilisation factors
Utilisation factor
Normal operation
0.67
Installation
0.78
Abnormal operation
1.00
3.8.2 Fatigue strength
Armour in cables exposed to dynamic loading shall be qualified with respect to fatigue. A procedure for the
qualification of power cables with respect to fatigue is given in Appendix A.
The safety factor on fatigue life, determined by calculation or testing, shall not be smaller than 10.
3.9 Anchoring of armour
3.9.1 Static strength
The utilisation factor of anchoring of armour in end terminations shall comply with Table 3-1. The utilisation
ratio shall be calculated as the applied load divided by the capacity of the termination.
3.9.2 Fatigue strength
Armour anchors subjected to significant fatigue loading shall be qualified with respect to fatigue strength by
testing. The safety factor on fatigue life, determined by calculation or testing, shall not be smaller than 10.
3.10 Radial compression – load carrying capacity
3.10.1 Radial compression – load cases
All radial loads on the cable cross section shall be considered. The design load cases shall include, but not be
limited to:
—
—
—
—
—
hydrostatic pressure
installation loads, e.g. clamping forces from caterpillar, temporary hang-off
contact forces in chutes
loads from clamps for anchors, buoyancy modules etc.
support reactions, e.g. over mid-water arches.
3.10.2 Radial compression – allowable load/stress/strain
The manufacturer shall specify the allowable compression loads, short term and long term, and/or allowable
compression strains as relevant, for each of the types of loads identified in accordance with 3.10.1.
The manufacturer shall specify radial compression creep data enabling the proper design of clamps with respect
to possible relaxation of clamp forces due to creep. Creep data shall reflect the service temperatures the cable
will experience.
Compression may lead to damage of the semi-conducting screen when compressed on the conductor. This
failure mode shall be considered when determining the allowable compression force.
Guidance note:
The maximum allowable compression force/strain may have a large impact on the choice of installation method,
installation equipment, design of ancillary equipment like clamps etc. A clear specification of the allowables is
therefore important at an early stage. The design of the cross section may be the subject to an iterative process between
manufacturer and purchaser.
---e-n-d---of---G-u-i-d-a-n-c-e---n-o-t-e---
3.11 Flexibility/Compliance
Possible requirements to the flexibility of the cable shall be clearly stated by the purchaser.
The manufacturer shall specify the flexibility or the bending stiffness of the cable, maximum and minimum,
for different temperatures as agreed with the manufacturer. The flexibility depends on the curvature of the
cable. The flexibility should as a minimum be stated for the minimum bending radius specified by the
manufacturer for installation and service.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-F401, February 2012
Sec.4. References – Page 10
3.12 Bending radius
Possible requirements to the allowable minimum bending radius of the cable, during installation and operation,
shall be clearly stated by the purchaser.
The manufacturer shall specify the minimum allowable bending radius for different temperatures for storage,
installation and operation.
3.13 Coiling
Cables which during manufacture, storage or installation may be subjected to coiling shall be qualified in
accordance with /8/.
4. References
/1/ ISO 13628-5. Petroleum and natural gas industries - Design and operation of subsea production systems Part 5: Subsea umbilicals. 2009.
/2/ IEC 60 502-1. Power cables with extruded insulation and their accessories for rated voltages from 1 kV
(Um = 1,2 kV) up to 30 kV (Um = 36 kV) - Part 1: Cables for rated voltages of 1 kV (Um = 1,2 kV) and 3 kV
(Um = 3,6 kV). 2009.
/3/ IEC 60 502-2. Power cables with extruded insulation and their accessories for rated voltages from 1 kV
(Um = 1,2 kV) up to 30 kV (Um = 36 kV) - Part 2: Cables for rated voltages from 6 kV (Um = 7,2 kV) up to
30 kV (Um = 36 kV). 2005.
/4/ IEC 60 228. Conductors of insulated cables. 2004.
/5/ Electra 189. Recommendations for Testing of Long AC Sub-marine Cables with Extruded Insulation for
System Voltage Above 30(36) to 150(170) kV.
/6/ IEC 60 050-461. International Electrotechnical Vocabulary - Part 461: Electric cables. 2008.
/7/ CENELEC HD 605 S2. Electric cables - Additional test methods. 2008.
/8/ Electra 171. Recommendations for Mechanical Tests on Sub-marine cables.
/9/ DNV RP-C203. Fatigue Design of Offshore Steel Structures.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-F401, February 2012
App.A Qualification with Respect to Fatigue – Page 11
APPENDIX A
QUALIFICATION WITH RESPECT TO FATIGUE
A.1 Limitations
This section covers qualification with respect fatigue loading in the high cycle regime.
Due to the limited strain levels normally expected in service, well known insulation and oversheath materials
with large strain to failures need not be subjected to a qualification procedure. The following materials are
considered to have adequate fatigue strength with respect to mechanical damage:
—
—
—
—
low density thermoplastic polyethylene (PE)
high density thermoplastic polyethylene (HDPE)
cross-linked polyethylene (XLPE)
ethylene propylene rubber (EPR),
unless significantly modified with fillers, additives or similar.
For other insulation materials a qualification with respect to fatigue shall be carried out. The qualification
program will be subject to agreement in each particular case.
A.2 Definitions
C
Curvature. C = 1/ρ
Cmax The maximum curvature during one deformation cycle = 1/ ρmin
Cmin The minimum curvature during one deformation cycle = 1/ ρmax
ρ
Bending radius. The bending radius shall be assigned a negative and a positive value when bending
occurs to either side of a straight line, see Figure A-1
Umbilical/cable
ρ<0
ρ>0
Figure A-1
Definitions of bending radius sign
ρstatic
ρmax
ρmin
ε
εmax
εmin
Rρ
The bending radius in the static equilibrium position of the cable
The maximum bending radius during one deformation cycle
The minimum bending radius during one deformation cycle
Strain
Maximum strain during one deformation cycle
Minimum strain during one deformation cycle
Rε
N
Strain ratio. Rμ = εmin/ εmax
Number of deformation cycles
D
Accumulated fatigue damage
k
ni
Number of strain/stress blocks
Number of cycles in strain block i
Curvature ratio. Rρ = 1 / ρ max
1 / ρ min
k
D=
i
ni
Ni
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-F401, February 2012
App.A Qualification with Respect to Fatigue – Page 12
Ni
m
Number of cycles to failure at strain range for strain block i
Slope of fatigue curve.
A.3 Input data
A detailed drawing of the cable cross section shall be available as well as specifications of all materials and
components included in the cross section. The specification shall be given on a level of detail that is sufficient
to carry out the qualification.
A.4 Pre-test straining
The cable and its components will be subjected to operations during which it may be deformed in ways that
may have an effect on the fatigue properties. Examples of such operations are:
—
—
—
—
—
cold work during closing and compacting of conductor
forming during cable manufacturing (e.g. metallic sheaths, screen or armour wire)
bending during reeling operations during manufacturing of the cable
bending when manufacturing the umbilical
bending during installation.
The effect of these operations on the mechanical properties shall be represented and/or simulated in a
representative manner, on cable samples or the individual components prior to testing.
A.5 Qualification principles
A.5.1 Qualification method
The different components in the cable cross section may be qualified separately, ref. A.6. For the subsequent
qualification of the complete cable cross section a reliable model of the cable cross section shall be established.
This model shall as a minimum describe how the local response of the components is dependent on the global
response of the cable, how the different components interact in the cable cross section, how relevant material
parameters scale with size, where relevant, etc. Important factors are friction between the components.
Calculation tools and assumptions for this purpose will in such a case also be subject to qualification.
Alternatively, qualification can be carried out on a complete cable cross section, i.e. all tests are carried out
on samples of the complete cable as delivered from the manufacturer, ref. A.7. (In some cases it may be
sufficient to carry out testing on complete cores, i.e. when it can be demonstrated that the components outside
the core do not have any effect on the test results.)
Irrespective of the approach that is chosen a number of electrical verification tests shall be carried out, ref. A.8.
For the mechanical tests relatively short samples may be used. For the electrical tests the length of the
specimens shall be as specified in the specified standards. Hence, excess cable length may have to be included
in the mechanical test specimens to carry out the subsequent electrical tests.
Guidance note:
Existing calculation tools applied to umbilicals might be possible to use to calculate the response of some of the
components in the cross section, e.g. conductor, screen wire, armour wire.
---e-n-d---of---G-u-i-d-a-n-c-e---n-o-t-e---
A.5.2 Load conditions
The fatigue load effect due to bending of the cable may be specified in terms of bending radius, i.e. a
deformation. Some of the materials, e.g. copper, may be deformed non-linearly. For such materials fatigue tests
and results should be performed in controlled strain and presented as strain vs. number of load cycles.
A.5.3 Load effect ratio – mean stress/strain
A cable will be subjected to a static curvature on which a dynamic curvature is superimposed and in addition
a tensile load. Consequently stress or strain ratio may vary from application to application and along the cable.
Stress/strain ratio shall be reflected in the qualification.
A.5.4 Temperature
Test conditions shall reflect the range of the service temperature whenever the temperature has a significant
influence on the properties. Alternatively, generally accepted methods for modifying material properties due
to temperature effects may be applied.
Fatigue testing may heat the test specimens, particularly at higher test frequencies and/or if the specimen has
some form of insulation, e.g. when testing a complete core or cable cross section. Further, the cable contains
visco-elastic materials and some materials may be deformed into their non-linear regime, leading to further
generation of heat. This shall be considered when carrying out the tests. Control of the test temperature shall
be established.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-F401, February 2012
App.A Qualification with Respect to Fatigue – Page 13
A.6 Qualification based on components
A.6.1 Qualification testing
A.6.1.1 Conductor
Test shall be carried out as uni-axial tensile tests on individual conductor strands (single wires). A fatigue
design curve shall be established as specified in A6.2. The failure criterion is broken strand. Due to the
relatively strong non-linear behaviour of copper the tests may have to be carried out in controlled strain.
The possible effect on the fatigue strength of wear and/or fretting between the strands shall be considered.
Fretting damage may occur due to the sliding between strands when the core is bent. The amount of fretting
depends also on the compressive force between the strands, due e.g. to external hydrostatic pressure, fitting of
clamps for buoyancy modules of anchors. It shall be demonstrated that wear/fretting is not significant with
respect to the fatigue strength. Otherwise alternative test methods shall be used, e.g. fatigue testing of the
complete conductor, e.g. in a completed core of cable.
A.6.1.2 Solid metallic tube sheath
Qualification can be based on small scale testing of samples taken from the tube. A fatigue design curve shall
be established as specified in A6.2. The failure criterion is broken specimen.
Test specimens shall be cut in the longitudinal direction of the tube and shall include the whole thickness. A
narrow gauge section may be included in the specimen.
Test specimens shall also include longitudinal weld(s), if any. Test shall be carried out on both base material
and welds.
A.6.1.3 Sheath – other configurations
Some types of sheaths, e.g. based on metallic foils, are not possible to fatigue test as individual components.
Qualification of such sheaths should be based on testing of cores or complete cable cross sections. Alternatively
testing could be carried out with the foil adhered to adjacent plastic layer for stabilisation of the foil, ref. A6.1.2.
Special attention shall be given to adhesive bonds used in foils. Such bonds will be subjected to a shear load
when the cable is bent. The effect of the fatigue shear stress shall be considered.
Other configurations of e.g. solid closed corrugated profiles may or may not be suitable to qualify based on
small scale testing. It may be impossible from a practical point of view, or the geometry may not lend itself to
reliable analysis of local stress/strain response. Such configurations should be tested either as a single
component or as part of the complete cable cross section depending on the possibility to generate reliable test
results.
A.6.1.4 Longitudinal armour, wire screen
Steel tensile armour will normally operate in the linear elastic regime. Fatigue testing can therefore be carried
out in load control and fatigue design curves can be presented based on stress range. A fatigue design curve
shall be established as specified in A6.2.
For other materials, e.g. copper screen wires, fatigue curves determined in load control may not be representative.
For such materials the requirements may have to be adapted, but shall follow the same principles.
Testing shall include welds for joining armour wire, if used in the dynamic section of the cable.
A.6.2 Fatigue design curve
A.6.2.1 Fatigue testing
Testing shall be carried out at minimum three different stress/strain range levels. For the lowest stress/strain
range it shall be aimed at obtaining fatigue lives of at least of the order of 2×106 load cycles. For the highest
stress/strain range it shall be aimed at obtaining fatigue lives of the order of 104 load cycles.
A minimum of 5 valid test results shall be obtained for each stress/strain range level.
A suitable method of gripping the specimens should be developed so that the specimens do not sustain damage
that may have an effect on the fatigue life. Alternatively, specimens failing in the gripping area may be accepted
if the fatigue design curve is based on these results.
The tests shall preferably be carried out at the R-ratio(s) and at the mean strain(s) that is relevant based on the
loading conditions. If the strain range to be tested is completely or partly on the compression side testing may
be impossible due to problems with the stability of the specimens.
A.6.2.2 Presentation and analysis of fatigue test results
Results shall be presented as stress or strain range, as applicable, versus number of cycles to failure. The results
shall be presented numerically and in plots that present the results in a representative manner.
R-values, pretension and specimen temperature shall be stated as well as any observations that may be relevant
for the evaluation of the results.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-F401, February 2012
App.A Qualification with Respect to Fatigue – Page 14
Fatigue design curve(s) shall be estimated based on the data using the least squares method (ref. Appendix D).
The design curve shall be given at the mean minus 2 standard deviations in log(N), i.e. representing a 97.5%
probability of survival.
For data for which complete information about the number of cycles to failure is not available (e.g. when testing
complete cross sections) the method given in Appendix E may be applied. The design curve shall represent a
97.5% probability of survival. It is also referred to /9/ that gives a procedure for how to analyse data of this type.
If the number of tested stress levels is sufficiently large the mean and design curves may be divided in more
than one regression line, thus increasing accuracy.
A.7 Qualification of complete cable cross section
A.7.1 Qualification testing
The qualification program shall address as a minimum the items given in A6.
Testing shall be conducted to establish fatigue design curves (SN-curves) for the different components listed
in A6 for which satisfactory fatigue life can not be demonstrated by alternative means.
Failure of the cable cross section is defined as failure of at least one of the components.
Guidance note:
For a cross section with a solid metallic sheath it may be the case that the sheath will be the component with the
shortest fatigue life by a significant margin. If this can be demonstrated by other means than testing of the cross
section, fatigue testing of the remaining components may not be necessary. Similar evaluations may be relevant for
other components.
---e-n-d---of---G-u-i-d-a-n-c-e---n-o-t-e---
A.7.2 Fatigue design curve
A.7.2.1 Test method
The test methods shown in Appendix B may be used. Other test methods providing reliable results may be
accepted.
Testing shall be carried out for realistic R-values, including the effect of pre-tension. Alternatively generally
acceptable methods for converting test results for R-values different from the test conditions may be used.
Dynamic axial tension shall be considered.
If 4-point-bending is used, due regard to the proper control of the bending radius shall be established, see
Appendix B.
The length of the gauge section shall be at least 6 times the outer diameter of the cable, in any case not smaller
than 400 mm. The length of the specimens and the configuration of the test fixture shall be such that end effects
are eliminated in the gauge section.
A.7.2.2 Number of tests
A fatigue design curve shall be established. The curve shall be given as allowable number of load cycles vs.
stress/strain/curvature range, as applicable. The curve shall be based on testing of at least three different strain/
curvature range levels. For the lowest range it shall be aimed at obtaining fatigue lives of at least 2×106 load
cycles. For the highest range it shall be aimed at obtaining fatigue lives of the order of 2×103 load cycles. Nonmetallic materials may require testing at additional strain levels.
A minimum of 5 valid test results shall be obtained for each strain/curvature range level.
If tests are carried out at more strain levels the design curve may be divided in more than one straight segment,
thus increasing the accuracy.
A.7.2.3 Detection of failure
A clear definition of failure shall be stated when reporting the results. This definition may be based on the
acceptance criteria of the component in question, A5, but may alternatively have to be defined based on what
can be detected during the tests. For metallic materials failure may be defined as e.g. breakdown of the stiffness
of the specimen, the observation of a macroscopic fatigue crack, the breakage of a core strand etc.
Detection techniques shall be established for all the relevant acceptance criteria. Detection techniques are
discussed In Appendix C. Where non-destructive detection techniques that provide a reliable indication of the
point of failure are not available conservative measure of the fatigue life of the specimens must be applied.
A.7.2.4 Presentation of test results and analysis
For the presentation and analyses of the results, see A6.2.2.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-F401, February 2012
App.A Qualification with Respect to Fatigue – Page 15
A.8 Electrical verification tests
A.8.1 General
There exists limited service experience from the use of LV and MV cables in subsea dynamic applications. In
order to ensure that all possible failure modes are covered it is also required that cable samples shall undergo
standard electrical tests followed by a prescribed fatigue load (in the form of a full scale bending test) followed
by the same standard electrical tests.
The following tests shall be carried out irrespective of which qualification alternative is chosen: testing of the
whole cross section or testing of components.
A.8.2 Test method – electrical verification tests – long specimens
The tests include verifying the electrical and some mechanical properties prior to and after the test sample being
subjected to a prescribed fatigue load.
Pre-fatigue electrical tests
The following tests, Table A-1, shall be carried out prior to fatigue loading.
Table A-1 Pre-fatigue Electrical Tests
Type of test
Test specification
Conductor resistance
IEC 60 502-2 Sec. 16.2
IEC 60 502-1 Sec. 15.2
Partial discharge
IEC 60 502-2 Sec. 16.3
Voltage
IEC 60 502-2 Sec. 16.4
IEC 60 502-1 Sec. 15.3
Acceptance criteria
IEC 60 502-2 Sec. 16.2
IEC 60 502-1 Sec. 15.2
IEC 60 502-2 Sec. 16.3
IEC 60 502-2 Sec. 16.4
IEC 60 502-1 Sec. 15.3
Application of fatigue loading
Subsequent to the Pre-fatigue electrical tests the specimens shall be subjected to dynamic loading. The methods
shown in Appendix B may be used for applying dynamic deformation to the specimens, with due regard to the
control of the curvature.
The length of the gauge section (the part of the specimen exposed to dynamic loading) shall be at least 6 times
the outer diameter of the cable, in any case not smaller than 400 mm. The length of the specimens and the
configuration of the test fixture shall be such that end effects have been eliminated in the gauge section. The
length of the specimens shall also be in accordance with the requirements of the specified IEC test standards.
All specimens shall be subjected to the same deformation in terms of range of curvature, strain range etc.
The curvature/strain range shall be based on the application for which cable cross section is to be qualified. The
fatigue load shall be described by a histogram or similar giving the curvature range and corresponding number
of cycles. The effective curvature ΔCeff shall be calculated according to the following equation:
k
ΔC eff =
1/ m
n
i
⋅ (ΔC i ) m
i =1
(A1)
N
The fatigue loading is thus given by the two parameters:
— the effective curvature range ΔCeff
— the total number of load cycles
The fatigue load that the cable will be exposed to in service can thus be represented by the expression:
(ΔC eff ) m ⋅ N
(A2)
The fatigue load used in the test shall be given by the following equation:
(ΔC test ) m ⋅ N test ≥ 10 ⋅ (ΔC eff ) m ⋅ N
(A3)
The range of curvature and the number of cycles in the test can in principle be chosen arbitrarily as long as the
equation above is fulfilled. However, Ntest shall not be taken smaller than 50 000.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-F401, February 2012
App.A Qualification with Respect to Fatigue – Page 16
In the preceding tests more than one slope m of the fatigue curve may have been obtained, e.g. it may vary from
component to component. The test condition shall be determined based on the most conservative choice of m.
Post-fatigue electrical tests
The following tests, Table A-2 shall be carried out subsequent to fatigue loading.
Table A-2 POST-FATIGUE ELECTRICAL TESTS
Type of test
Test specification
Electrical
Conductor resistance
IEC 60 502-2 Sec. 16.2
IEC 60 502-1 Sec. 15.2
Partial discharge
IEC 60 502-2 Sec. 16.3
Voltage
IEC 60 502-2 Sec. 16.4
IEC 60 502-1 Sec. 15.3
Mechanical/visual
Verification of cross section
The dimensions of all components in the
cross section shall be verified 1)
Visual inspection of conductor In fatigue loaded area
Visual inspection of sheath
In fatigue loaded area
Dye penetration examination of In fatigue loaded area
sheath
Visual inspection of armour
In fatigue loaded area
wires
Acceptance criteria
IEC 60 502-2 Sec. 16.2
IEC 60 502-1 Sec. 15.2
IEC 60 502-2 Sec. 16.3
IEC 60 502-2 Sec. 16.4
IEC 60 502-1 Sec. 15.3
All measurements to be within specified
tolerances.
No visual defects
No visual defects
No defects
No visual defects
1) The following shall be measured:
—
—
—
—
—
—
—
—
diameter or weight per unit length of conductor
inner diameter and thickness of conductor screen
inner and outer diameter of insulation
inner diameter and thickness of insulation screen
wire dimensions of screen
inner diameter and thickness of sheath
wire dimension of armour
inner diameter and thickness of external sheath.
A.8.3 Number of tests
At least 5 specimens shall be tested as described above.
A.8.4 Electrical verification tests
All specimens in the electrical verification test shall pass the acceptance criteria specified in A.8.2.
For applications where the fatigue load has not yet been established the qualification can be based on a
specified fatigue load ΔCeff ⋅ N. The cable is then qualified for fatigue loads that do not exceed the specified
fatigue load.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-F401, February 2012
App.B Test Methods – Fatigue Loading of Complete Cables – Page 17
APPENDIX B
TEST METHODS – FATIGUE LOADING OF COMPLETE CABLES
B.1 General
Two methods for fatigue testing are considered applicable for testing of complete cable cross sections:
— 4-point-bend testing
— bending against template.
These two methods are described in the following sections.
Alternative methods for applying a fatigue deformation and/or for detecting failure may be accepted subject to
a qualification of the reliability of the method(s) to impart a realistic fatigue deformation and/or to detect
failures(s).
B.2 4-point-bending
The principle for the method is shown in the Figure B-1. The method can be used on both short and long
specimens, but for practical reasons the gauge section will have to be relatively short.
The test set-up can be used for controlled displacement and controlled load and for different R-values.
The curvature of the cable can in principle be determined based on beam theory provided the length of the
specimen is long enough to eliminate end effects. Curvature as a function of displacement of the moving yoke
shall be calibrated. This can be accomplished by instrumenting one or more specimens with e.g. strain gauges
in order to obtain strain level at various positions as a function of yoke displacement. Effects of creep shall be
specially considered.
For long and or slender configurations 4-point-bending may not give sufficient control of the bending radius,
particularly when a tensile preload in the cable is used. Bending against templates may be necessary in such
cases.
Figure B-1
4-point bending
B.3 Bending against template
The principle for the method is shown in the figure B-2 and B-3. The method may for practical reasons not be
suited for long specimens.
The specimen is bent against preformed templates thus controlling the curvature directly. The test frequency
may be restricted to lower values than for 4-point-bending. Other arrangements using templates are possible.
The test method can be used for controlled displacement only. Varying R-values can be applied by modifying
the shapes of the templates.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-F401, February 2012
App.B Test Methods – Fatigue Loading of Complete Cables – Page 18
Figure B-2
Bending against former
Figure B-3
Bending against former
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-F401, February 2012
App.C Fatigue Testing Detection Techniques – Page 19
APPENDIX C
FATIGUE TESTING DETECTION TECHNIQUES
C.1 Metallic materials
Metallic materials are/may be used in the following components:
—
—
—
—
conductor core
sheath
screen
armour.
A number of different techniques for detecting defects are available, depending on the component in question.
Detection techniques are discussed below. Common for all or most of the techniques is that they are impossible
to automate or that automation will require a significant effort.
C.1.1 Visual inspection – dissection
Destructive detection method
Visual inspection from the outside of components inside the specimen during the course of the test is in
principle impossible, except for the external sheath. Visual inspection can be carried out after the specimen has
been taken apart for access to the component of interest, but this will prohibit further testing of the specimen.
Visual inspection can therefore be used for verifying that a defect has not been formed after a certain number
of cycles, but it will not give any indication of the remaining fatigue life. Or, it can be used to verify that a
defect has been formed, but not exactly at what number of load cycles. These uncertainties could in principle
be mitigated if a reasonably reliable way of back calculating defect growth rates is available. Such methods
may be included in the basis for the qualification.
The result from using visual inspection is illustrated in principle in Figure C-1 below. The cases where a defect
is found or no defect is found upon inspection are shown, the arrows indicating the uncertainty in determining
the “true” number of cycles to failure. Results of this type will increase the uncertainty in determining a SNcurve.
∆ε
Examination: no defect
Examination: defect
N
Figure C-1
Presentation of fatigue test results
C.1.2 Structural breakdown/Reduction of stiffness
Non-destructive detection method
Structural breakdown or a measurable reduction of the specimen stiffness is indicative of failure of one or more
of the load carrying components, e.g. a solid tube sheath or armour wires.
The method is most sensitive for the component giving the largest contribution to the stiffness of the cross
section, e.g. a solid metal tubular sheath. It is relatively less sensitive to failure of armour or screen wires. Such
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-F401, February 2012
App.C Fatigue Testing Detection Techniques – Page 20
wires are helically wound and relatively numerous such that a fairly large number of wires have to fail in close
proximity before a detectable change in stiffness occurs.
Similarly, it is questionable whether failure in the core can be detected by this method.
For early detection of fatigue cracks calibration of the detection method is probably required, i.e. the stiffness
as a function of crack size. The sensitivity to small cracks may be low.
Longitudinal cracks, e.g. in longitudinal weld seams, may not be possible to detect by this method due to their
relatively small effect on the stiffness.
C.1.3 Electrical resistance
Non-destructive detection method
Fatigue crack growth in metallic materials will eventually increase their electrical resistance. However, the
sensitivity of this technique may not be sufficiently high for many cable cross sections.
Currents applied to the sheath may pass through the semi-conducting layers under the sheath giving no or very
little appreciable increase in resistance as the length of the damage in the sheath is short, further reducing the
sensitivity of the method. Whether the sensitivity of this method is sufficient or not has to be determined from
case to case.
The resistance of wire screens may not increase appreciably until a large portion of the wires are broken in one
cross section.
The resistance of armour wire could be used as a detection method, provided the individual armour wires are
isolated from one another.
C.1.4 Eddy current
Non-destructive detection method
May be developed for use for detection of fatigue cracks in metal sheaths and screens and armour wires.
Depending on the level of development of the tools early detection of fatigue cracks may be possible. A
significant amount of development and calibration work may be necessary.
Eddy current can not be used, or is difficult to use, for components inside metallic screens or sheaths.
The method may require a significant investment for automating the inspection during testing.
C.1.5 X-ray
Non-destructive detection method
X-ray is a well developed technique for detection of defects in metallic materials. However, the ability of the
technique to discover defects in complete cables, in terms of sensitivity and discrimination between the
different components, has to be demonstrated in each individual case.
The method is not possible to automate.
C.1.6 Leakage
Non-destructive detection method
May tentatively be used for detection of through-wall defects on solid metal sheaths by applying compressed
air on the inside of the sheath. The method requires free passage of air inside and outside of the sheath. This
may not be possible to achieve for all cross section designs.
C.2 Plastic materials
Based on the discussion above the detection techniques that are considered possible to use are listed in the table
below. Development of detection techniques will form an important part of the qualification process. Where
suitable detection techniques are not available alternative means of determining the fatigue strength in a
conservative manner may be used.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-F401, February 2012
App.C Fatigue Testing Detection Techniques – Page 21
Table C-1 Defect detection techniques for metallic materials
Component
Failure mechanism
Core
Fatigue crack growth
Sheath
Fatigue crack in sheath –
circumferential crack
Fatigue crack in sheath –
longitudinal crack
Screen, wire
Fatigue crack growth
Armour
Fatigue crack growth
Possible detection technique
Visual inspection – dissection
Electrical resistance
(X-ray)
Stiffness
Leakage
Eddy current
Electrical resistance
(X-ray)
Visual inspection – dissection
Leakage
Eddy current
(X-ray)
Visual inspection – dissection
Eddy current
Electrical resistance
(X-ray)
Visual inspection – dissection
Eddy current
Electrical resistance
(X-ray)
Visual inspection – dissection
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-F401, February 2012
App.D Estimation of Fatigue Design Curves – Least Squares Method – Page 22
APPENDIX D
ESTIMATION OF FATIGUE DESIGN CURVES –
LEAST SQUARES METHOD
SN-type fatigue design curves can be expressed on the following form:
log N = log a - m log Δε
(D1)
where:
N
Δε
m
a
:
:
:
:
predicted number of cycles to failure for strain range Δε
strain range
inverse slope of SN-curve
intercept of log N axis by SN-curve
Using the least squares method the constants in the regression curve can be estimated by the following
equations:
n
m=
n
n
n log Δε i ⋅ log N i −  log Δε i  log N i
1
1
1


n (log Δε i ) 2 −  log Δε i 
1
1

n
log a =
n
n
n
1
1
2
 log N i − m log Δε i
(D2)
(D3)
n
where n is the number of data points/test results (log Ni; log Δεi) and:
Ni : number of cycles to failure in test i
Δεi : strain range in test i
The standard deviation s of log N is given by the following equation:
1/ 2
 n
2 
  [log N i − (log a − m log Δε i )] 
s= 1

n −1




(D4)
A fatigue design curve design curve can then be defined by the following equation, based on a 97.5%
probability of survival:
log N = log ā - m log Δε
where:
log ā = log a - 2⋅s.
DET NORSKE VERITAS AS
Recommended Practice DNV-RP-F401, February 2012
App.E Estimation of Fatigue Design Curves - Incomplete Observations of Number of Cycles to Failure – Page 23
APPENDIX E
ESTIMATION OF FATIGUE DESIGN CURVES INCOMPLETE OBSERVATIONS OF NUMBER OF CYCLES TO FAILURE
The procedure below applies to situation where the exact numbers of cycles to failure for the test specimens
are not known.
SN-type fatigue design curves can be expressed on the following form:
log N = log a - m log Δε
N
Δε
m
a
:
:
:
:
(E1)
predicted number of cycles to failure for strain range Δε
strain range
inverse slope of SN-curve
intercept of log N axis by SN-curve
For simplicity the equation above is written as:
y = A + Bx
(E2)
where:
x
y
A
B
=
=
=
=
log Δε
log N
log a
-m
In case of SN-curves it is often assumed that A is Normal distributed with a constant standard deviation equal
to σA. y is then also Normal distributed with a constant standard deviation σA and a median value given by E2.
The distribution density function for y is thus given by:
 ( y − A − Bx) 2 
exp−

2σ A2
2πσ A2


1
f y ( x) =
(E3)
with the cumulative probability function given by:
(E4)
Fy ( x) =  f y ( x)dx
Assume that the test observations consist of knf of SN-data (Xnfi;Ynfi) where no failure was observed on
inspection (i.e. the fatigue life is longer than the number of cycles applied in the test) and kf of SN-data (Xf;Yf)
where failure was observed on inspection (i.e. the fatigue life is shorter than the number of cycles applied in
the test).
The likelihood function for a sample including knf observations of non-failed specimens and kf observations
of failed specimens is then given by:
[
L = ∏ Fy (Ynfi ) ⋅ ∏ 1 − Fy (Y fi )
k nf
]
(E5)
kf
Since there is no way to determine E5 analytically, the estimators for A, B and σA may be found by maximising
E5 numerically.
DET NORSKE VERITAS AS