dnv design against accidental loads rp-c204 2010-10
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RECOMMENDED PRACTICE
DET NORSKE VERITAS
DNV-RP-C204
DESIGN AGAINST
ACCIDENTAL LOADSOCTOBER 2010
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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 andconsultancy services relating to quality of ships, offshore units and installations, and onshore industries worldwide, and carriesout research in relation to these functions.
DNV service documents consist of amongst other 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 EnergyO) Subsea Systems
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Recommended Practice DNV-RP-C204, October 2010
Changes – Page 3
CHANGES
• General
As of October 2010 all DNV service documents are primarily published electronically.
In order to ensure a practical transition from the “print” schemeto the “electronic” scheme, all documents having incorporated
amendments and corrections more recent than the date of thelatest printed issue, have been given the date October 2010.
An overview of DNV service documents, their update statusand historical “amendments and corrections” may be foundthrough http://www.dnv.com/resources/rules_standards/.
• Main changes
Since the previous edition (November 2004), this documenthas been amended, most recently in April 2005. All changeshave been incorporated and a new date (October 2010) has
been given as explained under “General”.
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Recommended Practice DNV-RP-C204, October 2010
Page 4 – Changes
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Recommended Practice DNV-RP-C204, October 2010
Contents – Page 5
CONTENTS
1. GENERAL .............................................................. 7
1.1 Introduction .............................................................7
1.2 Application ...............................................................7
1.3 Objectives.................................................................71.4 Normative references ..............................................71.4.1 DNV Offshore Standards (OS)...........................................71.4.2 DNV Recommended Practices (RP)...................................7
1.5 Definitions ................................................................7
1.6 Symbols.....................................................................8
2. DESIGN PHILOSOPHY....................................... 9
2.1 General .....................................................................9
2.2 Safety format............................................................9
2.3 Accidental loads.......................................................9
2.4 Acceptance criteria..................................................9
2.5 Analysis considerations.........................................10
3. SHIP COLLISIONS............................................. 10
3.1 General ...................................................................10
3.2 Design principles....................................................10
3.3 Collision mechanics ...............................................113.3.1 Strain energy dissipation...................................................113.3.2 Reaction force to deck ......................................................11
3.4 Dissipation of strain energy..................................11
3.5 Ship collision forces...............................................113.5.1 Recommended force-deformation relationships...............113.5.2 Force contact area for strength design of large diameter
columns.............................................................................133.5.3 Energy dissipation is ship bow .........................................13
3.6 Force-deformation relationships for denting oftubular members ...................................................14
3.7 Force-deformation relationships for beams........143.7.1 General..............................................................................143.7.2 Plastic force-deformation relationships including elastic,
axial flexibility..................................................................143.7.3 Support capacity smaller than plastic bending moment of
the beam............................................................................163.7.4 Bending capacity of dented tubular members ..................16
3.8 Strength of connections.........................................17
3.9 Strength of adjacent structure .............................17
3.10 Ductility limits........................................................17
3.10.1 General..............................................................................173.10.2 Local buckling .................................................................173.10.3 Tensile fracture.................................................................183.10.4 Tensile fracture in yield hinges.........................................18
3.11 Resistance of large diameter, stiffened columns.193.11.1 General..............................................................................193.11.2 Longitudinal stiffeners......................................................193.11.3 Ring stiffeners...................................................................193.11.4 Decks and bulkheads ........................................................19
3.12 Energy dissipation in floating productionvessels......................................................................19
3.13 Global integrity during impact ............................19
4. DROPPED OBJECTS ......................................... 19
4.1 General ...................................................................19
4.2 Impact velocity.......................................................20
4.3 Dissipation of strain energy..................................21
4.4 Resistance/energy dissipation...............................21
4.4.1 Stiffened plates subjected to drill collar impact ...............214.4.2 Stiffeners/girders .............................................................. 214.4.3 Dropped object .................................................................21
4.5 Limits for energy dissipation ...............................21
4.5.1 Pipes on plated structures.................................................214.5.2 Blunt objects.....................................................................21
5. FIRE...................................................................... 21
5.1 General................................................................... 21
5.2 General calculation methods................................22
5.3 Material modelling................................................22
5.4 Equivalent imperfections...................................... 22
5.5 Empirical correction factor.................................. 22
5.6 Local cross sectional buckling.............................. 22
5.7 Ductility limits ....................................................... 225.7.1 General.............................................................................. 22
5.7.2 Beams in bending .............................................................235.7.3 Beams in tension...............................................................23
5.8 Capacity of connections........................................23
6. EXPLOSIONS...................................................... 23
6.1 General................................................................... 23
6.2 Classification of response .....................................23
6.3 Recommended analysis models for stiffenedpanels...................................................................... 23
6.4 SDOF system analogy ...........................................25
6.5 Dynamic response charts for SDOF system ....... 26
6.6 MDOF analysis......................................................27
6.7 Classification of resistance properties ................276.7.1 Cross-sectional behaviour.................................................27
6.8 Idealisation of resistance curves ..........................28
6.9 Resistance curves and transformation factorsfor plates ................................................................28
6.9.1 Elastic - rigid plastic relationships.................................... 286.9.2 Axial restraint...................................................................296.9.3 Tensile fracture of yield hinges ........................................ 29
6.10 Resistance curves and transformation factorsfor beams................................................................29
6.10.1 Beams with no- or full axial restraint ...............................296.10.2 Beams with partial end restraint. ......................................326.10.3 Beams with partial end restraint - support capacity
smaller than plastic bending moment of member............. 346.10.4 Effective flange.................................................................346.10.5 Strength of adjacent structure ...........................................346.10.6 Strength of connections ....................................................346.10.7 Ductility limits..................................................................34
7. REFERENCES..................................................... 35
8. COMMENTARY................................................. 35
9. EXAMPLES ......................................................... 43
9.1 Design against ship collisions ............................... 439.1.1 Jacket subjected to supply vessel impact..........................43
9.2 Design against explosions ..................................... 449.2.1 Geometry.......................................................................... 449.2.2 Calculation of dynamic response of plate: .......................449.2.3 Calculation of dynamic response of stiffened plate.......... 44
9.3 Resistance curves and transformation factors ..449.3.1 Plates.................................................................................449.3.2 Calculation of resistance curve for stiffened plate ........... 459.3.3 Calculation of resistance curve for girder......................... 46
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Page 6 – Contents
9.4 Ductility limits .....................................................469.4.1 Plating ...............................................................................469.4.2 Stiffener: ...........................................................................469.4.3 Girder:...............................................................................47
9.5 Design against explosions - girder .......................47
9.5.1 Geometry, material and loads ...........................................479.5.2 Cross sectional of properties for the girder.......................489.5.3 Mass..................................................................................519.5.4 Natural period ...................................................................519.5.5 Ductility ratio ....................................................................529.5.6 Maximum blast pressure capacity.....................................52
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Recommended Practice DNV-RP-C204, October 2010
Page 7
1. General
1.1 Introduction
This Recommended Practice deals with design to maintain theload-bearing function of the structures during accidentalevents. The overall goal of the design against accidental loadsis to achieve a system where the main safety functions of the
installation are not impaired.The Recommended Practice has been developed for generalworld-wide application. Governmental legislation may includerequirements in excess of the provisions of this RecommendedPractice depending on type, location and intended service of the unit/installation.
The Design Accidental Loads and associated performance cri-teria are given in DNV-OS-A101. The Accidental Loads inthis standard are prescriptive loads. This Recommended Prac-tice may also be used in cases where the Design AccidentalLoads are determined by a formal safety assessment (seeDNV-OS-A101, Appendix C) or Quantified Risk Assessment(QRA).
The following main subjects are covered:
— Design philosophy — Ship Collisions — Dropped Objects — Fire — Explosions.
1.2 Application
The Recommended Practice is applicable to all types of float-ing and fixed offshore structures made of steel. The methodsdescribed are relevant for both substructures and topside struc-tures.
The document is limited to load-carrying structures and does
not cover pressurised equipment.1.3 Objectives
The objective with this Recommended Practice is to providerecommendations for design of structures exposed to acciden-tal events.
1.4 Normative references
The following standards include requirements which, throughreference in the text constitute provisions of this Recom-mended Practice. Latest issue of the references shall be usedunless otherwise agreed. Other recognised standards may beused provided it can be demonstrated that these meet or exceedthe requirements of the standards referenced below.
Any deviations, exceptions and modifications to the codes andstandards shall be documented and agreed between the sup-
plier, purchaser and verifier, as applicable.
1.4.1 DNV Offshore Standards (OS)
The latest revision of the following documents applies:
1.4.2 DNV Recommended Practices (RP)
The latest revision of the following documents applies:
1.5 Definitions
Load-bearing structure: That part of the facility whose mainfunction is to transfer loads.
Accidental Event: An undesired incident or condition which, incombination with other conditions (e.g.: weather conditions,failure of safety barrier, etc.), determines the accidentaleffects.
Accidental Effect: The result of an accidental event, expressedin terms of heat flux, impact force and energy, acceleration,etc. which is the basis for the safety evaluations.
Design Accidental Event (DAE): An accidental event, whichresults in effects that, the platform should be designed to sus-tain.
Acceptance criteria: Functional requirements, which are con-cerned with the platforms' resistance to accidental effects. Thisshould be in accordance with the authority's definition of acceptable safety levels.
Active protection: Operational loads and mechanical equip-ment which are brought into operation when an accident isthreatening or after the accident has occurred, in order to limitthe probability of the accident and the effects thereof, respec-tively. Some examples are safety valves, shut down systems,water drenching systems, working procedures, drills for cop-ing with accidents, etc.
Passive protection: Protection against damage by means of distance, location, strength and durability of structural ele-ments, insulation, etc.
Event control: Implementation of measures for reducing the probability and consequence of accidental events, such aschanges and improvements in equipment, working procedures,active protection devices, arrangement of the platform, person-nel training, etc.
Indirect design: Implementation of measures for improvingstructural ductility and resistance without numerical calcula-tions and determination of specific accidental effects.
Direct design: Determination of structural resistance, dimen-sions, etc. on basis of specific design accidental effects.
Load: Any action causing load effect in the structure.
Characteristic load: Reference value of a load to be used indetermination of load effects when using the partial coefficientmethod or the allowable stress method.
Load effect: Effect of a single load or combination of loads onthe structure, such as stress, stress resultant (internal force andmoment), deformation, displacement, motion, etc.
Resistance: Capability of a structure or part of a structure toresist load effect.
Characteristic resistance: The nominal capacity that may beused for determination of design resistance of a structure or structural element. The characteristic value of resistance is to
be based on a defined percentile of the test results.
Design life: The time period from commencement of construc-tion until condemnation of the structure.
Limit state: A state where a criterion governing the load-carry-ing ability or use of the structure is reached.
DNV-OS-A101 Safety Principles and Arrangements
DNV-OS-C101 Design of Offshore Steel Structures,General (LRFD Method)
DNV-OS-C102 Structural Design of Offshore Ships
DNV-OS-C103 Structural Design of Column StabilisedUnits (LRFD)
DNV-OS-C104 Structural Design of Self-Elevating Units(LRFD)
DNV-OS-C105 Structural Design of TLPs (LRFD)
DNV-OS-C106 Structural Design of Deep Draught Floating Units (LRFD)
DNV-OS-C301 Stability and Watertight Integrity ofOffshore Units
DNV-RP-C201 Buckling Strength of Plated Structures
DNV-RP-C202 Buckling Strength of Shells
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1.6 Symbols
A Cross-sectional area
Ae Effective area of stiffener and effective plate flange
As Area of stiffener
A p Projected cross-sectional area
Aw Shear area of stiffener/girder
B Width of contact area
CD Hydrodynamic drag coefficient
D Diameter of circular sections, plate stiffness
E Young's Modulus of elasticity,(for steel 2.1⋅105 N/mm2)
E p Plastic modulus
Ekin Kinetic energy
Es Strain energy
F Lateral load, total load
G Shear modulus
H Non-dimensional plastic stiffness
I Moment of inertia, impulse
J Mass moment of inertia
K l Load transformation factor
K m Mass transformation factor
K l m Load-mass transformation factor
L Girder length
M Total mass, cross-sectional moment
MP Plastic bending moment resistance
NP Plastic axial resistance
Sd Design load effect
T Fundamental period of vibration
N Axial force
NSd Design axial compressive force
NRd Design axial compressive capacity
NP Axial resistance of cross section
R Resistance
R D Design resistance
R 0 Plastic collapse resistance in bending
V Volume, displacement
WP Plastic section modulus
W Elastic section modulus
a Added mass
as Added mass for ship
ai Added mass for installation
b Width of collision contact zone
bf Flange widthc Factor
cf Axial flexibility factor
cl p Plastic zone length factor
cs Shear factor for vibration eigenperiod
cQ Shear stiffness factor
cw Displacement factor for strain calculation
d Smaller diameter of threaded end of drill collar
dc Characteristic dimension for strain calculation
Generalised load
f u Ultimate material tensile strength
f y Characteristic yield strength
g Acceleration of gravity, 9.81 m/s2
hw Web height for stiffener/girder
i Radius of gyration
k Stiffness, characteristic stiffness, plate stiffness, factor
Generalised stiffness
k e Equivalent stiffnessk l Bending stiffness in linear domain for beam
Stiffness in linear domain including shear deformation
k Q Shear stiffness in linear domain for beam
Temperature reduction of effective yield stress formaximum temperature in connection
Plate length, beam length
m Distributed mass
ms Ship mass
mi Installation mass
meq Equivalent mass
Generalised mass
p Explosion pressure
r Radius of deformed area, resistance
r c Plastic collapse resistance in bending for plate
r g Radius of gyration
s Distance, stiffener spacing
sc Characteristic distance
se Effective width of plate
t Thickness, time
td Duration of explosion
tf Flange thickness
tw Web thickness
vs Velocity of ship
vi Velocity of installation
vt Terminal velocity
w Deformation, displacement
wc Characteristic deformation
wd dent depth Non-dimensional deformation
x Axial coordinate
f
k
'
1k
θy,k
l
m
w
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2. Design Philosophy
2.1 General
The overall goal for the design of the structure against acciden-tal loads is to prevent an incident to develop into an accidentdisproportional to the original cause. This means that the mainsafety functions should not be impaired by failure in the struc-ture due to the design accidental loads. With the main safetyfunctions is understood:
— usability of escapeways,
— integrity of shelter areas,
— global load bearing capacity
In this section the design procedure that is intended to fulfil
this goal is presented.The design against accidental loads may be done by direct cal-culation of the effects imposed on the structure, or indirectly, by design of the structure as tolerable to accidents. Examplesof the latter are compartmentation of floating units which pro-vides sufficient integrity to survive certain collision scenarioswithout further calculations.
The inherent uncertainty of the frequency and magnitude of theaccidental loads, as well as the approximate nature of the meth-ods for determination of accidental load effects, shall be recog-nised. It is therefore essential to apply sound engineering judgement and pragmatic evaluations in the design.
Typical accidental events are:
— Ship collision
— Dropped objects
— Fire
— Explosion
2.2 Safety format
The requirements to structures exposed for accidental loads aregiven in DNV-OS-C101 Section 7.
The structure should be checked in two steps:
— First the structure will be checked for the loads to which itis exposed due to the accidental event
— Secondly in case the structural capacity towards ordinaryloads is reduced as a result of the accident then the strengthof the structure is to be rechecked for ordinary loads.
The structure should be checked for all relevant limit states.The limit states for accidental loads are denoted AccidentalLimit States (ALS). The requirement may be written as
where:
For check of Accidental limit states (ALS) the load and mate-rial factor should be taken as 1.0.
The failure criterion needs to be seen in conjunction with theassumptions made in the safety evaluations.
The limit states may need to be alternatively formulated to beon the form of energy formulation, as acceptable deformation,or as usual on force or moment.
2.3 Accidental loads
The accidental loads are either prescriptive values or definedin a Formal Safety Assessment. Prescriptive values may begiven by authorities, the owner or found in DNV OffshoreStandard DNV-OS-A101.
Usually the simplification that accidental loads need not to becombined with environmental loads is valid.
For check of the residual strength in cases where the accident
lead to reduced load carrying capacity in the structure thecheck should be made with the characteristic environmentalloads determined as the most probable annual maximum value.
2.4 Acceptance criteria
Examples of failure criteria are:
— Critical deformation criteria defined by integrity of pas-sive fire protection. To be considered for walls resistingexplosion pressure and shall serve as fire barrier after theexplosion.
— Critical deflection for structures to avoid damage to proc-ess equipment (Riser, gas pipe, etc). To be considered for structures or part of structures exposed to impact loads asship collision, dropped object etc.
— Critical deformation to avoid leakage of compartments. To be considered in case of impact against floating structureswhere the acceptable collision damage is defined by theminimum number of undamaged compartments to remainstable.
y Generalised displacement, displacement amplitude
yel Generalised displacement at elastic limit
z Distance from pivot point to collision point
z plast Smaller distance from flange to plastic neutral axis
α Plate aspect parameter
β Cross-sectional slenderness factor
ε Yield strength factor, strain
εcr Critical strain for rupture
εy Yield strain
η Plate eigenperiod parameter
Displacement shape function
Reduced slenderness ratio
μ Ductility ratio
ν Poisson's ratio, 0.3
θ Angleρ Density of steel, 7860 kg/m3
ρw Density of sea water, 1025 kg/m3
τ Shear stress
τcr Critical shear stress for plate plugging
ξ Interpolation factor
ψ Plate stiffness parameter
φ
λ
(2.1)
Sd = Design load effect
R d = Design resistance
Sk = Characteristic load effect
γf = partial factor for loads
R = Characteristic resistance
γM = Material factor
dd R S ≤
f k γS
M
k
γ
R
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2.5 Analysis considerations
The mechanical response to accidental loads is generally con-cerned with energy dissipation, involving large deformationsand strains far beyond the elastic range. Hence, load effects(stresses forces, moments etc.) obtained from elastic analysisand used in ultimate limit state (ULS) checks on componentlevel are generally not applicable, and plastic methods of anal-
ysis should be used.Plastic analysis is most conveniently based upon the kinemat-ical approach, taking into account the effect of the strengthen-ing (membrane tension) or softening (compression) caused byfinite deformations, where applicable.
The requirements in this RP are generally derived from plasticmethods of analysis, including the effect of finite deforma-tions.
Plastic methods of analysis are valid for materials that canundergo considerable straining and during this process exhibitconsiderable strain hardening. If the material is ductile as such,i.e. it can be strained significantly, but has little strain harden-ing, the member tends to behave brittle in a global sense (i.e.with respect to energy dissipation), and plastic methods should
be used with great caution.
A further condition for application of plastic methods to mem- bers undergoing large, plastic rotations is compact cross-sec-tions; typically type I cross-sections (refer DNV-OS-C101,Table A1). The methods may also be utilised for type II sec-tions provided that the detrimental effect of local buckling istaken into account. Note that for members subjected to signif-icant tensile straining, the tendency for local buckling may beoverridden by membrane tension for large deformations.
The straining, and hence the amount of energy dissipation, islimited by fracture. This key parameter is associated with con-siderable uncertainty, with respect to both physical occurrenceas well as modelling in theoretical analysis. If good and vali-dated models for prediction of fracture are not available, safe
and conservative assumptions for ductility limits should beadopted.
If non-linear, dynamic finite elements analysis is applied, itshall be verified that all behavioural effects and local failuremodes (e.g. strain rate, local buckling, joint overloading, and joint fracture) are accounted for implicitly by the modellingadopted, or else subjected to explicit evaluation.
3. Ship Collisions
3.1 General
The requirements and methods given in this section have his-
torically been developed for jackets. They are generally validalso for jack-up type platforms, provided that the increasedimportance of global inertia effects are accounted for. Column-stabilised platforms and floating production and storage ves-sels (FPSOs) consist typically plane or curved, stiffened pan-els, for which methods for assessment of energy dissipation in braced platforms (jackets and jack-ups) sometimes are not rel-evant. Procedures especially dedicated to assessment of energydissipation in stiffened plating are, however, also given basedon equivalent beam-column models.
The ship collision load is characterised by a kinetic energy,governed by the mass of the ship, including hydrodynamicadded mass and the speed of the ship at the instant of impact.Depending upon the impact conditions, a part of the kinetic
energy may remain as kinetic energy after the impact. Theremainder of the kinetic energy has to be dissipated as strainenergy in the installation and, possibly, in the vessel. Generallythis involves large plastic strains and significant structuraldamage to the installation, the ship or both. The strain energydissipation is estimated from force-deformation relationships
for the installation and the ship, where the deformations in theinstallation shall comply with ductility and stability require-ments.
The load bearing function of the installation shall remain intactwith the damages imposed by the ship collision load. In addi-tion, damaged condition should be checked if relevant, seeSection 2.2.
The structural effects from ship collision may either be deter-mined by non-linear dynamic finite element analyses or byenergy considerations combined with simple elastic-plasticmethods.
If non-linear dynamic finite element analysis is applied alleffects described in the following paragraphs shall either beimplicitly covered by the modelling adopted or subjected tospecial considerations, whenever relevant.
Often the integrity of the installation can be verified by meansof simple calculation models.
If simple calculation models are used the part of the collisionenergy that needs to be dissipated as strain energy can be cal-culated by means of the principles of conservation of momen-tum and conservation of energy, refer Section 3.3.
It is convenient to consider the strain energy dissipation in theinstallation to take part on three different levels:
— local cross-section — component/sub-structure — total system
Interaction between the three levels of energy dissipation shall be considered.
Plastic modes of energy dissipation shall be considered for cross-sections and component/substructures in direct contactwith the ship. Elastic strain energy can in most cases be disre-garded, but elastic axial flexibility may have a substantialeffect on the load-deformation relationships for components/
sub-structures. Elastic energy may contribute significantly ona global level.
3.2 Design principles
With respect to the distribution of strain energy dissipationthere may be distinguished between, see Figure 3-1:
— strength design — ductility design — shared-energy design
Figure 3-1Energy dissipation for strength, ductile and shared-energy design
Strength design implies that the installation is strong enough toresist the collision force with minor deformation, so that the
ship is forced to deform and dissipate the major part of theenergy.
Ductility design implies that the installation undergoes large, plastic deformations and dissipates the major part of the colli-sion energy.
Strength
design
Shared-energydesign
Ductile
design
Relative strength - installation/ship
ship
installation
E n e r g y d i s s i p a t i o n
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Shared energy design implies that both the installation andship contribute significantly to the energy dissipation.
From calculation point of view strength design or ductilitydesign is favourable. In this case the response of the «soft»structure can be calculated on the basis of simple considera-tions of the geometry of the «rigid» structure. In shared energydesign both the magnitude and distribution of the collisionforce depends upon the deformation of both structures. Thisinteraction makes the analysis more complex.
In most cases ductility or shared energy design is used. How-ever, strength design may in some cases be achievable with lit-tle increase in steel weight.
3.3 Collision mechanics
3.3.1 Strain energy dissipation
The collision energy to be dissipated as strain energy may -depending on the type of installation and the purpose of theanalysis - be taken as:
Compliant installations
Fixed installations
Articulated columns
ms = ship massas = ship added massvs = impact speedmi = mass of installationai = added mass of installationvi = velocity of installationJ = mass moment of inertia of installation (including
added mass) with respect to effective pivot pointz = distance from pivot point to point of contact
In most cases the velocity of the installation can be disre-garded, i.e. vi = 0.
The installation can be assumed compliant if the duration of impact is small compared to the fundamental period of vibra-tion of the installation. If the duration of impact is compara-tively long, the installation can be assumed fixed.
Floating platforms (semi-submersibles, TLP’s, productionvessels) can normally be considered as compliant. Jack-upsmay be classified as fixed or compliant. Jacket structures cannormally be considered as fixed.
3.3.2 Reaction force to deck
In the acceleration phase the inertia of the topside structure
generates large reaction forces. An upper bound of the maxi-mum force between the collision zone and the deck for bottomsupported installations may be obtained by considering the platform compliant for the assessment of total strain energydissipation and assume the platform fixed at deck level whenthe collision response is evaluated.
Figure 3-2Model for assessment of reaction force to deck
3.4 Dissipation of strain energy
The structural response of the ship and installation can for-mally be represented as load-deformation relationships asillustrated in Figure 3-3. The strain energy dissipated by theship and installation equals the total area under the load-defor-mation curves.
Figure 3-3
Dissipation of strain energy in ship and platform
As the load level is not known a priori an incremental proce-dure is generally needed.
The load-deformation relationships for the ship and the instal-lation are often established independently of each other assum-ing the other object infinitely rigid. This method may have,however, severe limitations; both structures will dissipatesome energy regardless of the relative strength.
Often the stronger of the ship and platform will experience lessdamage and the softer more damage than what is predictedwith the approach described above. As the softer structuredeforms the impact force is distributed over a larger contactarea. Accordingly, the resistance of the strong structureincreases. This may be interpreted as an "upward" shift of theresistance curve for the stronger structure (refer Figure 3-3 ).
Care should be exercised that the load-deformation curves cal-culated are representative for the true, interactive nature of thecontact between the two structures.
3.5 Ship collision forces
3.5.1 Recommended force-deformation relationships
Force-deformation relationships for supply vessels with a dis- placement of 5000 tons are given in Figure 3-4 for broad side-, bow-, stern end and stern corner impact for a vessel withstern roller.
The curves for broad side and stern end impacts are based upon
(3.1)
(3.2)
(3.3)
ii
ss
2
s
i
2
ssss
am
am1
v
v1
)va(m2
1E
++
+
⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −
+=
2
ssss )va(m2
1E +=
J
zm1
v
v1
)a(m2
1E2
s
2
s
i
sss
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ −
+=
(3.4)
Collision response Model
dws dwi
R iR s
Ship Installation
Es,sEs,i
∫∫ +=+=maxi,maxs, w
0ii
w
0ssis,ss,s dwR dwR EEE
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penetration of an infinitely rigid, vertical cylinder with a givendiameter and may be used for impacts against jacket legs (D =1.5 m) and large diameter columns (D = 10 m).
The curve for stern corner impact is based upon penetration of an infinitely rigid cylinder and may be used for large diameter column impacts.
In lieu of more accurate calculations the curves in Figure 3-4may be used for square-rounded columns.
The curve for bow impact is based upon collision with an infi-nitely rigid, plane wall and may be used for large diameter col-umn impacts, but should not be used for significantly different
collision events, e.g. impact against tubular braces.
For beam -, stern end – and stern corner impacts against jacket braces all energy shall normally be assumed dissipated by the brace, refer Ch.8, Comm. 3.5.2.
For bow impacts against jacket braces, reference is made toSection 3.5.3.
For supply vessels and merchant vessels in the range of 2-5000 tons displacement, the force deformation relationshipsgiven in Figure 3-5 may be used for impacts against jacket legswith diameter 1.5 m – 2.5 m.
Figure 3-4
Recommended-deformation curve for beam, bow and stern impact
Figure 3-5
Force -deformation relationship for bow with and without bulb (2-5.000 dwt)
0
10
20
30
40
50
0 1 2 3 4Indentation (m)
I m p a c t f o r c e
( M N )
Broad side
D = 10 m
= 1.5 m
Stern end
D = 10 m
= 1.5 m
Bo
Stern corner
D
D
Bow0
10
20
30
40
50
0 1 2 3 4Indentation (m)
I m p a c t f o r c e
( M N )
Broad side
D = 10 m
= 1.5 m
Stern end
D = 10 m
= 1.5 m
Bo
Stern corner
D
D
Bow
0
20
40
60
80
0 1 2 3 4 5
Deformation [m]
E n e r g y [ M J ]
0
10
20
30
F o r c e [ M N ]
Contact forceEnergy
no bulb
with bulb
curve - plane wall
0
20
40
60
80 40
Energy
Design-
0
20
40
60
80
0 1 2 3 4 5
Deformation [m]
E n e r g y [ M J ]
0
10
20
30
F o r c e [ M N ]
Contact forceEner gy
no bulb
with bulb
curve - plane wall
0
20
40
60
80 40
Energy
Design-
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Figure 3-6Force -deformation relationship for tanker bow impact(~ 125.000 dwt)
Figure 3-7Force -deformation relationship and contact area for the bulbousbow of a VLCC (~ 340.000 dwt)
Force-deformation relationships for tanker bow impact aregiven in Figure 3-6 for the bulbous part and the superstructure,respectively, and for the bulb of a VLCC in Figure 3-7. Thecurves may be used provided that the impacted structure (e.g.stern of floating production vessels) does not undergo substan-tial deformation i.e. strength design requirements are compliedwith. If this condition is not met interaction between the bowand the impacted structure shall be taken into consideration.
Non-linear finite element methods or simplified plastic analy-sis techniques of members subjected to axial crushing shall beemployed, see Ch.7 /3/, /4/.
3.5.2 Force contact area for strength design of large diam-eter columns.
The basis for the curves in Figure 3-4 is strength design, i.e.limited local deformations of the installation at the point of contact. In addition to resisting the total collision force, largediameter columns have to resist local concentrations (subsets)of the collision force, given for stern corner impact in Table 3-1 and stern end impact in Table 3-2.
If strength design is not aimed for - and in lieu of more accurateassessment (e.g. nonlinear finite element analysis) - all strainenergy has to be assumed dissipated by the column, corre-sponding to indentation by an infinitely rigid stern corner.
3.5.3 Energy dissipation is ship bow
For typical supply vessels bows and bows of merchant vesselsof similar size (i.e. 2-5000 tons displacement), energy dissipa-tion in ship bow may be taken into account provided that thecollapse resistance in bending for the brace, R 0, see Section 3.7is according to the values given in Table 3-3. The figures are
valid for normal bows without ice strengthening and for bracediameters < 1.25 m. The values should be used as step func-tions, i.e. interpolation for intermediate resistance levels is notallowed. If contact location is not governed by operation con-ditions, size of ship and platform etc., the values for arbitrarycontact location shall be used. (see also Ch.8, Comm. 3.5.3).
In addition, the brace cross-section must satisfy the following
0
10
20
30
40
50
60
70
0 1 2 3 4 5 6
Deformation [m]
F o r c e [ M N ]
0
2
4
6
8
10
12
C o n t a c t d i m e n s i o n [ m ]Bulb force
a
b
a
b
0
10
20
30
40
50
60
70
0 1 2 3 4 5
Deformation [m]
F o r c
e [ M N ]
0
2
4
6
8
10
12
14
16
18
C o n t a c t d i m e n s i o n [ m ]
Force
superstructure
a
b
a
b
0
20
40
60
80
100
120
140
160
0 1 2 3 4 5 6 7 8Deformation [m]
F o r c e [ M N ]
0
100
200
300
400
500
600
700
800
E n e r g y [ M J ]
Force
Energy
0
2
4
6
8
10
12
0 1 2 3 4 5 6 7 8Deformation [m]
C o n t a c t d i m e n s i o n s [ m ]
a
b
a
b
Table 3-1 Local concentrated collision force -evenly distributedover a rectangular area. Stern corner impact
Contact area Force (MN)
a (m) b (m)
0.35 0.65 3.0
0.35 1.65 6.4
0.20 1.15 5.4
Table 3-2 Local concentrated collision force -evenly distributed
over a rectangular area. Stern end impact
Contact area Force (MN)
a (m) b (m)
0.6 0.3 5.6
0.9 0.5 7.5
2.0 1.1 10
Table 3-3 Energy dissipation in bow versus brace resistance
Contact location Energy dissipation in bow
if brace resistance R0
> 3 MN > 6 MN > 8 MN > 10 MN
Above bulb 1 MJ 4 MJ 7 MJ 11 MJ
First deck 0 MJ 2 MJ 4 MJ 17 MJ
First deck - oblique brace 0 MJ 2 MJ 4 MJ 17 MJ
Between forcastle/firstdeck
1 MJ 5 MJ 10 MJ 15 MJ
Arbitrary location 0 MJ 2 MJ 4 MJ 11 MJ
a
b
b
a
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compactness requirement
where factor is the required resistance in [MN] given in Table3-3.
See Section 3.6 for notation.If the brace is designed to comply with these provisions, spe-cial care should be exercised that the joints and adjacent struc-ture is strong enough to support the reactions from the brace.
3.6 Force-deformation relationships for denting oftubular members
The contribution from local denting to energy dissipation issmall for brace members in typical jackets and should beneglected.
The resistance to indentation of unstiffened tubes may be takenfrom Figure 3-8. Alternatively, the resistance may be calcu-lated from Equation (3.6):
Figure 3-8Resistance curve for local denting
NSd = design axial compressive force NRd = design axial compressive resistanceB = width of contactareawd = dent depth
The curves are inaccurate for small indentation, and theyshould not be used to verify a design where the dent damage isrequired to be less than wd / D > 0.05.
The width of contact area is in theory equal to the height of thevertical, plane section of the ship side that is assumed to be incontact with the tubular member. For large widths, anddepending on the relative rigidity of the cross-section and theship side, it may be unrealistic to assume that the tube is sub- jected to flattening over the entire contact area. In lieu of moreaccurate calculations it is proposed that the width of contactarea be taken equal to the diameter of the hit cross-section (i.e.B/D = 1).
3.7 Force-deformation relationships for beams
3.7.1 General
The response of a beam subjected to a collision load is initiallygoverned by bending, which is affected by and interacts withlocal denting under the load. The bending capacity is alsoreduced if local buckling takes place on the compression side.As the beam undergoes finite deformations, the load carryingcapacity may increase considerably due to the development of
membrane tension forces. This depends upon the ability of adjacent structure to restrain the connections at the member ends to inward displacements. Provided that the connectionsdo not fail, the energy dissipation capacity is either limited bytension failure of the member or rupture of the connection.
Simple plastic methods of analysis are generally applicable.Special considerations shall be given to the effect of:
— elastic flexibility of member/adjacent structure, — local deformation of cross-section, — local buckling, — strength of connections, — strength of adjacent structure, and — fracture.
3.7.2 Plastic force-deformation relationships includingelastic, axial flexibility
Relatively small axial displacements have a significant influ-ence on the development of tensile forces in members under-going large lateral deformations. An equivalent elastic, axialstiffness may be defined as
k node = axial stiffness of the node with the considered mem- ber removed. This may be determined by introduc-ing unit loads in member axis direction at the end
nodes with the member removed.Plastic force-deformation relationship for a central collision(midway between nodes) may be obtained from:
— Figure 3-9 for tubular members — Figure 3-10 for stiffened plates in lieu of more accurate
analysis.
The following notation applies:
(3.5)
(3.6)
factor 3
2Dtf 0.51.5
y ⋅≥
0
2
4
6
8
10
12
14
16
18
20
0 0.1 0.2 0.3 0.4 0.5
wd/D
R / ( k R c )
2
1
0.5
0
b/D =
Rd
Sd
Rd
Sd
Rd
Sd
Rd
Sd
2
1
2
yc
c
d
1
c
N
N0.60k
0.6 N
N0.20.2
N
N21.0k
0.2 N
N1.0k
D
B3.5
1.925c
D
B1.222c
t
D
4
tf R
D
wkc
R
R 2
≤=
<<⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −−=
≤=
+=
+=
=
⎟⎟ ⎠
⎞⎜⎜⎝
⎛ =
(3.7)
plastic collapse resistance in bending forthe member, for the case that contact pointis at midspan
non-dimensional deformation
non-dimensional spring stiffness
c1 = 2 for clamped beams
2EAk
1
k
1
node
l+=
l
P1Mc4R 0 =
cwc
ww
1
=
lAf
kw4cc
y
2
c1=
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For non-central collisions the force-deformation relationshipmay be taken as the mean value of the force-deformationcurves for central collision with member half length equal tothe smaller and the larger portion of the member length,respectively.
For members where the plastic moment capacity of adjacentmembers is smaller than the moment capacity of the impactedmember the force-deformation relationship may be interpo-lated from the curves for pinned ends and clamped ends:
For non-central collisions the force-deformation relationshipmay be taken as the mean value of the force-deformation
curves for central collision with member half length equal tothe smaller and the larger portion of the member length,respectively.
For members where the plastic moment capacity of adjacentmembers is smaller than the moment capacity of the impacted
member the force-deformation relationship may be interpo-lated from the curves for pinned ends and clamped ends:
where
i = adjacent member no i
j = end number {1,2}MPj,i = Plastic bending resistance for member number i at
end j.
Elastic, rotational flexibility of the node is normally of moder-ate significance.
Figure 3-9Force-deformation relationship for tubular beam with axial flexibility
c1 = 1 for pinned beams
characteristic deformation for tubular beams
characteristic deformation for stiffened plating
WP = plastic section modulus= member length
2
Dw c =
A
W2.1w P
c =
l
(3.8)
(3.9)
=Plastic resistance by bending action of beam account-ing for actual bending resistance of adjacent members
(3.10)
(3.11)
pinnedclamped R ζ)(1R R −+= ζ
11M4R ξ0
P
actual
0 ≤−=≤
l
actual
0R
l
P2P1Pactual
0
2M2M4MR
++=
∑ ≤=i
iPj,Pj MMM
Bending & membrane
Membrane only
k k
F (collision load)
w
0
0,5
1
1,5
2
2,5
3
3,5
44,5
5
5,5
6
6,5
0 0,5 1 1,5 2 2,5 3 3,5 4
Deformation
R / R 0
1
0.1
0.2
0,3
0.5
0.05c=∞
w
Bending & membrane
Membrane only
k k
F (collision load)
w
Bending & membrane
Membrane only
k k
F (collision load)
w
0
0,5
1
1,5
2
2,5
3
3,5
44,5
5
5,5
6
6,5
0 0,5 1 1,5 2 2,5 3 3,5 4
Deformation
R / R 0
1
0.1
0.2
0,3
0.5
0.05c=∞
w
0
0,5
1
1,5
2
2,5
3
3,5
44,5
5
5,5
6
6,5
0 0,5 1 1,5 2 2,5 3 3,5 4
Deformation
R / R 0
1
0.1
0.2
0,3
0.5
0.05c=∞
w
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Figure 3-10Force-deformation relationship for stiffened plate with axial flexibility
3.7.3 Support capacity smaller than plastic bendingmoment of the beam
For beams where the plastic moment capacity of adjacentmembers is smaller than the moment capacity of the impacted
beam, the force-deformation relationship, R *, may be derivedfrom the resistance curve, R, for beams where the plastic
moment capacity of adjacent members is larger than themoment capacity of the impacted beam (Section 3.7.2), usingthe expression:
where
R 0
= Plastic bending resistance with clamped ends (c1
= 2) – moment capacity of adjacent members larger thanthe plastic bending moment of the beam
= Plastic bending resistance - moment capacity of adja-cent members at one or both ends smaller than the plas-tic bending moment of the beam
i = adjacent member no i j = end number {1,2}MPj,i= Plastic bending resistance for member no. iwlim = limiting non-dimensional deformation where the
membrane force attains yield, i.e. the resistance curve,R, with actual spring stiffness coefficient, c, intersects
with the curve for c = ∞. If c = ∞, for
tubular beams and for stiffened plate
3.7.4 Bending capacity of dented tubular members
The reduction in plastic moment capacity due to local dentingshall be considered for members in compression or moderatetension, but can be neglected for members entering the fully
plastic membrane state.
Conservatively, the flat part of the dented section according tothe model shown in Figure 3-11 may be assumed non-effec-tive. This gives:
wd = dent depth as defined in Figure 3-11.
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
5
0 0,5 1 1,5 2 2,5 3 3,5 4
Deformation
R / R 0
1
0
0.1
0.20.5
w
Bending & membrane
Membrane only
k k
F (collision load)
w∞=c
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
5
0 0,5 1 1,5 2 2,5 3 3,5 4
Deformation
R / R 0
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
5
0 0,5 1 1,5 2 2,5 3 3,5 4
Deformation
R / R 0
1
0
0.1
0.20.5
ww
Bending & membrane
Membrane only
k k
F (collision load)
w∞=c
Bending & membrane
Membrane only
k k
F (collision load)
w
Bending & membrane
Membrane only
k k
F (collision load)
ww∞=c ∞=c ∞=c
,
(3.12)
(3.13)
(3.14)
lim
*
00
*)R (R R R
w
w−+= 0.1
lim
≤w
w
R R * = 0.1
lim
≥w
w
*0R
l
P2P1P*0
2M2M4MR
++=
∑ ≤=i
PiPj,Pj MMM
(3.15)
lim2
w wπ =
lim 1.2w w=
⎟ ⎠
⎞⎜⎝
⎛ −=
=
−=
D
2w1arccosθ
tDf M
sinθ
2
1
2
θcos
M
M
d
2
yP
P
red
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Figure 3-11Reduction of moment capacity due to local dent
3.8 Strength of connections
Provided that large plastic strains can develop in the impactedmember, the strength of the connections that the member frames into should be checked.
The resistance of connections should be taken from ULSrequirements in relevant standards.
For braces reaching the fully plastic tension state, the connec-tion shall be checked for a load equal to the axial capacity of the member. The design axial stress shall be assumed equal tothe ultimate tensile strength of the material.
If the axial force in a tension member becomes equal to theaxial capacity of the connection, the connection has to undergogross deformations. The energy dissipation will be limited andrupture should be considered at a given deformation. A safeapproach is to assume failure (disconnection of the member)once the axial force in the member reaches the axial capacityof the connection.
If the capacity of the connection is exceeded in compressionand bending, this does not necessarily mean failure of themember. The post-collapse strength of the connection may betaken into account provided that such information is available.
3.9 Strength of adjacent structure
The strength of structural members adjacent to the impactedmember/sub-structure must be checked to see whether theycan provide the support required by the assumed collapsemechanism. If the adjacent structure fails, the collapse mecha-nism must be modified accordingly. Since, the physical behav-iour becomes more complex with mechanisms consisting of anincreasing number of members it is recommended to consider a design which involves as few members as possible for eachcollision scenario.
3.10 Ductility limits
3.10.1 General
The maximum energy that the impacted member can dissipatewill – ultimately - be limited by local buckling on the compres-sive side or fracture on the tensile side of cross-sections under-going finite rotation.
If the member is restrained against inward axial displacement,any local buckling must take place before the tensile strain dueto membrane elongation overrides the effect of rotationinduced compressive strain.
If local buckling does not take place, fracture is assumed to
occur when the tensile strain due to the combined effect of rotation and membrane elongation exceeds a critical value.
To ensure that members with small axial restraint maintainmoment capacity during significant plastic rotation it is recom-mended that cross-sections be proportioned to section type Irequirements, defined in DNV-OS-C101.
Initiation of local buckling does, however, not necessarily
imply that the capacity with respect to energy dissipation isexhausted, particularly for type I and type II cross-sections.The degradation of the cross-sectional resistance in the post-
buckling range may be taken into account provided that suchinformation is available, refer Ch.8, Comm. 3.10.1.
For members undergoing membrane stretching a lower boundto the post-buckling load-carrying capacity may be obtained
by using the load-deformation curve for pure membraneaction.
3.10.2 Local buckling
Tubular cross-sections:
Buckling does not need to be considered for a beam with axialrestraints if the following condition is fulfilled:
where
axial flexibility factor
dc = characteristic dimension = D for circular cross-sectionsc1 = 2 for clamped ends = 1 for pinned endsc = non-dimensional spring stiffness, refer Section 3.7.2.
κ ≤ 0.5 = the smaller distance from location of collisionload to adjacent joint
If this condition is not met, buckling may be assumed to occur when the lateral deformation exceeds
For small axial restraint (c < 0.05) the critical deformation may be taken as
Stiffened plates/ I/H-profiles:
In lieu of more accurate calculations the expressions given for circular profiles in Equation (3.19) and (3.20) may be usedwith
0
0,2
0,4
0,6
0,8
1
0 0,2 0,4 0,6 0,8 1
wd/D
M r e d /
M P
D
wd
(3.16)
(3.17)
(3.18)
(3.19)
(3.20)
dc = characteristic dimension for local buckling, equalto twice the distance from the plastic neutral axis in
bending to the extreme fibre of the cross-section
= h height of cross-section for symmetric I –profiles
= 2hw for stiffened plating (for simplicity)
3
1
2
c1
yf
d
κ
c
f 14cβ ⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟ ⎠
⎞⎜⎜⎝
⎛ ≤
l
yf 235
tDβ =
2
f
c1
cc ⎟
⎟
⎠
⎞⎜⎜
⎝
⎛
+
=
l l
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜⎝
⎛
−−=
2
c3
1
yf
f c d
κ
βc
f 14c
112c
1
d
w l
2
c3
1
y
c d
κ
βc
3.5f
d
w⎟⎟
⎠
⎞⎜⎜⎝
⎛ =
l
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For flanges subjected to compression;
For webs subjected to bending
bf = flange widthtf = flange thicknesshw = web heighttw = web thickness
3.10.3 Tensile fracture
The degree of plastic deformation or critical strain at fracturewill show a significant scatter and depends upon the followingfactors:
— material toughness — presence of defects — strain rate — presence of strain concentrations
The critical strain for plastic deformations of sections contain-ing defects need to be determined based on fracture mechanicsmethods. Welds normally contain defects and welded jointsare likely to achieve lower toughness than the parent material.For these reasons structures that need to undergo large plasticdeformations should be designed in such a way that the plasticstraining takes place outside the weld. In ordinary full penetra-tion welds, the overmatching weld material will ensure thatminimal plastic straining occurs in the welded joints even incases with yielding of the gross cross section of the member.In such situations, the critical strain will be in the parent mate-rial and will be dependent upon the following parameters:
— stress gradients — dimensions of the cross section — presence of strain concentrations — material yield to tensile strength ratio — material ductility
Simple plastic theory does not provide information on strainsas such. Therefore, strain levels should be assessed by means
of adequate analytic models of the strain distributions in the plastic zones or by non-linear finite element analysis with asufficiently detailed mesh in the plastic zones. (For informa-tion about mesh size see Ch.8, Comm. 3.10.4.)
When structures are designed so that yielding take place in the parent material, the following value for the critical averagestrain in axially loaded plate material may be used in conjunc-tion with nonlinear finite element analysis or simple plasticanalysis
where:
3.10.4 Tensile fracture in yield hinges
When the force deformation relationships for beams given inSection 3.7.2 are used rupture may be assumed to occur whenthe deformation exceeds a value given by
where the following factors are defined;
Displacement factor
plastic zone length factor
axial flexibility factor
non-dimensional plastic stiffness
The characteristic dimension shall be taken as:
For small axial restraint (c < 0.05) the critical deformation may
type I cross-sections (3.21)
type II and type III cross-sections (3.22)
type I cross-sections (3.23)
type I and type III cross-sections (3.24)
(3.25)
t = plate thickness
= length of plastic zone. Minimum 5t
y
f f
f 235
t b2.5β =
y
f f
f 235
t b3β =
y
ww
f 235
th0.7β =
y
ww
f 235
th0.8β =
l
t65.00.02 +=cr ε
l
(3.26)
(3.27)
(3.28)
(3.29)
(3.30)
c1 = 2 for clamped ends= 1 for pinned ends
c = non-dimensional spring stiffness, refer Sec-tion 3.7.2
κ ≤ 0.5 the smaller distance from location ofcollision load to adjacent joint
W = elastic section modulus
WP = plastic section modulus
εcr = critical strain for rupture (see Table 3-4 forrecommended values)
= yield strain
f y = yield strength
f cr = strength corresponding to εcr
dc = D diameter of tubular beams
= 2hw twice the web height for stiffened plates (se·t > As)
= h height of cross-section for symmet-ric I-profiles
= 2 (h − z plast) for unsymmetrical I-profiles
z plast = smaller distance from flange to plastic neutral axis ofcross-section
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −+= 1
c
εc4c1
2c
c
d
w
1
cr f w
f
1
c
2
ccr
y
P
lplp
1
wd
κ
ε
ε
W
W14c
3
11c
c
1c ⎟⎟
⎠
⎞⎜⎜⎝
⎛ ⎟⎟
⎠
⎞⎜⎜⎝
⎛ ⎟⎟
⎠
⎞⎜⎜⎝
⎛ −+⎟
⎠
⎞⎜⎝
⎛ −=
l
1HWW1
εε
HW
W1
ε
ε
c
Py
cr
Py
cr
lp
+⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −
⎟⎟
⎠
⎞⎜⎜⎝
⎛ −
=
2
f
c1
cc ⎟
⎟ ⎠
⎞⎜⎜⎝
⎛
+=
⎟⎟
⎠
⎞⎜⎜⎝
⎛
−
−==
ycr
ycr p
εε
f f
E
1
E
EH
l l
E
f ε
y
y =
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be taken as
The critical strain εcr and corresponding strength f cr should beselected so that idealised bi-linear stress-strain relation givesreasonable results, see Ch.8, Commentary. For typical steelmaterial grades the following values are proposed:
3.11 Resistance of large diameter, stiffened columns
3.11.1 General
Impact on a ring stiffener as well as midway between ring stiff-eners shall be considered.
Plastic methods of analysis are generally applicable.
3.11.2 Longitudinal stiffeners
For ductile design the resistance of longitudinal stiffeners inthe beam mode of deformation can be calculated using the pro-cedure described for stiffened plating, Section 3.7.
For strength design against stern corner impact, the plastic bending moment capacity of the longitudinal stiffeners has tocomply with the requirement given in Figure 3-12, on theassumption that the entire load given in Table 3-1 is taken byone stiffener.
Figure 3-12Required bending capacity of longitudinal stiffeners
3.11.3 Ring stiffeners
In lieu of more accurate analysis the plastic collapse load of aring-stiffener can be estimated from:
where
=characteristic deformation of ring stiffener
D = column radiusMP = plastic bending resistance of ring-stiffener including
effective shell flangeWP = plastic section modulus of ring stiffener including
effective shell flangeAe = area of ring stiffener including effective shell flange
Effective flange of shell plating: Use effective flange of stiff-ened plates, see Chapter 6.
For ductile design it can be assumed that the resistance of thering stiffener is constant and equal to the plastic collapse load,
provided that requirements for stability of cross-sections arecomplied with, refer Section 3.10.2.
3.11.4 Decks and bulkheads
Calculation of energy dissipation in decks and bulkheads hasto be based upon recognised methods for plastic analysis of deep, axial crushing. It shall be documented that the collapsemechanisms assumed yield a realistic representation of the truedeformation field.
3.12 Energy dissipation in floating production ves-sels
For strength design the side or stern shall resist crushing force
of the bow of the off-take tanker. In lieu of more accurate cal-culations the force-deformation curve given in Section 3.5.2may be applied. (See Ch.8, Comm. 3.12 on strength design of stern structure)
For ductile design the resistance of stiffened plating in the beam mode of deformation can be calculated using the proce-dure described in Section 3.7.2. (See Ch.8, Comm. 3.12 onresistance of stiffened plating)
3.13 Global integrity during impact
Normally, it is unlikely that the installation will turn into a glo- bal collapse mechanism under direct collision load, becausethe collision load is typically an order of magnitude smaller than the resultant design wave force.
Linear analysis often suffices to check that global integrity ismaintained.
The installation should be checked for the maximum collisionforce.
For installations responding predominantly statically the max-imum collision force occurs at maximum deformation.
For structures responding predominantly impulsively the max-imum collision force occurs at small global deformation of the
platform. An upper bound to the collision force is to assumethat the installation is fixed with respect to global displace-ment. (e.g. jack-up fixed with respect to deck displacement).
4. Dropped Objects
4.1 General
The dropped object load is characterised by a kinetic energy,governed by the mass of the object, including any hydrody-namic added mass, and the velocity of the object at the instan-tof impact. In most cases the major part of the kinetic energyhas to be dissipated as strain energy in the impacted componentand, possibly, in the dropped object. Generally, this involveslarge plastic strains and significant structural damage to theimpacted component. The strain energy dissipation is esti-mated from force-deformation relationships for the componentand the object, where the deformations in the component shallcomply with ductility and stability requirements.
The load bearing function of the installation shall remain intactwith the damages imposed by the dropped object load. In addi-tion, damaged condition should be checked if relevant, seeSection 2.2.
Dropped objects are rarely critical to the global integrity of the
(3.31)
Table 3-4 Proposed values for εcr and H for different steel
grades
Steel grade εcr H
S 235 20 % 0.0022
S 355 15 % 0.0034
S 460 10 % 0.0034
(3.32)
cr w
c
εcd
w=
0
1
2
3
1 2 3 4
Distance between ring stiffeners (m)
P l a s t i c b e n d i n g c a p a c i t y
( M N m )
0
1
2
3
1 2 3 4
Distance between ring stiffeners (m)
P l a s t i c b e n d i n g c a p a c i t y
( M N m )
Dw
M F
c
P 240 =
e
P c
A
W w =
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installation and will mostly cause local damages. The major threat to global integrity is probably puncturing of buoyancytanks, which could impair the hydrostatic stability of floatinginstallations. Puncturing of a single tank is normally covered
by the general requirements to compartmentation and water-tight integrity given in DNV-OS-C301.
The structural effects from dropped objects may either bedetermined by non-linear dynamic finite element analyses or
by energy considerations combined with simple elastic-plasticmethods as given in Sections 4.2 - 4.5.
If non-linear dynamic finite element analysis is applied alleffects described in the following paragraphs shall either beimplicitly covered by the modelling adopted or subjected tospecial considerations, whenever relevant.
4.2 Impact velocity
The kinetic energy of a falling object is given by:
and
a = hydrodynamic added mass for considered motion
For impacts in air the velocity is given by
s = travelled distance from drop pointv = vo at sea surface
For objects falling rectilinearly in water the velocity dependsupon the reduction of speed during impact with water and thefalling distance relative to the characteristic distance for theobject.
Figure 4-1Velocity profile for objects falling in water
The loss of momentum during impact with water is given by
F(t) = force during impact with sea surface
After the impact with water the object proceeds with the speed
Assuming that the hydrodynamic resistance during fall inwater is of drag type the velocity in water can be taken fromFigure 4-1 where
ρw = density of sea water
Cd = hydrodynamic drag coefficient for the object in theconsidered motion
m = mass of object
A p = projected cross-sectional area of the object
V = object displacement
The major uncertainty is associated with calculating the loss of
momentum during impact with sea surface, given by Equation(4.4). However, if the travelled distance is such that the veloc-ity is close to the terminal velocity, the impact with sea surfaceis of little significance.
Typical terminal velocities for some typical objects are given
(in air) (4.1)2
kin mv2
1E =
(in water) (4.2)
(4.3)
( ) 2
kin vam2
1E +=
2gsv =
s
-3
-2
-1
0
1
2
3
4
5
6
7
0 0,5 1 1,5 2 2,5 3 3,5 4
Velocity [v/vt]
In water
In air
D i
s t a n c e [ s / s c
]
ss
-3
-2
-1
0
1
2
3
4
5
6
7
0 0,5 1 1,5 2 2,5 3 3,5 4
Velocity [v/vt]
In water
In air
D i
s t a n c e [ s / s c
]
-3
-2
-1
0
1
2
3
4
5
6
7
0 0,5 1 1,5 2 2,5 3 3,5 4
Velocity [v/vt]
In water
In air
D i
s t a n c e [ s / s c
]
(4.4)
=
terminal velocity for the
object (drag force and buoyancy force balance thegravity force)
∫=Δ dt
0F(t)dtvm
Δvvv 0 −=
pdw
wt
ACρV)ρ2g(mv −=
= characteristic distance
)m
Vρ2g(1
)ma(1v
ACρ
ams
w
2t
pdw −
+=
+=c
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in Table 4-1.
Rectilinear motion is likely for blunt objects and objects whichdo not rotate about their longitudinal axis. Bar-like objects(e.g. pipes) which do not rotate about their longitudinal axismay execute lateral, damped oscillatory motions as illustratedin Figure 4-1.
4.3 Dissipation of strain energy
The structural response of the dropped object and the impactedcomponent can formally be represented as load-deformationrelationships as illustrated in Figure 4-2. The part of the impactenergy dissipated as strain energy equals the total area under
the load-deformation curves.
As the load level is not known a priori an incremental approachis generally required.
Often the object can be assumed to be infinitely rigid (e.g. axialimpact from pipes and massive objects) so that all energy is to
be dissipated by the impacted component.
Figure 4-2Dissipation of strain energy in dropped object and installation
If the object is assumed to be deformable, the interactive nature
of the deformation of the two structures should be recognised.
4.4 Resistance/energy dissipation
4.4.1 Stiffened plates subjected to drill collar impact
The energy dissipated in the plating subjected to drill collar impact is given by
where:
f y = characteristic yield strength
R = πdtτ = contact force for τ ≤τ cr refer Section 4.5.1 for τ cr
For validity range of design formula reference is given to Ch.8,Comm. 4.4.1.
Figure 4-3Definition of distance to plate boundary
4.4.2 Stiffeners/girders
In lieu of more accurate calculations stiffeners and girders sub- jected to impact with blunt objects may be analysed withresistance models given in Section 6.10.
4.4.3 Dropped objectCalculation of energy dissipation in deformable droppedobjects shall be based upon recognised methods for plasticanalysis. It shall be documented that the collapse mechanismsassumed yield a realistic representation of the true deformationfield.
4.5 Limits for energy dissipation
4.5.1 Pipes on plated structures
The maximum shear stress for plugging of plates due to drillcollar impacts may be taken as
f u = ultimate material tensile strength
4.5.2 Blunt objects
For stability of cross-sections and tensile fracture, refer Sec-tion 3.10.
5. Fire
5.1 General
The characteristic fire structural load is temperature rise inexposed members. The temporal and spatial variation of tem-
perature depends on the fire intensity, whether or not the struc-tural members are fully or partly engulfed by the flame and towhat extent the members are insulated.
Structural steel expands at elevated temperatures and internalstresses are developed in redundant structures. These stresses
Table 4-1 Terminal velocities for objects falling in water
Item Mass[kN]
Terminal velocity[m/s]
Drill collar Winch,Riser pump
28250100
23-24
BOP annular preventer 50 16
Mud pump 330 7
(4.5)
(4.6)
: stiffness of plateenclosed by hinge circle
∫∫ +=+=max,maxo, w
0ii
w
0oois,os,s dwR dwR EEE
i
dwo dwi
R iR o
Object Installation
Es,oEs,i
2
i2
spm
m0.481
2k
R E ⎟⎟
⎠
⎞⎜⎜⎝
⎛ +=
( )⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
+
⎟ ⎠ ⎞⎜
⎝ ⎛ +−+
=2
2
2
yc1
2r d6.256c
r d51
tπf 2
1k
= mass of plate enclosed by hinge circle
m = mass of dropped object
ρ p = mass density of steel plate
d = smaller diameter at threaded end of drillcollar
r = smaller distance from the point of impact tothe plate boundary defined by adjacentstiffeners/girders, refer Figure 4-3.
(4.7)
⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −−
−= 2r
d12.5
ec
tπr ρm 2 pi =
r r r
⎟
⎠
⎞⎜
⎝
⎛ +=
d
t0.410.42f τ ucr
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are most often of moderate significance with respect to globalintegrity. The heating causes also progressive loss of strengthand stiffness and is, in redundant structures, accompanied byredistribution of forces from members with low strength tomembers that retain their load bearing capacity. A substantialloss of load-bearing capacity of individual members and sub-assemblies may take place, but the load bearing function of theinstallation shall remain intact during exposure to the fire load.
In addition, damaged condition should be checked if relevant,see Section 2.2.
Structural analysis may be performed on either
— individual members — subassemblies — entire system
The assessment of fire load effect and mechanical responseshall be based on either
— simple calculation methods applied to individual mem- bers,
— general calculation methods,
or a combination.
Simple calculation methods may give overly conservativeresults. General calculation methods are methods in whichengineering principles are applied in a realistic manner to spe-cific applications.
Assessment of individual members by means of simple calcu-lation methods should be based upon the provisions given inCh.7 /2/ Eurocode 3 Part 1.2. /2/ .
Assessment by means of general calculation methods shall sat-isfy the provisions given in Ch.7 /2/ Eurocode 3 Part1.2, Sec-tion 4.3.
In addition, the requirements given in this section for mechan-ical response analysis with nonlinear finite element methods
shall be complied with.
Assessment of ultimate strength is not needed if the maximumsteel temperature is below 400°C, but deformation criteria mayhave to be checked for impairment of main safety functions.
5.2 General calculation methods
Structural analysis methods for non-linear, ultimate strengthassessment may be classified as
— stress-strain based methods — stress-resultants based (yield/plastic hinge) methods
Stress-strain based methods are methods where non-linear material behaviour is accounted for on fibre level.
Stress-resultants based methods are methods where non-linear material behaviour is accounted for on stress-resultants level based upon closed form solutions/interaction equation for cross-sectional forces and moments.
5.3 Material modelling
In stress-strain based analysis temperature dependent stress-strain relationships given in Ch.7 /2/ Eurocode 3, Part 1.2, Sec-tion 3.2 may be used.
For stress resultants based design the temperature reduction of the elastic modulus may be taken as k E,θ according to Ch.7 /2/ Eurocode 3. The yield stress may be taken equal to the effec-tive yield stress, f y,θ. The temperature reduction of the effec-tive yield stress may be taken as k y,θ.
Provided that the above requirements are complied with creepdoes need explicit consideration.
5.4 Equivalent imperfections
To account for the effect of residual stresses and lateral distor-
tions compressive members shall be modelled with an initial,sinusoidal imperfection with amplitude given by
Elastic-perfectly plastic material model, refer Figure 6-4 :
Elasto-plastic material models, refer Figure 6-4 :
α = 0.5 for fire exposed members according to columncurve c, Ch.7 /2/ Eurocode 3
i = radius of gyrationz0 = distance from neutral axis to extreme fibre of cross-
sectionWP = plastic section modulusW = elastic section modulus
A = cross-sectional areaI = moment of inertiae* = amplitude of initial distortion
= member length
The initial out-of-straightness should be applied on each phys-ical member. If the member is modelled by several finite ele-ments the initial out-of-straightness should be applied asdisplaced nodes.
The initial out-of-straightness shall be applied in the samedirection as the deformations caused by the temperature gradi-ents.
5.5 Empirical correction factor
The empirical correction factor of 1.2 should be accounted for in calculating the critical strength in compression and bendingfor design according to Ch.7 /2/ Eurocode 3, refer Ch.8,Comm. A.5.5.
5.6 Local cross sectional buckling
If shell modelling is used, it shall be verified that the softwareand the modelling is capable of predicting local buckling withsufficient accuracy. If necessary, local shell imperfectionshave to be introduced in a similar manner to the approachadopted for lateral distortion of beams
If beam modelling is used local cross-sectional buckling shall be given explicit consideration.
In lieu of more accurate analysis cross-sections subjected to
plastic deformations shall satisfy compactness requirementsgiven in DNV-OS-C101:
type I: Locations with plastic hinges (approximately full plastic utilization)
type II: Locations with yield hinges (partial plastification)
If this criterion is not complied with explicit considerationsshall be performed. The load-bearing capacity will be reducedsignificantly after the onset of buckling, but may still be signif-icant. A conservative approach is to remove the member fromfurther analysis.
Compactness requirements for type I and type I cross-sectionsmay be disregarded provided that the member is capable of developing significant membrane forces.
5.7 Ductility limits
5.7.1 General
The ductility of beams and connections increase at elevatedtemperatures compared to normal conditions. Little informa-
α π 0
y*
z
i
E
f 1e=
l
αAI
W
E
f 1α
z
i
E
f 1
W
We py
0
y*
p
π π ==
l
l
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tion exists.
5.7.2 Beams in bending
In lieu of more accurate analysis requirements given in Section3.10 shall be complied with.
5.7.3 Beams in tension
In lieu of more accurate analysis an average elongation of 3%of the member length with a reasonably uniform temperaturecan be assumed.
Local temperature peaks may localise plastic strains. It is con-sidered to be to the conservative side to use critical strains for steel under normal temperatures. See Section 3.10 and 3.10.4.
5.8 Capacity of connections
In lieu of more accurate calculations the capacity of the con-nection can be taken as:
R θ = k y,θ R 0where
R 0 = capacity of connection at normal temperature
k y,θ = temperature reduction of effective yield stress for max-imum temperature in connection
6. Explosions
6.1 General
Explosion loads are characterised by temporal and spatial pres-sure distribution. The most important temporal parameters arerise time, maximum pressure and pulse duration.
For components and sub-structures the explosion pressureshall normally be considered uniformly distributed. On globallevel the spatial distribution is normally non-uniform both withrespect to pressure and duration.
The response to explosion loads may either be determined bynon-linear dynamic finite element analysis or by simple calcu-lation models based on Single Degree Of Freedom (SDOF)analogies and elastic-plastic methods of analysis.
If non-linear dynamic finite element analysis is applied alleffects described in the following paragraphs shall either beimplicitly covered by the modelling adopted or subjected tospecial considerations, whenever relevant.
In the simple calculation models the component is transformedto a single spring-mass system exposed to an equivalent load pulse by means of suitable shape functions for the displace-ments in the elastic and elastic-plastic range. The shape func-tions allow calculation of the characteristic resistance curve
and equivalent mass in the elastic and elastic-plastic range aswell as the fundamental period of vibration for the SDOF sys-tem in the elastic range.
Provided that the temporal variation of the pressure can beassumed to be triangular, the maximum displacement of thecomponent can be calculated from design charts for the SDOFsystem as a function of pressure duration versus fundamental period of vibration and equivalent load amplitude versus max-imum resistance in the elastic range. The maximum displace-ment must comply with ductility and stability requirements for the component.
The load bearing function of the installation shall remain intactwith the damages imposed by the explosion loads. In addition,damaged condition should be checked if relevant, see Section2.2.
6.2 Classification of response
The response of structural components can conveniently beclassified into three categories according to the duration of theexplosion pressure pulse, td, relative to the fundamental periodof vibration of the component, T:
Impulsive domain:
The response is governed by the impulse defined by
Hence, the structure may resist a very high peak pressure pro-
vided that the duration is sufficiently small. The maximumdeformation, wmax, of the component can be calculated itera-tively from the equation
where
R(w)= force-deformation relationship for the componentmeq = equivalent mass for the component.
Quasi-static-domain:
The response is governed by the peak pressure and the risetime of the pressure relative to the fundamental period of vibra-
tion. If the rise time is small the maximum deformation of thecomponent can be solved iteratively from the equation:
If the rise time is large the maximum deformation can besolved from the static condition
Dynamic domain:
The response has to be solved from numerical integration of
the dynamic equations of equilibrium.6.3 Recommended analysis models for stiffened pan-els
Various failure modes for a stiffened panel are illustrated inFigure 6-1. Suggested analysis model and reference to applica- ble resistance functions are listed in Table 6.1. Application of a Single Degree of Freedom (SDOF) model in the analysis of stiffeners/girders with effective flange is implicitly based onthe assumption that dynamic interaction between the plateflange and the profile can be neglected.
Impulsive domain td/T < 0.3
Dynamic domain 0.3 < td/T < 3
Quasi-static domain 3 < td/T
(6.1)
(6.2)
(6.3)
(6.4)
( )∫=dt
0dttFI
( )∫=maxw
0eq dwwR 2mI
( )∫=maxw
0max
max dwwR F
1w
)R(wF maxmax =
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Figure 6-1Failure modes for two-way stiffened panel
Table 6-1 Analysis models
Failure mode Simplifiedanalysis model Resistance models Comment
Elastic-plastic deformation of plate SDOF Section 6.9
Stiffener plastic – plate elastic
SDOF Stiffener: Section 6.10.1-2.Plate: Section 6.9.1
Elastic, effective flange of plate
Stiffener plastic – plate plastic
SDOF Stiffener: Section 6.10.1-2.Plate: Section 6.9
Effective width of plate at mid span. Elastic, effectiveflange of plate at ends.
Girder plastic – stiffener and plating elastic
SDOF Girder: Section 6.10.1-2Plate: Section 6.9
Elastic, effective flange of plate with concentrated loads(stiffener reactions). Stiffener mass included.
Girder plastic – stiffener elastic – plate plastic
SDOF Girder: Section 6.10.1-2Plate: Section 6.9
Effective width of plate at girder mid span and ends.Stiffener mass included
Girder and stiffener plastic – plate elastic
MDOF Girder and stiffener:Section 6.10.1-2Plate: Section 6.9
Dynamic reactions of stiffeners→ loading for girders
Girder and stiffener plastic – plate plastic
MDOF Girder and stiffener:Section 6.10.1-2Plate: Section 6.9
Dynamic reactions of stiffeners→ loading for girders
By girder/stiffener plastic is understood that the maximum displacement wmax exceeds the elastic limit wel
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6.4 SDOF system analogy
Biggs method:
For many practical design problems it is convenient to assumethat the structure - exposed to the dynamic pressure pulse - ulti-mately attains a deformed configuration comparable to thestatic deformation pattern. Using the static deformation patternas displacement shape function, i.e.
the dynamic equations of equilibrium can be transformed to anequivalent single degree of freedom system:
The equilibrium equation can alternatively be expressed as:
where
The natural period of vibration for the equivalent system in thelinear resistance domain is given by
The response, y(t), is - in addition to the load history - entirelygoverned by the total mass, load-mass factor and the character-istic stiffness.
For a linear system, the load mass factor and the characteristicstiffness are constant k = k 1. The response is then alternativelygoverned by the eigenperiod and the characteristic stiffness.
For a non-linear system, the load-mass factor and the charac-teristic stiffness depend on the response (deformations). Non-linear systems may often conveniently be approximated byequivalent bi-linear or tri-linear systems, see Section 6.8. Insuch cases the response can be expressed in terms of (see Fig-ure 6-6 for definitions):
k 1 = characteristic stiffness in the initial, linear resistancedomain
yel = displacement at the end of the initial, linear resistancedomain
T = eigenperiod in the initial, linear resistance domain
and, if relevant,
k 3 = normalised characteristic resistance in the third linear resistance domain.
Characteristic stiffness is given explicitly or can be derivedfrom load-deformation relationships given in Section 6.10. If the response is governed by different mechanical behaviour relevant characteristic stiffness must be calculated.
For a given explosion load history the maximum displacement,ymax, is found by analytical or numerical integration of equa-tion (6.6).
For standard load histories and standard resistance curvesmaximum displacements can be presented in design charts.Figure 6-2 shows the normalised maximum displacement of a
SDOF system with a bi- (k 3 = 0) or tri-linear (k 3 > 0) resistancefunction, exposed to a triangular pressure pulse with zero risetime. When the duration of the pressure pulse relative to theeigenperiod in the initial, linear resistance range is known, themaximum displacement can be determined directly from thediagram as illustrated in Figure 6-2.
(6.5)
φ(x) = displacement shape func-tion
y(t) = displacement amplitude
= generalized mass
= generalized load
= generalized elastic bend-ing stiffness
= generalized plastic bend-ing stiffness(fully developed mecha-nism)
= generalized membranestiffness(fully plastic: N = NP)
m = distributed mass
Mi = concentrated massq = explosion load
Fi = concentrated load (e.g.support reactions)
xi = position of concentratedmass/load
(6.6)
=load-mass transformation factorfor uniform mass
=load-mass transformation factorfor concentrated mass
=mass transformation factor for uni-form mass
=
mass transformation factor for
concentrated mass
( ) ( ) ( )tyxt,xw φ=
( )tf yk m =+ y&&
( ) ∑∫ +=i
2
ii
2φMdxxmφm
l
( )∫ ∑+=l i
iiφFdxxq(t)φ)t(f
( )∫=l
dxxEIφk 2
xx,
0k =
( )∫=l
dxx Nφk 2
x,
( )ii xxφφ ==
F(t)K(y)yy)MK M(K ccm,uum, =++ &&ll
l
l
K
K K
um,
um, =
l
l
K
K K
um,
um, =
u
2
um,M
dx(x)m
K
∫= l
ϕ
c
i
2
cm,M
M
K
∑=
ϕ i
=load transformation factor foruniformly distributed load
=load transformation factor forconcentrated load
= total uniformly, distributed mass
= total concentrated mass
=total load in case of uniformlydistributed load
=total load in case of concentratedload
= equivalent stiffness
(6.7)
F
(x)dxq
K
∫= l
l
ϕ
F
F
K i
∑=
iiϕ
l
∫=l
mdxM u
c i
i
M M= ∑
∫=l
qdxF
i
i
F F= ∑
lk
k k e =
e
ccm,uum,
k
MK MK 2
k
m2T
ll +
== π π
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Figure 6-3Dynamic response of a SDOF system to a triangular load (rise time = 0.50 td)
6.6 MDOF analysis
SDOF analysis of built-up structures (e.g. stiffeners supported
by girders) is admissible if
— the fundamental periods of elastic vibration are suffi-ciently separated
— the response of the component with the smallest eigenpe-riod does not enter the elastic-plastic domain so that the
period is drastically increased
If these conditions are not met, then significant interaction between the different vibration modes is anticipated and amulti degree of freedom analysis is required with simultaneoustime integration of the coupled system.
6.7 Classification of resistance properties
6.7.1 Cross-sectional behaviour
Figure 6-4Bending moment-curvature relationships
Elasto-plastic : The effect of partial yielding on bendingmoment is accounted for
Elastic-perfectly plastic: Linear elastic up to fully plastic bend-ing moment
The simple models described herein assume elastic-perfectly plastic material behaviour.
Note: Even if the analysis is based upon elastic-perfectly plas-tic behaviour, the material has to exhibit strain hardening in
practice for the theory to be valid. The effect of strain harden-ing on the plastic, cross-sectional resistances may beaccounted for by using an equivalent (increased) yield stress.If this is considered relevant, and the material is utilised
beyond ultimate strain, it is often justified to use an equivalentyield stress equal to the mean of the lower yield stress and theultimate stress.
In the clauses for the ductility limits the effect of strain hard-ening is accounted for.
0.1
1
10
100
0.1 1 10
td/T
y m a x / y e l
= 1.1
Rel/Fmax= 0.
= 1.0
= 0.9
= 1.2= 1.5
=0.1
= 0.7
= 0.6= 0.5Rel/Fmax=0.05 = 0.3
yel y
R
Rel
F
Fmax
td0.50td
k1
k3 = 0.5k1 =0.2k1 =0.1k1
k3 = 0
k3 = 0.1k1
k3 = 0.2k1
k3 = 0.5k1
Elastic-perfectly plastic
elasto-plastic
Moment
Curvature
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Component behaviour
Figure 6-5Resistance curves
Elastic: Elastic material, small deformations
Elastic-plastic (determinate): Elastic-perfectly plastic mate-rial. Statically determinate system. Bending mechanism fullydeveloped with occurrence of first plastic hinge(s)/yield lines.
No axial restraint. Elastic-plastic (indeterminate): Elastic perfectly plastic mate-rial. Statically indeterminate system. Bending mechanismdevelops with sequential formation of plastic hinges/yieldlines. No axial restraint. For simplified analysis this systemmay be modelled as an elastic-plastic (determinate) systemwith equivalent initial stiffness. In lieu of more accurate anal-ysis the equivalent stiffness should be determined such that thearea under the resistance curve is preserved.
Elastic-plastic with membrane: Elastic-perfectly plastic mate-rial. Statically determinate (or indeterminate). Ends restrainedagainst axial displacement. Increase in load-carrying capacitycaused by development of membrane forces.
6.8 Idealisation of resistance curves
The resistance curves in 6.7 are idealised. For simplifiedSDOF analysis the resistance characteristics of a real non-lin-ear system may be approximately modelled. An example witha tri-linear approximation is illustrated in Figure 6-6. The stiff-ness in the k 3 phase may have some contribution from strainhardening, but in most cases the predominant effect is devel-opment of membrane forces when member ends are restrainedform inward displacement.
Figure 6-6Representation of a non-linear resistance by an equivalent tri-lin-ear system
In lieu of more accurate analysis the resistance curve of elastic- plastic systems may be composed by an elastic resistance anda rigid-plastic resistance as illustrated in Figure 6-7.
Figure 6-7Construction of elastic -plastic resistance curve
6.9 Resistance curves and transformation factors forplates
6.9.1 Elastic - rigid plastic relationships
In lieu of more accurate calculations rigid plastic theory com- bined with elastic theory may be used.
In the elastic range the stiffness and fundamental period of vibration of a clamped plate under uniform lateral pressure can be expressed as:
The factors ψ and η are given in Figure 6-8.
Figure 6-8Coefficients ψ and η.
R R R R
w w w w
k 1k 1k 1k 1
k 3k 2
k 2k 2
Elastic Elastic-plastic
(determinate)
Elastic-plastic
(indeterminate)
Elastic-plastic
with membrane
R R R R
w w w w
k 1k 1k 1k 1
k 3k 2
k 2k 2
Elastic Elastic-plastic
(determinate)
Elastic-plastic
(indeterminate)
Elastic-plastic
with membrane
k 2=0
k 1
R
k 3
w
R el
wel
k 2=0
k 1
R
k 3
w
R el
wel
r = k 1w=
resistance-displacement relationship for plate centre
= plate stiffness
= natural period of vibration
= plate bending stiffness
Elastic
+ =
Rigid-plastic Elastic-plastic with membrane
41s
Dψk =
D
tsρ
η
2πT
4
=
( )2
3
ν112
tED
−=
0
100
200
300
400
500
600
700
800
1 1.5 2 2.5 3l/s
ψ
0
5
10
15
20
25
30
35
40
ψ
η
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In the plastic range the resistance (r) of plates with edges fullyrestrained against inward displacement and subjected to uni-form pressure can be taken as:
l (>s) = plate lengths = plate widtht = plate thicknessr c = plastic resistance in bending for plates with no axial
restraint= non-dimensional displacement parameter
Figure 6-9Plastic load-carrying capacities of plates as a function of lateraldisplacement
6.9.2 Axial restraint
In Equation (6.8) the beneficial effect of membrane stiffeningis represented by the term containing the non-dimensional dis-
placement parameter . Great caution should be exercisedwhen assuming the presence of the membrane effect, becausethe membrane forces must be anchored in the adjacent struc-ture. For plates located in the middle of a continuous platefield, the boundaries have often considerable restraint against
pull-in. If the plate is located close to the boundary, the edgesare often not sufficiently stiffened to prevent pull-in of edges.
Unlike stiffeners no simple method with a clear physical inter- pretation exists to quantify the effect of flexibility on the resist-
ance of plates under uniform pressure. In the formulations usedin this RP the flexibility may be split into two contributions
1) Pull-in of edges
2) Elastic straining of the plate
The effect of flexibility may be taken into account in anapproximate way by means of plate strip theory and the proce-dure described in Section 3.7.2. The relative reduction of the
plate’s plastic resistance, with respect to the values given inEquation (6.8), is taken equal to the relative reduction of theresistance for a beam with rectangular cross-section (platethickness x unit width) and length equal to stiffener spacing,using the diagram for α = 2 (Figure 6-12). The elastic straining
of the plate is accounted for by the 2nd term in Equation (6.8).In lieu of more accurate calculation, the effect of pull-in, given
by the first term in Equation (6.8) may be estimated by remov-ing the plate and apply a uniformly distributed unit in-planeforce normal to the plate edges. The axial stiffness should betaken as the inverse of the maximum in-plane displacement of the long edge.
In lieu of more accurate calculation, it should be conserva-tively assumed that no membrane effects exist for a platelocated close to an unsupported boundary, i.e. the resistanceshould be taken as constant and equal to the resistance in bend-ing, r = r c over the allowable displacement range.
In lieu of more accurate calculations, it is suggested to assessthe relative reduction of the resistance for a uniformly loaded
plate located some distance from an unsupported boundarywith c = 1.0.
If membrane forces are taken into account it must be verifiedthat the adjacent structure is strong enough to anchor the fully
plastic membrane tension forces.
6.9.3 Tensile fracture of yield hinges
In lieu of more accurate calculations the procedure describedin Section 3.10.4 may be used for a beam with rectangular cross-section (plate thickness x unit width) and length equal tostiffener spacing.
6.10 Resistance curves and transformation factorsfor beams
Provided that the stiffeners/girders remain stable against local buckling, tripping or lateral torsional buckling stiffened plates/girders may be treated as beams. Simple elastic-plastic meth-ods of analysis are generally applicable. Special considerationsshall be given to the effect of:
— Elastic flexibility of member/adjacent structure — Local deformation of cross-section — Local buckling — Strength of connections — Strength of adjacent structure — Fracture
6.10.1 Beams with no- or full axial restraint
Equivalent springs and transformation factors for load andmass for various idealised elasto-plastic systems are shown inTable 6-2. For more than two concentrated loads, equal inmagnitude and spacing, use values for uniform loading.
Shear deformation may have a significant impact on the elasticflexibility and eigenperiod of beams and girders with a shortspan/web height ratio (L/hw), notably for clamped ends. Theeigenperiod and stiffness in the linear domain including shear deformation may be calculated as:
and
where
(6.8)
Pinned ends:
Clamped ends:
= plate aspect parameter
( )1w
3α9
2α3αw1
r
r 2
2
c
≤⎟⎟ ⎠
⎞⎜⎜⎝
⎛
−−+
+=
( ) 1w1w3
1α3α2α1w2
r r
2c
>⎟⎟ ⎠ ⎞⎜⎜
⎝ ⎛ ⎟⎟
⎠ ⎞⎜⎜
⎝ ⎛ −
−−+=
22
2
y
cα
t6f r
t
w2w
l ==
22
2
y
cα
t12f r
t
ww
l ==
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−⎟ ⎠
⎞
⎜⎝
⎛
+= l l l
ss
3
s
α
2
w
0
1
2
3
4
5
6
0 0.5 1 1.5 2 2.5 3
Relative displacement
R e s i s t a n c e [ r / r c ]
l/s = 100
5
3 2
1
w
w
(6.9)
(6.10)
⎟⎟ ⎠
⎞⎜⎜⎝
⎛ +⎟
⎟ ⎠
⎞⎜⎜⎝
⎛ +
+==
w
2
g
s'
1
ccm,uum,
A
A
G
E1
L
r πc1
k
MK MK 2
k
m2T
ll
π π
L
GAck ,
k
1
k
1
k
1 w
Q1'
1
=+=
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cs = 1.0 for both ends simply supported= 1.25 for one end clamped and one end simply sup-
ported = 1.5 for both ends clampedL = length of beam/girder E = elastic modulusG = shear modulusA = total cross-sectional area of beam/girder
Aw = shear area of beam/girder k Q = shear stiffness for beam/girder k 1 = bending stiffness of beam/girder in the linear domain
according to Table 6-2r g = radius of gyration
M ps = plastic bending capacity of beam at supportM pm= plastic bending capacity of beam at midspan
and regardless of rotational boundary conditions the followingvalues may be used
cQ = 8 for uniformly distributed loads = 4 for one concentrated loads
= 6 for two concentrated loads
The dynamic reactions according to Table 6-2 become increas-ingly inaccurate for loads with short duration and/or high mag-nitudes.
Table 6-2 Transformation factors for beams with various boundary and load conditions
Load case
Resistancedomain
Load Factor
K l
Mass factor K m
Load-mass factor K lm Maximum
resistance Rel
Linear stiffness
k 1
Dynamic reaction
V Concen-
tratedmass
Uni- formmass
Concen-tratedmass
Uniformmass
Elastic 0.64 0.50 0.78
Plastic bending
0.50 0.33 0.66 0
Plasticmembrane
0.50 0.33 0.66
Elastic 1.0 1.0 0.49 1.0 0.49
Plastic bending
1.0 1.0 0.33 1.0 0.33 0
Plasticmembrane
1.0 1.0 0.33 1.0 0.33
Elastic 0.87 0.76 0.52 0.87 0.60
Plastic bending
1.0 1.0 0.56 1.0 0.56 0
Plasticmembrane
1.0 1.0 0.56 1.0 0.56
Load case Resist-
ancedomain
Load Fac-
tor K l
Mass factor K m
Load-mass factor K lm Maximum
resistance Rel
Linear stiffness
k 1
Equiva-lent lin-
ear stiffness
k e
Dynamic reaction
V Concen-tratedmass
Uniformmass
Con-cen-
tratedmass
Uniformmass
Elastic 0.53 0.41 0.77
Elasto- plastic
bending0.64 0.50 0.78
Plastic bending
0.50 0.33 0.66 0
Plasticmem- brane
0.50 0.33 0.66
F=pL
L
8 M
L
p 384
53
EI
L0 39 011. . R F +
8 M
L
p 0 38 012. . R F el +
4 N
L
P
L
y N max P 2
L/2
F
L/2
4 M
L
p 483
EI
L0 78 0 28. . R F −
4 M
L
p 0 75 0 25. . R F el −
4 N
L
P
L
y N max P 2
L/3 L/3 L/3
F/2 F/2
6 M
L
p 56 43
. EI
L0 525 0 025. . R F −
6 M
L
p 0 52 0 02. . R F el −
6 N
L
P
L
y N max P 3
F=pL
L
12 M
L
ps 3843
EI
L F R 14.036.0 +
( )8 M M
L
ps Pm+ 384
53
EI
L0 39 011. . R F el +
( )8 M M
L
ps Pm+ 0 38 012. . R F el +
4 N
L
P
L
y N max p2
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Where:
q = explosion load per unit length = ps for stiffeners = p for girders
m1, m2 and m3 are factors for deriving the equivalent stiffness:
Elastic 1.0 1.0 0.37 1.0 0.37
Plastic bending
1.0 1.0 0.33 1.0 0.33 0
Plasticmem- brane
1.0 1.0 0.33 1.0 0.33
Elastic 080 0.64 0.41 0.80 0.51
Elasto- plastic
bending0.87 0.76 0.52 0.87 0.60
Plastic bending
1.0 1.0 0.56 1.0 0.56 0
Plasticmem- brane
1.0 1.0 0.56 1.0 0.56
Elastic 0.58 0.45 0.78
Elasto- plastic bending
0.64 0.50 0.78
Plastic bending
0.50 0.33 0.66 0
Plasticmem- brane
0.50 0.33 0.66
Elastic 1.0 1.0 0.43 1.0 0.43
Elasto- plastic
bending1.0 1.0 0.49 1.0 0.49
Plastic bending
1.0 1.0 0.33 1.0 0.33 0
Plasticmem- brane
1.0 1.0 0.33 1.0 0.33
Elastic 0.81 0.67 0.45 0.83 0.55
Elasto- plastic
bending0.87 0.76 0.52 0.87 0.60
Plastic bending
1.0 1.0 0.56 1.0 0.56 0
Plasticmem- brane
1.0 1.0 0.56 1.0 0.56
Load case Resist-
ancedomain
Load Fac-
tor K l
Mass factor K m
Load-mass factor K lm Maximum
resistance Rel
Linear stiffness
k 1
Equiva-lent lin-
ear stiffness
k e
Dynamic reaction
V Concen-
tratedmass
Uniformmass
Con-cen-
tratedmass
Uniformmass
F
L/2L/2
( )4 M M
L
ps Pm+ 1923
EI
L13
48m
L
EI ⋅⎟
⎠
⎞⎜⎝
⎛
0 71 0 21. . R F −
( )4 M M
L
ps Pm+ 0 75 0 25. . R F el −
4 N
L
P
L
y N max P 2
L/3 L/3 L/3
F/2 F/2
9 ps M
L3
260 EI
L
13
212m
L
EI ⋅⎟
⎠
⎞⎜⎝
⎛
0.48 0.02 R F +
( )6 ps Pm M M
L
+3
56.4 EI
L0.52 0.02el R F −
( )6 ps Pm M M
L
+0.52 0.02el R F −
6 P N
L
V2 V1
F=pL
L
8 M
L
ps 1853
EI
L23
160m
L
EI ⋅⎟
⎠
⎞⎜⎝
⎛
V R F 1 0 26 0 12= +. .
V R F 2 0 43 019= +. .
( )4 2 M M
L
ps Pm+ 384
5 3
EI
L
0 39 011. . R F
M L Ps
+
±
( )4 2 M M
L
ps Pm+ 0 38 012. . R F
M L Ps
+
±
4 N
L
P
L
y N max P 2
V1
L/2 L/2
F
V2
16
3
M
L
Ps 1073
EI
L23
160m
L
EI ⋅⎟
⎠
⎞⎜⎝
⎛
V R F 1 0 25 0 07= +. .V R F 2 054 014= +. .
( )2 2 M M
L
ps Pm+ 483
EI
L
0 78 0 28. . R F
M L Ps
−
±
( )2 2 M M
L
ps Pm+ 0 75 0 25. . R F
M L Ps
−
±
4 N
L
P
L
y N max P 2
V1
L/3 L/3 L/3
F/2 F/2
V2
6 M
L
Ps 1323
EI
L
33
122
m L
EI
⋅⎟ ⎠
⎞
⎜⎝
⎛
V R F 1 017 017= +. .
V R F 2 0 33 0 33= +. .
( )2 3 M M
L
ps Pm+ 563
EI
L
0 525 0 025. . R F
M L Ps
−±
( )2 3 M M
L
ps Pm+ 0 52 0 02. . R F
M L
el
Ps
−
±
6 N
L
P
L
y N max P 3
l
25.05.1
1 ++
= pm ps
ps
M M
M m
5.02
5.12 ++=
pm ps
ps
M M
M m
5.03
23 +
+=
pm ps
ps
M M
M m
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6.10.2 Beams with partial end restraint.
Relatively small axial displacements have a significant influ-ence on the development of tensile forces in members under-going large lateral deformations. Equivalent elastic, axialstiffness may be defined as
k node = axial stiffness of the node with the considered member removed. This may be determined by introducing unit loads inmember axis direction at the end nodes with the member removed.
Plastic force-deformation relationship for a beam under uni-form pressure may be obtained from Figure 6-10, Figure 6-11or Figure 6-12 if the plastic interaction between axial force and
bending moment can be approximated by the following equa-tion:
In lieu of more accurate analysis α = 1.2 can be assumed for stiffened plates and H or I beams. For members with tubular section α = 1.5. For rectangular sections and plates α = 2.0 can
be assumed.
Figure 6-10Plastic load-deformation relationship for beam with axial flexibility (α = 1.2)
(6.11)
(6.12)
= plastic collapse resistance in bending forthe member with uniform load.
= member length
2EAk
1
k
1
node
l+=
21for 1 N
N
M
Mα
p p
<<=⎟⎟
⎠
⎞⎜⎜
⎝
⎛ + α
l
py1
0
Wf 8cR =
l
= non-dimensional deformation
=characteristic beam height for beamsdescribed by plastic interaction equation(6.12).
= non-dimensional spring stiffness
c1 = 2 = for clamped beams
c1 = 1 = for pinned beams
WP = plastic section modulus for the cross sec-tion of the beam
W p = zgAg = plastic section modulus for stiffened plate for set > As
A = As + st = total area of stiffener and plate flange
Ae = As + set = effective cross-sectional area of stiffenerand plate flange,
zg = distance from plate flange to stiffenercentre of gravity.
As = stiffener area
s = stiffener spacing
se = effective width of plate flange see Sec-tion 6.10.4
c1wc
ww =
A
αWw
p
c =
lAf
kw4cc
y
2
c1=
α = 1.2
0
1
2
3
4
5
6
0 0,5 1 1,5 2 2,5 3 3,5 4
Deformation
1
0
0.10.2
0.5c = ∞
w
Bending & membrane
Membrane only
k k
F (explosion load)
w
R / R 0
α = 1.2
0
1
2
3
4
5
6
0 0,5 1 1,5 2 2,5 3 3,5 4
Deformation
1
0
0.10.2
0.5c = ∞
w
Bending & membrane
Membrane only
k k
F (explosion load)
w
R / R 0
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Figure 6-11Plastic load-deformation relationship for beam with axial flexibility (α = 1.5)
Figure 6-12Plastic load-deformation relationship for beam with axial flexibility (α = 2)
For members where the plastic moment capacity of adjacentmembers is smaller than the moment capacity of the exposedmember the force-deformation relationship may be interpo-lated from the curves for pinned ends and clamped ends:
where
α = 1.5
0
1
2
3
4
5
6
7
0 0,5 1 1,5 2 2,5 3 3,5 4
Deformation
0.10.20.5
1
0
w
c = ∞
Bending & membraneMembrane only
k k
F (explosion load)
w
R / R 0
α = 1.5
0
1
2
3
4
5
6
7
0 0,5 1 1,5 2 2,5 3 3,5 4
Deformation
0.10.20.5
1
0
w
c = ∞
Bending & membraneMembrane only
k k
F (explosion load)
w
R / R 0
α = 2
0
1
2
3
4
5
6
7
8
9
0 0,5 1 1,5 2 2,5 3 3,5 4
Deformation
1
0.5
0.20.1
0
w
c = ∞
Bending & membrane
Membrane only
k k
F (explosion load)
w
R / R 0
α = 2
0
1
2
3
4
5
6
7
8
9
0 0,5 1 1,5 2 2,5 3 3,5 4
Deformation
1
0.5
0.20.1
0
w
c = ∞
Bending & membrane
Membrane only
k k
F (explosion load)
w
R / R 0
(6.13) pinnedclamped ζ)R (1ζR R −+=
(6.14)11
M8
R ζ0
p
actual
0≤−=≤
l
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i = adjacent member no i
j = end number {1,2}
MPj,i = Plastic bending moment for member no. i.
Elastic, rotational flexibility of the node is normally of moder-ate significance
6.10.3 Beams with partial end restraint - support capacitysmaller than plastic bending moment of member
For beams where the plastic moment capacity of adjacentmembers is smaller than the moment capacity of the impacted beam, the force-deformation relationship, R*, may be derivedfrom the resistance curve, R, for beams where the plasticmoment capacity of adjacent members is larger than themoment capacity of the impacted beam (Section 3.7.2), usingthe expression:
where
R 0 = Plastic bending resistance with clamped ends (c1 = 2) – moment capacity of adjacent members larger thanthe plastic bending moment of the beam
= Plastic bending resistance - moment capacity of adja-cent members at one or both ends smaller than the plas-tic bending moment of the member
i = adjacent member no i
j = end number {1,2}MPj,i = Plastic bending resistance for member no. i.
wlim = limiting non-dimensional deformation where themembrane force attains yield, i.e. the resistancecurve, R, with actual spring stiffness coefficient, c,intersects with the curve for c = ∞. If c = ∞,
for tubular beams and for stiffened plate
6.10.4 Effective flange
In order to analyse stiffened plate as a beam the effective width
of the plate between stiffeners need to be determined. Theeffective width needs to be reduced due to buckling and/or shear lag.
Shear lag effects may be neglected if the length is more than2.5 times the width between stiffeners. For guidance see Ch.8,
Commentary.
Determination of effective flange due to buckling can be madeas for buckling of stiffened plates see DNV-RP-C201.
The effective width for elastic deformations may be used whenthe plate flange is on the tension side.
In most cases the flange will experience varying stress with
parts in compression and parts in tension. It may be undulyconservative to use the effective width for the section with thelargest compression stress to be valid for the whole member length. For continuous stiffeners it will be reasonable to use themean value between effective width at the section with thelargest compression stress and the full width. For simple sup- ported stiffeners with compression in the plate it is judged to be reasonable to use the effective width at midspan for the totallength of the stiffener.
6.10.5 Strength of adjacent structure
The adjacent structure must be checked to see whether it can provide the support required by the assumed collapse mecha-nism for the member/sub-structure
6.10.6 Strength of connections
The capacity of connections can be taken from recognisedcodes.
The connection shall be checked for the dynamic reactionforce given in Table 6-2.
For beams with axial restraint the connection should also bechecked for lateral - and axial reaction in the membrane phase:
— If the axial force in a tension member exceeds the axialcapacity of the connection the member should be assumeddisconnected.
— If the capacity of the connection is exceeded in compres-sion and bending, this does not necessarily mean failure of the member. The post-collapse strength of the connectionmay be taken into account provided that such informationis available.
6.10.7 Ductility limits
Reference is made to Section 3.10.
The local buckling criterion in Section 3.10.2 and tensile frac-ture criterion given in Section 3.10.3 may be used with:
dc = characteristic dimension equal to twice the distancefrom the plastic neutral axis in bending to the extreme
fibre of the cross-sectionc = non-dimensional axial spring stiffness calculated in
Section 6.10.2.
Alternatively, the ductility ratios in Table 6-3 may be used.
= Collapse load in bending for beam accounting foractual bending resistance of adjacent members
(6.15)
(6.16)
, (6.17)
,
(6.18)
(6.19)
actual
0R
l
p2 p1 pactual
0
4M4M8MR
++=
∑ ≤=i
PPj,iPj MMM
lim
*00
* )R (R R R w
w−+= 0.1
lim
≤w
w
R R * = 0.1
lim
≥w
w
*0R
* P P1 P20
4M 2M 2MR
+ +=
l
∑ ≤=i
PiPj,Pj MMM
lim2
w wπ
= Table 6-3 Ductility ratios μ - beams with no axial restraint
Boundaryconditions
Load Cross-section type 1)
Type I Type II Type III
Cantilevered ConcentratedDistributed
67
45
22
Pinned ConcentratedDistributed
612
48
23
Fixed ConcentratedDistributed
64
43
22
1) Crossecton types are defined in DNV-OS-C101, Table A3, Appendix A
el y
ymax=μ
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7. References
8. Commentary
Comm. 2.3 General
The structural design is seen as having acceptable safetyagainst accidental loads when the design accidental loads areless than the design resistance. This is similar to the check of the structure for ordinary loads but with the following differ-
ences: As ordinary loads are either permanent or occur fre-quent it will not be acceptable that the load lead to reduced loadcarrying capacity while the short duration and the low proba-
bility of accidental loads make this an acceptable assumptions.E.g. a blast wall need not be capable of resisting another explo-sion after a blast, but if the wall is used as a fire barrier it needto serve as such after the blast.
Comm. 3.1 General
For typical installations, the contribution to energy dissipationfrom elastic deformation of component/substructures in directcontact with the ship is very small and can normally beneglected. Consequently, plastic methods of analysis apply.
However, elastic elongation of the hit member as well as axial
flexibility of the nodes to which the member is connected, havea significant impact on the development of membrane forces inthe member. This effect has to be taken into account in theanalysis, which is otherwise based on plastic methods. The dia-grams in Section 3.7.2 are based on such an approach.
Depending on the structure size/configuration as well as thelocation of impact elastic strain energy over the entire structuremay contribute significantly.
Comm. 3.2 Design principles
The transition from essentially strength behaviour to ductileresponse can be very abrupt and sensitive to minor changes inscantlings. E.g. integrated analyses of impact between the sternof a supply vessel and a large diameter column have shownthat with moderate change of (ring - and longitudinal) stiffener
size and/or spacing, the energy dissipation may shift from pre-dominantly platform based to predominantly vessel based.Due attention should be paid to this sensitivity when the calcu-lation procedure described in Section 3.5 is applied.
Comm. 3.3 Collision mechanics
The added mass is due to the hydrodynamic pressure induced by the forced motion of water particles on the wet surface of the ship. By solving the velocity potential for the fluid on the
body surface, the added mass is determined by means of 2-D(strip theory) or 3-D techniques. The added mass is frequencydependent, and thus varies with time during a collision, but aconstant value is recommended for simple analysis.
The fraction of collision energy to be dissipated as strainenergy for shuttle tanker impact on FPSO stern is shown inFigure 8-1. Note the strong dependency of the mass ratio; thelarger the mass of shuttle tanker, the lesser of the collisionenergy must be dissipated as strain energy. (However, pro-vided that the speed of the shuttle tanker is constant, the abso-lute value of the strain energy increases)
The relative size may differ considerably for the approach phase (shuttle tanker in ballast, FPSO fully loaded) and thedeparture phase (shuttle tanker fully loaded, FPSO in ballast).To illustrate this, possible values are listed in Table 8-1. In thisexample both the FPSO and shuttle tanker are large comparedto typical North Sea conditions. The same added mass coeffi-cient applies to both vessels. It is observed that the fraction of energy to be dissipated as strain energy varies between 0.33(departure) and 0.71 (approach). This indicates that theapproach phase may be particularly critical with respect to theconsequences of collision.
Figure 8-1Fraction of collision energy to be dissipated as strain energy forshuttle tanker impact on an FPSO.
Comm. 3.5.1 Recommended force-deformation relationships
The force-deformation relationship for impacts from supplyvessels/merchant vessels against jacket legs have been elabo-rated because of the need to consider high-energy impacts (col-lision energy ~50 MJ) for some installations in the North Sea.The likelihood of a central impact against a leg is obviously notvery large, but has still been considered because loss of a legcould be critical for some platforms. Experience has shown,however, that many large North Sea jackets have sufficientstrength to crush the bow. Reference is made to Amdahl andJohansen (2001).
/1/ NORSOK Standard N-003 Action and Action Effect
/2/ NS-ENV 1993-1 Eurocode 3: Design of Steel structuresPart 1-2. General rules - Structural fire design
/3/ Amdahl, J.: “Energy Absorption in Ship-PlatformImpacts”, UR-83-34, Dept. Marine Structures, Norwe-
gian Institute of Technology, Trondheim, 1983./4/ SCI 1993: Interim Guidance Notes for the Design and
Protection of Topside Structures against Explosion andFire
/5/ Amdahl, J.: “Mechanics of Ship-Ship Collisions- BasicCrushing Mechanics”. West Europene Graduate Schoolof Marine Technology, WEGEMT , Copenhagen, 1995
/6/ Skallerud, B. and Amdahl, j.: “Nonlinear Analysis ofOffshore Structures”, Research studies Press, UK 2002
/7/ Amdahl, J. and Johansen, A.: “High-Energy Ship Colli-sion with Jacket Legs” ISOPE, Stavanger, 2001
/8/ Moan, T., Amdahl, J., Wang, X. and Spencer, J.: “ RiskAssessment of FPSOs, with Emphasis on Collisions”,
SNAME Annual Meeting, Boston, 2002/9/ Skallerud, B. and Amdahl, j.: “Nonlinear Analysis of
Offshore Structures”, Research studies Press, UK 2002
/10/ Amdahl, J. and Johansen, A.: “High-Energy Ship Colli-sion with Jacket Legs” ISOPE, Stavanger, 2001 Table 8-1 Fraction of collision energy for strain energy
dissipation
Vessel size [dwt]
Approach phase Departure phase
Shuttle tanker 150.000 370.000
FPSO 320.000 160.000
Strain energy fraction 0.71 0.33
0
0,2
0,4
0,6
0,8
1
1,2
0 1 2 3 4
Mass ratio [(ms+ as)/(mi+ ai)]
S t r a i n e n e r g y f r a c t i o n
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The curve for bow impact in Figure 3-4 has been derived on theassumption of impacts against an infinitely rigid wall. Some-times the curve has been used erroneously to assess the energydissipation in bow-brace impacts.
Experience from small-scale tests Ch.7, /3/ indicates that the bow undergoes very little deformation until the brace becomesstrong enough to crush the bow. Hence, the brace absorbs most
of the energy. When the brace is strong enough to crush the bow the situation is reversed; the brace remains virtuallyundamaged.
On the basis of the tests results and simple plastic methods of analysis, force-deformation curves for bows subjected to(strong) brace impact were established in Ch.7, /3/ as a func-tion of impact location and brace diameter. These curves arereproduced in Figure 8-2. In order to fulfil a strength designrequirement the brace should at least resist the load level indi-cated by the broken line (recommended design curve). For
braces with a diameter to thickness ratio < 40 it should be suf-ficient to verify that the plastic collapse load in bending for the
brace is larger than the required level. For larger diameter tothickness ratios local denting must probably be taken intoaccount.
Normally sized jacket braces are not strong enough to resist thelikely bow forces given in Figure 8-2, and therefore it has to beassumed to absorb the entire strain energy. For the same rea-sons it has also to be assumed that the brace has to absorb allenergy for stern and beam impact with supply vessels.
Figure 8-2Load-deformation curves for bow-bracing impact, Ch.7, /3/
Comm. 3.5.2 Force contact area for strength design of largediameter columns.
Figure 8-3Distribution of contact force for stern corner/large diameter col-umn impact
Figure 8-3 shows an example of the evolution of contact force
intensity during a collision between the stern corner of a supplyvessel and a stiffened column. In the beginning the contact is
concentrated at the extreme end of the corner, but as the corner
deforms it undergoes inversion and the contact ceases in the
central part. The contact area is then, roughly speaking, bounded by two concentric circles, but the distribution is une-
ven.
The force-deformation curves given in Figure 3-4 relate to
total collision force for stern end - and stern corner impact ,
respectively. Table 3-1 and Table 3-2 give the anticipatedmaximum force intensities (local force and local contact areas,
i.e. subsets of the total force and total area) at various stages of
deformation.
The basis for the design curves is integrated, non-linear finite
element analysis of stern/column impacts.
The information given in 3.5.2 may be used to perform
strength design. If strength design is not achieved numerical
analyses have shown that the column is likely to undergo
severe deformations and absorb a major part of the strainenergy. In lieu of more accurate calculations (e.g. non-linear
FEM) it has to be assumed that the column absorbs all strain
energy.
Comm. 3.5.3 Energy dissipation is ship bow.
The requirements in this paragraph are based upon considera-
tions of the relative resistance of a tubular brace to local dent-
ing and the bow to penetration of a tubular beam. A
fundamental requirement for penetration of the brace into the bow is, first - the brace has sufficient resistance in bending,
second - the cross-section does not undergo substantial local
deformation. If the brace is subjected to local denting, i.e.
undergoes flattening, the contact area with the bow increasesand the bow inevitably gets increased resistance to indentation.
The provisions ensure that both requirements are complied
with.
Figure 8-8 indicates the level of the various contact locations.
Figure 8-4 shows the minimum thickness as a function of brace
diameter and resistance level in order to achieve sufficient
resistance to penetrate the ship bow without local denting. Itmay seem strange that the required thickness becomes smaller
for increasing diameter, but the brace strength, globally as well
as locally, decreases with decreasing diameter.
Local denting in the bending phase can be disregarded pro-
vided that the following relationship holds true:
Figure 8-5 shows brace thickness as a function of diameter and
length diameter ratio that results from Equation (8.1). The
thickness can generally be smaller than the values shown, and
still energy dissipation in the bow may be taken into account, but if Equation (8.1) is complied with denting does not need to
be further considered.
The requirements are based upon simulation with LS-DYNA
for penetration of a tube with diameter 1.0 m. Great cautionshould therefore be exercised in extrapolation to diameters
substantially larger than 1.0 m, because the resistance of the
bow will increase. For brace diameters smaller than 1.0 m, the
requirement is conservative.
4
2.01.51.00.50
8
12
Indentation [m]
Impact force [MN]
Between stringers (D= 0.75) m
On a stringer (D= 0.75 m)
Between a stringer (D= 1.0 m)
Recommended design curve for brace impact
Total collision force
distributed over this
Area with high force
intensity
Deformed stern corner
(8.1)2
2
1
D 10.14
t c D
⎛ ⎞≤ ⎜ ⎟⎝ ⎠
l
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Figure 8-4Required thickness versus grade and resistance level of brace topenetrate ship bow without local denting
Figure 8-5Brace thickness yielding little local deformation in the bendingphase
Comm. 3.7.3 Support capacity smaller than plastic bending moment of the beam
The procedure is illustrated in Figure 8-6.
Elastic, rotational flexibility of the node is normally of moder-ate significance.
Figure 8-6Derivation of force-deformation relationship for beam with endmoments less than beam plastic moment.
The procedure given is essentially the same as the one used in NORSOK N-004, but is formulated differently. The bendingmoment boundary condition is important in the bending phase,
but has no influence on the resistance in the pure membrane
phase. Between these extremities, simple linear interpolation isused.
Comm. 3.10.1 General
If the degradation of bending capacity of the beam cross-sec-tion after buckling is known the load-carrying capacity may beinterpolated from the curves with full bending capacity and no
bending capacity according to the expression:
= Collapse load with full bending contribution
= Collapse load with no bending contribution
Comm. 3.10.4 Tensile fracture in yield hinges
The rupture criterion is calculated using conventional beamtheory. A linear strain hardening model is adopted. For a can-tilever beam subjected to a concentrated load at the end, thestrain distribution along the beam can be determined from the
bending moment variation. In Figure 8-7 the strain varia-tion, , is shown for a circular cross-section for agiven hardening parameter. The extreme importance of strainhardening is evident; with no strain hardening the high strainsare very localised close to the support (x = 0), with strain hard-ening the plastic zone expands dramatically.
On the basis of the strain distribution the rotation in the plastic
zone and the corresponding lateral deformation can be deter-mined.
If the beam response is affected by development of membraneforces it is assumed that the membrane strain follows the samerelative distribution as the bending strain. By introducing thekinematic relationships for beam elongation, the maximummembrane strain can be calculated for a given displacement.
Figure 8-7Axial variation of maximum strain for a cantilever beam with cir-cular cross-section
Adding the bending strain and the membrane strain allowsdetermination of the critical displacement as a function of thetotal critical strain.
Figure 8-8 shows deformation at rupture for a fully clamped beam as a function of the axial flexibility factor c.
0
20
40
60
80
0,6 0,8 1 1,2 1,4
Diameter [m]
T
h i c k n e s s [ m m ]
fy = 235 MPa, 6 MN
fy = 235 MPa, 3 MN
fy = 355 MPa, 6 MN
fy = 355 MPa, 3 MN
fy = 420 MPa, 6 MN
fy = 420 MPa, 3 MN
0
20
40
60
80
100
0,6 0,8 1 1,2 1,4Diameter [m]
T h i c k n e s s [ m m ]
L/D =20
L/D =25
L/D =30
α = 1.5
0
1
2
3
4
5
6
7
0 1 2 3 4
Deformation w
R / R 0
R * / R 0
c =0.5
wlim
R/R 0
R*/R 0
*0 0R / R
(8.2)
= Plastic collapse load in bending with reducedcross-sectional capacities. This has to beupdated along with the degradation of cross-
sectional bending capacity.
)1)(()()(01
ξ ξ −+= == w Rw Rw R P M P M
)(1
w R P M =
)(0
w R P M =
)0(1
,
==
= w R
R
P M
red P M ξ
red,PMR
Ycr εεε =
0
5
10
15
20
25
30
35
40
45
50
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
x/
S t r a i n
Hardening parameter H = 0.005
Maximum strain
εcr /εY
= 50 = 40
= 20
No hardening
P
l
x
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Figure 8-8
Maximum deformation for a tubular fully clamped beam
(H=0.005)
The plastic stiffness factor H is determined from the stress-strain relationship for the material. The equivalent linear stiff-ness shall be determined such that the total area under thestress-strain curve up to the critical strain is preserved (The two
portions of the shaded area shall be equal), refer Figure 8-9. Itis un-conservative and not allowable to use a reduced effectiveyield stress and a plastic stiffness factor as illustrated in Figure8-10.
Figure 8-9
Determination of plastic stiffness
Figure 8-10
Erroneous determination of plastic stiffness
The accuracy of the calculation model for tensile fracture inyield hinges has been investigated by Amdahl and Skallerud(2002). The maximum strain as a function of lateral displace-ment (Equation (3.22)) for a tubular beam is compared with themaximum strain from finite element calculations in Figure8-11. The beam is assumed to be clamped and fixed againstinward axial displacement, l = 25 m, D = 1 m, t = 0.06 m, f y =
300 MPa, H = 0.00287 (i.e. ultimate stress f u = 390 MPa for atultimate strain εu = 0.15). The mesh size for USFOS shell andABAQUS is 0.25 ⋅ 0.39 m and for ABAQUS fine mesh0.05 ⋅ 0.195 m. The element used in ABAQUS analyses is theS4R reduced integration element .
Figure 8-11Strain versus displacement of clamped beam
It is observed that the strain estimated in ABAQUS analysisdepend significantly on the mesh size evidencing the need for a mesh-size-dependent fracture strain criterion. The NORSOK criterion agrees fairly well with FEM calculations when a fine
mesh is used. The criterion is conservative, as desired. Thestrain calculation in the USFOS beam element assumes a yield
plateau followed by parabolic type hardening. Only the fineABAQUS mesh captures the yield plateau effect.
Comm. 3.12 Energy dissipation in floating production vessels
Figure 8-12Design of an impact resistant stern – collision with a VLCC.
Calculation of energy dissipation in stringers, decks and bulk-heads subjected to gross, axial crushing shall be based uponrecognised methods for plastic analysis, e.g. Ch.7, /3/ andCh.7, /4/. It shall be documented that the folding mechanismsassumed yield a realistic representation of the true deformationfield.
The force deformation relationships given in Figure 3-6 may
be used to design a collision resistant stern of an FPSO. Inorder to be impact resistant, stringers and frames must be fairlyclosely spaced, typically in the range of 1.5 – 2 m. Given therelative dimensions of the girder system and the bulb cross-section, as illustrated in Figure 8-12, it is reasonable to applythe total collision force as uniformly distributed line loads onthe stringers and frames. The integrity of the stringers andframes can then be checked in a FEM analysis. Moderate localyielding should be accepted.
The stern structure must resist the collision force during allstages of the collision process. Normally, it suffices to analysea few collision force and contact area situations.
It is normally neither practical nor necessary to design the plat-ing and stiffeners such that their response is elastic. Large plas-tic deformations can be accepted, but fracture of the platingshould not occur (Note: provided that strength design is aimedfor). In lieu of more accurate calculations, the contact forcemay be considered uniformly distributed over the plate field,and the resistance may be assessed using the provisions givenfor the resistance of plates and stiffeners to explosion loads.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 20 40 60 80 100 120
εcr/εy
w / D
l/D = 30
c= 0
= 0.05 = 0.5
= 1000
l/D = 20
c = 0
= 0.05 = 0.5
= 1000
E
H E
εcr
f cr
E
H E
εcr
f cr
H E
f
ε
0%
5%
10%
15%
20%
0.0 0.5 1.0 1.5 2.0
Displacement [m]
S t r a i n
NORSOK
ABAQUS fine
USFOS beam
ABAQUS
USFOS shell
1 6 0 0
1 6 0 0
1600
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Even if the stiffeners are allowed to deform under extreme col-lision loads, they should be sufficiently robust to initiate crush-ing of the bulb. Engineering judgment must be applied, but itis recommended to design the stiffeners according to require-ments for ships navigating in ice; DNV Ice Class POLAR.
With respect to deformation resistance of stiffened plating, seenext paragraph.
The ductile resistance of stiffened plates may be analysed con-sidering the side as an assembly of plate/stiffeners. The resist-ance of individual stiffeners with associated plate flange can becalculated with the methods given in Section 6.3 using rela-tionships for a concentrated force, see example in Ch.8,Comm. 9.3. The resistance of the various stiffeners will bemobilised according to the geometry (raking) of the impacting
bow.
Unless the frame spacing is long or the stiffener height is small,fracture will take place before noticeable membrane stiffeninghas taken place. The initiation of fracture does not necessarily
imply that the resistance is totally lost, because fracture takes place in the top flange while the strain on the plate side is con-siderably smaller .
The above procedure neglects the effect of membrane forcestransverse to the stiffeners. Depending on the geometry of the
panel this contribution may be substantial.
Collisions with FPSOs have been studied in-depth in a paper
by Moan et.al. (2002). Force-deformation relationships aregiven for supply vessels/merchant vessels, 18.000 tons chemi-cal tanker and a 42.000 tons tanker and a shuttle tanker. Thecollision risk for all categories of vessels is discussed exten-sively. The consequences of a collision with a shuttle tanker servicing the FPSO are especially considered.
Figure 8-13 shows the force-deformation relationship for sup- ply vessel/merchant vessel colliding with the side of an FPSO.It is interesting to see that the force level for bow without bulbis smaller than the bow force-deformation curve given in Fig-ure 3-4.
Figure 8-13Force-deformation relationship for supply vessel/merchant vessel impact against FPSO side
Comm. 4.4.1 Stiffened plates subject to drill collar impact
The validity for the energy equation 4.6 is limited to7 < 2 r/d < 41, t/d < 0.22 and mi/m < 0.75.
The formula neglect the local energy dissipation which can beadded as Eloc = R·0.2 t.
In case of hit near the plate edges the energy dissipation will below and may lead to unreasonable plate thickness. The failurecriterion used for the formula is locking of the plate. In manycases locking may be acceptable as long as the falling object isstopped. If the design is based on a hit in the central part of a
plate with use of the smaller diameter in the treaded part in thecalculations, no penetration of the drill collar will take place atany other hit location due to the collar of such dropped objects.
Comm. 5.1 General
For redundant structures thermal expansion may cause buck-ling of members below 400°C. Forces due to thermal expan-sion are, however, purely internal and will be released once themember buckles. The net effect of thermal expansion is there-
fore often to create lateral distortions in heated members. Inmost cases these lateral distortions will have a moderate influ-ence on the ultimate strength of the system.
As thermal expansion is the source of considerable inconven-ience in conjunction with numerical analysis it would tempting
to replace its effect by equivalent, initial lateral member distor-tions. There is however, not sufficient information to supportsuch a procedure at present.
Comm. 5.5 Empirical correction factor
In Ch.7 /2/ Eurocode 3 an empirical reduction factor of 1.2 is
applied in order to obtain better fit between test results and col-umn curve c for fire exposed compressive members. In thedesign check this is performed by multiplying the design axialload by 1.2. In non-linear analysis such a procedure is imprac-tical. In non-linear space frame, stress resultants based analysisthe correction factor can be included by dividing the yieldcompressive load and the Euler buckling load by a factor of 1.2. (The influence of axial force on member’s stiffness isaccounted for by the so-called Livesly’s stability multipliers,which are functions of the Euler buckling load.) In this way thereduction factor is applied consistently to both elastic andelasto-plastic buckling.
The above correction factor comes in addition to the reductioncaused by yield stress and elastic modulus degradation at ele-
vated temperature if the reduced slenderness is larger than 0.2.Comm. 6.2 Classification of response
Equation (6.2) is derived using the principle of conservation of momentum to determine the kinetic energy of the componentat the end of the explosion pulse. The entire kinetic energy is
0
5
10
15
20
25
30
0 1 2 3Bow Displacement [m]
E n e r g y [ M J ]
0
5
10
15
20
25
30
F o r c e [ M N ]
Energy superstr.
Energy bulb
Total force
Force superstr.
Force bulb
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then assumed dissipated as strain energy.
Equation (6.3) is based on the assumption that the explosion pressure has remained at its peak value during the entire defor-mation and equates the external work with the total strainenergy. In general, the explosion pressure is not balanced byresistance, giving rise to inertia forces. Eventually, these iner-tia forces will be dissipated as strain energy.
Equation (6.4) is based on the assumption that the pressureincreases slowly so that the static condition (pressure balanced
by resistance) applies during the entire deformation.
Comm. 6.4 SDOF system analogy
The displacement at the end of the initial, linear resistancedomain yel will generally not coincide with the displacement atfirst yield. Typically, yel represents the displacement at the ini-tiation of a plastic collapse mechanism. Hence, yel is larger than the displacement at first yield for two reasons:
i) Change from elastic to plastic stress distribution over beam cross-section
ii) Bending moment redistribution over the beam (redundant beams) as plastic hinges form
Figure 8-14Iso-damage curve for ymax/yel = 10. Triangular pressure
Figure 8-14 is derived from the dynamic response chart for aSDOF system subjected to a triangular load with zero rise timegiven in Figure 6-3.
In the example it is assumed that from ductility considerationsfor the assumed mode of deformation a maximum displace-ment of ten times elastic limit is acceptable. Hence the line
represents the upper limit for the
displacement of the component. From the diagram it is seenthat several combinations of pulses characterised by Fmax andtd may produce this displacement limit. Each intersection witha response curve (e.g. k 3 = 0) yields a normalized pressure
and a normalised impulse
By plotting corresponding values of normalised impulse andnormalised pressure the iso-damage curve given in Figure 8-14is obtained.
If the displacement shape function changes as a non-linear structure undergoes deformation the transformation factorschange. In lieu of accurate analysis an average value of thecombined load-mass transformation factor can be used:.
μ = ymax
/yel
ductility ratio
Since μ is not known a priori iterative calculations may be nec-essary.
Dynamic response charts for a SDOF system with triangular pressure pulses with rise time different from td/2 are given inFigure 8-15 to Figure 8-17.
0
1
2
3
4
5
6
7
8
9
10
11
0 1 2 3 4 5 6 7 8 9 10 11
Impulse I/(RT)
P
r e s s u r e
F / R
Pressure asymptote
I m p u l s i v e a s y m p t o t
Iso-damage curve for ymax/yelastic = 10
Elastic-perfectly plastic resistance
(8.3)
10
y
y
y el
max
el
allow ==
el
max
R
F
R
F=
T
t
FR 2
1
TR
tF2
1
RT
I d
max
elel
dmax
⋅==
( )μ
μ plastic
m
elastic
maverage
m
k k k ll
l
1−+=
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Figure 8-15Dynamic response of a SDOF system to a triangular load (rise time=0)
Figure 8-16Dynamic response of a SDOF system to a triangular load (rise time = 0.15td)
0.1
1
10
100
0.1 1 10
td/T
y m a x / y e l
=0.1 = 0.7= 0.6= 0.5Rel/Fmax=0.05 = 0.3
= 1.1
= 1.0
= 0.9
Rel/Fmax= 0.8
= 1.2
= 1.5
yel y
R
Rel
F
Fmax
td
k1
k3 = 0.5k1 =0.2k1 =0.1k1k3 = 0
k3 = 0.1k1
k3 = 0.2k1
k3 = 0.5k1
0.1
1
10
100
0.1 1 10
td/T
y m a x / y e l
=0.1 = 0.7= 0.6= 0.5Rel/Fmax=0.05 = 0.3
= 1.1
= 1.0
= 0.9
Rel/Fmax= 0.8
= 1.2
= 1.5
yel y
R
Rel
F
Fmax
td0.15td
k1
k3 = 0.5k1 =0.2k1 =0.1k1k3 = 0
k3 = 0.1k1
k3 = 0.2k1
k3 = 0.5k1
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Figure 8-17Dynamic response of a SDOF system to a triangular load (rise time = 0.30t d)
Comm.6.7.1.1 Component behaviour
For beams the characteristic linear stiffness given for theelasto-plastic resistance domain in Table 6-2 is derived fromthe equal area principle on the assumption that the supportmoment is equal to the plastic bending moment of the beam.
Comm. 6.7.1.1 Component behaviour
For deformations in the elastic range the effective width (shear lag effect) of the plate flange, se, of simply supported or clamped stiffeners/girders may be taken from Figure 8-18.
Figure 8-18
Effective flange for stiffeners and girders in the elastic range
Comm. 6.10.7 Ductility limits
The table is taken from Ch.7, Reference /4/. The values are based upon a limiting strain, elasto-plastic material and cross-sectional shape factor 1.12 for beams and 1.5 for plates. Strainhardening and any membrane effect will increase the effectiveductility ratio. The values are likely to be conservative.
9. Examples
9.1 Design against ship collisions
9.1.1 Jacket subjected to supply vessel impact
The location of contact is at brace mid-span and the force acts parallel to global x-axis. The brace dimensions are 762 x 28.6
mm. From linear elastic analysis it is found that the stiffness of nodes 508 and 628 against displacement in the brace directionis 736 MN/m and 51 MN/m respectively, when the brace isremoved. The unequal stiffness may be represented by twoequal springs, each with stiffness:
0.1
1
10
100
0.1 1 10
td/T
y m a x / y e l
=0.1 = 0.7= 0.6= 0.5Rel/Fmax=0.05 = 0.3
= 1.1= 1.0
= 0.9
Rel/Fmax= 0.
= 1.2= 1.5
yel y
R
Rel
F
Fmax
td0.30td
k1
k3 = 0.5k1 =0.2k1 =0.1k1
k3 = 0
k3 = 0.1k1
k3 = 0.2k1
k3 = 0.5k1
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8
/s
s e
/ s
n > 6
n = 5n = 4
n = 3
Uniform distribution or
nF
= L
nF
= 0.6L
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Figure 9-1Jacket subjected to ship impact
The axial stiffness of the brace is given by
and is large compared to the stiffness of the node. This yieldsan effective stiffness of
Assuming clamped ends (c1 = 2) the non-dimensional springstiffness comes out to be
The resulting end restraint is quite flexible. This is particularlydue to low stiffness in node 628, in spite of the support by the
adjacent braces. Hence, the build-up of tension force will bedelayed compared to a full axial fixity.
The collapse load in bending is calculated assuming clampedconditions at both ends. This is a good approximation at thelower end but slightly optimistic at the upper end.
The load-deformation characteristics for the brace are obtained by interpolation of the curves given in Figure 3-7. The result isdepicted in Figure 9-2. The response predicted by means of thenonlinear analysis program USFOS is also plotted. It appearsthat the simplified approach performs very well when axialflexibility is taken into account. The loss of stiffness predicted
by USFOS at large displacements is due to initiation of failureof adjacent members at node 628. Collapse of these memberstakes place at a load level of 2.8 MN.
It must also be verified that the capacity of the joints is suffi-cient to support the force state in the brace both in the bendingmode of deformation and in the membrane tension state. Fig-ure 9-3 displays the simulated bending moment-axial forceinteraction history in the brace and shows that the membraneforce becomes substantial, but doe not attain the fully plasticaxial force. In lieu of accurate calculations, it should be assumethat the fully plastic tension is developed.
Provided that the joints and adjacent structure are capable of supporting the brace ends, the energy dissipation is limited byfracture due to excessive straining of the brace. Fracture crite-ria are given Section 3.10.3. Using the fracture criterion in Sec-tion 3.10.3 there is obtained wcrit = 2.2 m and a correspondingenergy dissipation E = 6 MJ.
Figure 9-2Load versus lateral deformation of the contact point
Figure 9-3Axial force-bending moment interaction in brace
Tensile fracture in jacket brace
Tensile fracture of the brace considered in is estimated. Thecharacteristic dimension is, dc = D = 0.762 m. For steel gradeS 355 a strain hardening coefficient of H = 0.0034 is used, refer Table 3-3. c1 = 2 (clamped ends are assumed), the collision
occurs at mid span, hence κ = 0.5, and κ /dc = 15.3. The non-dimensional spring stiffness is c = 0.18 and W/WP = π /4. Thisyields wcrit = 2.2 m.
Because of the large κ /dc – ratio, the brace is capable of deforming almost three times its diameter.
628
508
762 x 28.6 mm
l= 23.3 m
m MN K node /9551
1
736
12
1
=⎟ ⎠
⎞⎜⎝
⎛ +=
−
m MN EA
/12343.23
0286.0762.0101.2225
=⋅⋅⋅⋅⋅
= π
l
MN/m881234
1
95
11=+=
K
18.03.230286.0355
762.08822
Af
Kw4cc
y
2
c1 ≅⋅⋅⋅
⋅⋅===
π π ll t f
Kd
y
MN9.13.23
0286.0)0286.0762.0(355244R
21
0 =⋅−⋅⋅⋅
==l
P M c
0
2
4
6
8
10
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Displacement [m]
I m p a c t f o r c
e [ M N ]
0
2
4
6
8
10
E n e r g y d i s s i p a t i o n [ M J ]
USFOS
Simple model
Energy dissipation
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0 Normalised moment M/MP
N o r m a l i s e d f o r c e N / N P
l
l
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9.2 Design against explosions
9.2.1 Geometry
The geometry of the structure is outlined in Figure 9-4. The plate, stiffeners and girders will be assessed. The main dimen-sions are:
t = 10 mm
s = 500 mml = 2000 mm
Stiffener dimension Hp 180
Figure 9-4Geometry
9.2.2 Calculation of dynamic response of plate:
The dynamic response of the plate considered in Section 9.3.1is studied. The plate is subjected to a triangular pressure pulsewith duration of 20 msecs. The peak pressure is f max = 2.5MPa. Assume that the resistance curve for c = 1.0 in Figure 9-7 applies. This yields r el/f max = 0.3. The curve is redrawn
below along with approximate relationships
Alternative 1- static analysis: The eigenperiod of the plateaccording to Section 9.3.1 with η = 25 is T = 4.0 msecs. Hencetd/T = 5. This is a fairly long duration and static behaviour can
be assumed. The maximum deflection is determined directlyfrom Figure 9-7, i.e. wmax = 27 mm.
Alternative 2 - tri-linear resistance: By inspection of thedynamic response charts and the resistance curve for the plateit is noticed that none of the tri-linear curves apply very well.The best fit is obtained with k 3 = 0.5 k 1, but this underestimatesthe resistance for large deformations. From the response chartfor td/T = 5 there is read ymax/yel ~ 4.8. This yields wmax = 4.8· 6.15 = 30 mm.
Alternative 3 – equivalent linear resistance: For large defor-mations the stiffness is fairly linear. Assume that the averagestiffness is linear and equal to 65 % of the elastic stiffness, i.e.k = 0.65 · 123 = 80 MPa/m. In this case the r el can be set arbi-trarily, but it should be ensured that the response is such thatymax/yel < 1.0, and it is practical to select a given r el/f max ratiofor which a response curve is provided. Hence assume r el/f max= 1.5, which gives r el = 47.3 mm and then it follows r. The
eigenperiod is adjusted by toaccount for less stiffness. This yields td/Tmod = 4.0. From theresponse chart there is obtained ymax/yel ~ 0.7. This yieldswmax = 0.7 · 47,3 = 33 mm.
All these methods yield approximately the same result. Thestatic approach is quite good, but there is a slight dynamicamplification > 1 in the present case.
The plate must be checked with respect to rupture, see Section9.4.1.
It is noticed that if no membrane force can be taken intoaccount, i.e. c = 0, then ymax/yel >> 100 and the plate will failcompletely.
9.2.3 Calculation of dynamic response of stiffened plate
The dynamic response of the stiffened plate considered in Sec-tion 9.3.2, subjected to a triangular explosion pulse with dura-tion 20 msecs and peak pressure f max = 2.5 MPa, is studied.The collapse resistance is R 0 = R el = 0.58 MN, and no mem-
brane stiffening can be assumed, i.e. k 3 = 0. As the plate/stiff-ener undergoes a phase with elasto-plastic bending, theresistance is approximated by a linear elastic-perfectly plastic
model, with equivalent stiffness of 208 MN/m and wel = 2.8mm. The critical deformation at rupture wcrit = 36 mm, hencethe ductility ratio is μ = ymax/yel = 36/2.8 = 13.
The total mass is 108 kg. The load-mass factor is ~ 0.77 and0.66 in the elastic/elasto-plastic and plastic bending phase,respectively. Using Equation (8.3) the average load-mass fac-tor becomes and theeigenperiod is:
This gives td/T = 5.4. By inspection of Figure 6-3 it is found
that for μ = 13 and td/T =5.4 → R el/Fmax ≅ 0.75 (in other words, because of limited pulse duration it is possible to “over-load” the stiffener by 33% compared to the static collapseresistance in bending).
The maximum peak pressure the stiffener can resist is:
Consequently; the stiffener is not strong enough to resist theexplosion pressure without rupture (see discussion in Section9.3.2 as concerns rupture of stiffener).
It is a fairly common experience that stiffeners are more likelyto be critical with respect to explosion loads than the plating
between stiffeners.
9.3 Resistance curves and transformation factors
9.3.1 Plates.
Generation of elastic–plastic resistance curve is illustrated for a plate with the following particulars: Length, l = 2 m, width,s = 0.5 m, thickness, t = 10 mm, yield stress f y = 355 MPa. Itis assumed that the plate is a part of a continuous plate field.Large deformations are expected so that the plate will yieldalong the boundaries. Then clamped boundaries are assumed.
The rigid – plastic curve is given by Equation (8.3). The col-lapse resistance in bending is r c = 0.76 MPa. The resistancecurve for fully fixed boundaries are indicated by the line “Platec = inf” in Figure 9-6. Below, the curve will be adjusted for the
effect of in-plane flexibility using the procedure described inSection 6.8.2.
First, the resistance of a plate-strip is calculated, using infor-mation given in Section 6.9.2 with α = 2 (rectangular cross-section). Clamped boundaries with c1 = 2 are assumed also for
t = 10
Stiffener Hp180
Girder
0
1
2
3
4
5
0 10 20 30 40 50
Deformation [mm ]
R e s i s t a n c e [ M p a ]
Plate c = 1.0
Tri-linear
Eq. linear Static
msecs0.565.01mod
== T T
( )( )0.77 13 1 0.66 /13 0.67averagelmk = + − =
smk
M k T
averagelm
sec7.321
== π
.MPa8.075.0
1==
l s
R f el
crit
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the plate strip. The collapse resistance in bending for the platestrip is r c = 0.57 MPa.
The characteristic beam height is.
The resistance curve for the plate strip is shown in Figure 9.6for fully fixed boundaries , and for two values of the non-dimensional spring stiffness, c = 1.0 and c = 0.3. It is observedthat the difference between the plate strip and the plate solutionis small for the present fairly large aspect ration, notably whenthe membrane effect predominates
On the assumption that the plate experiences the same relativereduction of the resistance due to axial flexibility as does the
plate strip, resistance curves for the plate with non-dimen-sional spring stiffness, c = 1.0, and c = 0.3 can be generated asshown in Figure 9-6.
The next step is to assess the flexibility factor c:
If the flexibility of the adjacent structure is neglected, account-ing only for the 2nd term in Equation (6.11), there is obtained
This yields a non-dimensional spring stiffness, c = 0.95.
Figure 9-5Approximate determination of flexibility by means of membraneanalysis
In order to assess the influence of the flexibility of the adjacentstructure, a membrane analysis is performed with the plateremoved, see Figure 9-5. A constant stress of 100 MPa isapplied perpendicular the boundaries. The maximum deforma-tion obtained, at the mid-point of the long edges, is 0.25 mm.This yields an equivalent stiffness of k node = 100·0.010·1/
0.25·10-3
= 4000 MN/m. When both effects are accounted for,the resulting stiffness becomes k = (1/8400 +1/4000)-1 =2710 MN/m and c = 0.31. Hence, the plate resistance may beassessed reasonably well by means of the curves for either c =1.0 or c = 0.3.
Finally, the linear elastic solution up to the collapse resistancein bending, r c, is added to the rigid-plastic solution. Using theinformation given in Section 6.9.1, ψ = 400, and k 1 = 123 MPa/m. The deformation corresponding to r = r c is wel = 6.15 mm.The resulting resistance curves are shown in Figure 9.7.
Figure 9-6Derivation of rigid-plastic resistance curves for a plate
Figure 9-7Elastic-plastic resistance for a plate with various degrees of axialflexibility.
9.3.2 Calculation of resistance curve for stiffened plate
The plate considered in Section 9.3.1 is stiffened with HP 180x8 stiffeners with yield stress f y = 355 MPa. The girder spacingis 2.0 m. It is assumed that the stiffener is continuous, so thatyield hinges can form at the connections to the girder, hence c1= 2. The area of the stiffener As= 1.88·10-2 m2 and the distanceto the centroid is zg = 0.109 m.
From Figure 8-18 it is found that the plate flange is approxi-mately 80% for a uniformly distributed load when
/s = 0.6⋅2.0/0.5 = 2.4. The effective area of the plate flangeis 0.8 s t = 4·10-3 m2 > As. Hence, it may be assumed that the
plastic neutral axis for the effective section lies at the stiffener web toe. This yields the plastic section modulus WP = As zg =
2.05·10-3 m3 and collapse resistance in bending
The characteristic beam height is.
The moment of inertia for stiffener with effective plate flangeis I = 2.28 10-5 m4. The initial elastic stiffness is taken fromTable 6-2:
This yields a lateral “elastic” deformation of wel = 2.5 mm for R = R 0.
The resistance curve for the stiffener with associated plateflange is shown in Figure 9.8 for various degrees of axial flex-ibility (Note elastic part not included!).
21
412 2 t
t
t
A
W w P
c =
⋅⋅
== α
∞=c
m MN s
Et EA
k /8400
122
=
⋅
== l
Inwarddisplacement
Uniform stress field applied along
boundary of removed plate
0
1
2
3
4
5
0 10 20 30 40 50
Deformation [mm]
R e
s i s t a n c e [ M p a ]
Plate c = inf
Plate c = 1.0
Plate c = 0.3
Strip c = inf
Strip c = 1.0Strip c = 0.3
0
1
2
3
4
5
0 10 20 30 40 50
Deformation [mm]
R e s i s t a n c e [ M p a ]
Plate c = inf
Plate c = 1.0
Plate c = 0.3
l
MN58.08 1
0 ==l
P yW f c R
wc
α W P
A------------ α z g 1.2 0.109⋅ 0.13m= = = =
MN/m230L
384EIk
3 ==
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For uniformly loaded, clamped beams there will be an elasto- plastic bending phase between the occurrence of first plastichinge and final formation of final collapse mechanism. Toaccount for this effect, the initial stiffness may be modified onthe basis of equal area principle. The equivalent elastic stiff-ness is obtained from Table 6-2 with m1 = 1:
and wel = 3.2 mm for R = R 0.
It is noticed that the stiffener must undergo a substantial plasticdeformation before membrane strengthening becomes signifi-cant according to the present model. Whether this is achievabledepends on the ductility of the stiffener, refer Section 9.4.2.
Recent investigations indicate that the model adopted for stiff-ened plate is considerably conservative, which may warrant amore accurate nonlinear finite element analysis if the stiffener response becomes critical.
Figure 9-8Resistance curve for stiffener with associated plate flange.
9.3.3 Calculation of resistance curve for girder
What is the maximum pressure a steel girder can resist prior torupture, when the explosion load is triangular, with equal riseand decay time, and the duration is 0.33 s?
The girder has the following dimensions:
Length L = 12 m, web height, hw = 1.5 m, web thickness, tw =13 mm, top flange breadth, btop = 0.45 m, top flange thicknessttop = 19 mm. The girder spacing is 2 m and the plate thicknessis 10 mm. For simplicity it is assumed that the plate flange isfully effective. The girder has a distributed load of intensity 10kN/m2 and mounted equipment with mass 1.8·105 kg. Theequipment load acts equally at two points located L/3 frommember ends. The girder is simply supported at one end andclamped at the other end. At the clamped end fully plastic
bending moment of the girder can be assumed. There is noaxial restraint. Yield stress f y = 355 MPa, acceleration of grav-ity g = 10 m/s2, density of steel 7.86⋅103 kg/m3.
The following is obtained for the girder:
Moment of inertia I = 1.84⋅10-2 m4, elastic section modulus, W= 1.96⋅10-2 m3, plastic section modulus, WP = 2.51⋅10-2 m3,total cross-sectional area 0.048 m2. The total distributed mass,including mass of girder is 0.29⋅10-5 kg, so the concentratedmass predominates. Hence, transformation factors for two con-
centrated loads in Table 6.2 are used.
The equivalent stiffness in the elasto-plastic range (m3 = 1) is.
The plastic bending resistance is
and wel* = 21.8 mm. However, the functional loads amount to1.8 + 0.29 = 2.09 MN (including steel weight), so 21.8·2.09/5.95 = 7.6 mm is already utilised and only R el = 5.95-2.09 =3.86 MN and wel = 14.1 mm is available in the equivalent elas-
tic range. The limiting deformation for rupture calculated in9.4.3 is wmax = 95 mm, yielding ductility ratio μ = w/max / wel= 95/14.1 = 6.7.
When calculating the load-mass factor the change in transfor-mation factor from the elastic to plastic regime may beaccounted for, see Ch.8, Comm. 6.4. The factor for distributedmass and concentrated mass is
k lmaverage,u = (0.55 + (6.7 − 1) ⋅ 0.56) / 6.7 = 0.56
and
k lmaverage,c = (0.83 + (6.7 − 1) ⋅ 1.0) / 6.7 = 0.975,
respectively. The eigenperiod becomes
and hence td/T= 0.33/0.166 ~ 2. From Figure 6-3 there is readR el/Fmax = 0.7 for coordinates (2,6.7). Hence, the girder canresist a dynamic load of Fmax = 3.86/0.7 = 5.5 MN, corre-sponding to a peak pressure of f max= 0.23 MPa.
Example girder:
The neutral axis for the girder studied in Section 9.3.3 islocated 0.315 m from the plate flange. This yields a character-istic dimension dc = 2 ⋅ (1.5 − 0.315) = 2.37 m. The criticallocation is at the clamped side, whereby κ =1/3. Clamped endyields c1 = 2 for the fracture check. With H = 0.0034 and c =0, there is obtained w/dc = 0.069 and w = 0.095 m.
9.4 Ductility limits
9.4.1 Plating
Rupture of the plating for the example considered in Section9.2.2 may be estimated by means of the procedure given inSection 3.10.4, using the plate strip analogy. The characteristicdimension is, dc = t = 10 mm. For steel grade S 355 a strainhardening coefficient of H = 0.0034 is used, refer Table 3-4. κ= 0.5, c1 = 2 (clamped ends) and κ /dc = 0.5 s/t = 25. Thisyields the following values for the critical deformation, wcrit,depending on the spring stiffness c, see Table 9.1 (Note: theelastic deformation r el = 6.15 mm is added to the valuesobtained). By inspection of Figure 9-7 it is noticed that thefully plastic membrane state according to this procedure is
attained in all cases but c = 0.
9.4.2 Stiffener:
Rupture is calculated for the stiffened plate considered in sec-tion 9.2.3 using the procedure given in Section 3.10.4. Thesteel grade is S 355 with a strain hardening coefficient of H =0.0034, refer Table 3-3. Clamped conditions are assumed, i.e.c1 = 2. The shape factor (somewhat arbitrarily) set to 1.5. Thecharacteristic dimension of the stiffened plate is dc = 2hw =0.36 m. This yields λ/dc = 5.56, only. This critical deformation
becomes wcrit
= 0.1dc = 36 mm, almost independent of the
spring stiffness c (Note: ductility ratio is μ = 36/2.2 = 16). Thisfairly small value is due to the low κλ/dc – ratio for the stiff-ener. The stiffener is far from entering the membrane stiffen-ing phase, so that any discussion of the possibility for membrane forces to develop is irrelevant.
MN/m184L
307EIk
3
==
α = 1.2
0.0
0.5
1.0
1.5
2.0
0 0.1 0.2 0.3 0.4 0.5
Deformation w [m]
R [ M N ]
c = inf
c = 1.0
c = 0.5
c = 0.2
c = 0.1
MN/m274L
122EIk
3 ==
Table 9-1 Ductility limit as a function of the spring stiffness
c ∞ 1.0 0.3 0
wcrit [mm] 35 51 59 76
*
85.95 MN Pm
el
M R
L= =
T 2p0.56 2.9 10
4⋅⋅ 0.975 1.8 10
5⋅ ⋅+
274 106
⋅----------------------------------------------------------------------------------- 0.166s= =
l
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If the stiffener is free against rotation and/or has a longer spanmembrane effects may become important prior to rupture.
Observe that rupture is calculated for the location subjected tothe largest strains, i.e. at the stiffener top flange. Rupture in thetop flange is not necessarily critical with respect to intactnessto explosion loads, because the plate side experiences far lessstrains. It is likely that the plate will remain intact beyond the
deformation limit corresponding to rupture in the top flange. Asignificant part of the contribution to resistance from the stiff-ener is lost, but the plating between girders may have a signif-icant residual resistance after failure of stiffeners provided thatthe plate does not disintegrate. It is, however, difficult to pro-vide validated, closed form solution for this situation.
A stiffener subjected to pressure on the plate side may tripabout the weld toe at mid span. In this case the assumptionsused in the strain calculation model are no longer valid.
9.4.3 Girder:
The neutral axis for the girder studied in Ch.8, Comm. 6.10 islocated 0.315 m from the plate flange. This yields a character-istic dimension dc = 2 ⋅ (1.5 − 0.315) = 2.37 m. The critical
location at the clamped side, whereby κ =1/3. Clamped endyields c1 = 2 for the fracture check. With H = 0.0034 and c =0, there is obtained w/dc = 0.069 and w = 0.095 m.
9.5 Design against explosions - girder
9.5.1 Geometry, material and loads
The geometry of the structure is outlined in Figure 9-4. Themain dimensions are:
Plate thickness: t = 14 mmStiffener dimension: HP240x10, simulated as an L-profile
with dimension L240x39x10x29Stiffener spacing: s = 800 mmStiffener length: l = 3200 mmGirder dimension: T-girder with dimension: 870x300x10
x20Girder length: L = 12000 mm
The material properties are as follow:
Permanent loads and live loads are as follow:
Figure 9-9Geometry
Yield strength: f y = 420 MPa
Strain rate factor: γε = 1.0
Effective yield strength: f y = f y· γε = 420 MPa
Modulus of elasticity: E = 2.1·105 MPa
Material density: ρ = 7850 kg/m3
Poisson’s ratio: ν = 0.3
Max. plastic strain: 1.0% (maximum allowable, corre-spond to cross section class 3 or 4, seesub-section 9.5.2)
Permanent loads: pP = 10.0 kN/m2
Live loads: pL = 5.0 kN/m2
Explosion pulse period:
td = 0.15 sec (triangular loadwith a rise time =0.50·td)
t = 14
Bulkhead
Girder:
800 (typ.)
10
Stiffener: Hp240
Girder: TG870x300x10x20
Bulkhead
12000
Stiffener:
870
300
20
10240
39
29
3200(typ.)
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9.5.2 Cross sectional of properties for the girder
Effective plate flange according to DNV Classification Note30.1 (July 1995), sub-section 3.4.3 and 3.5.4:
Determination of cross section class, Ref. NS3472:2001, Sec- tion 12.1:
In the following calculations, a plate flange width larger than cross sectional class 3 will not be considered, i.e.:
Gross sectional properties:
Effective area of plate flange: A p = l e·t = 303.2·14 = 4245.1 mm2
Area of girder flange: Af = bfg·t fg = 300·20 = 6000.0 mm2
Total area of girder web: Aw = hwg·t wg = 850·10 = 8500.0 mm2
Total area (gross section): AG = A p+ Af + Aw = 4245.1+6000+8500 = 18745.1 mm2
Distance to neutral axis (from bottom of girder flange):
Web height in tension:ht = z 0-t fg = 403.6-20.0 = 383.6mm
Web height in compression:hc = hwg-ht = 850.0-383.6 = 466.4mm
56.251.2
420
14
800=⋅=⋅=
E E
f
t
s y β
58.056.2
8.0
56.2
8.18.08.122
=−=−= β β
xC
mm s
l C sl xe 6.784
56.2
111
800
32001.058.0800
1111.0
2
2
2
2 =
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟ ⎠
⎞⎜⎝
⎛ +⋅⎟ ⎠
⎞⎜⎝
⎛ −⋅+⋅=⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟ ⎠
⎞⎜⎜⎝
⎛ +⋅⎟
⎠
⎞⎜⎝
⎛ −⋅+⋅= β
Web: , i.e. class 3 (bending considered)
Bottom Flange: , i.e. class 2 (bending & axial)
Plate Flange: , i.e. class 4 (bending & axial)
6.113420/235
)10/850(
/235
)/(==
y
wg wg
f
t h
7.9420/235
)20
)10300(5.0(
/235
))(5.0
(
=−⋅
=
−⋅
y
fg
wg fg
f
t
t b
0.37420/235
)14
)106.784(5.0(
/235
))(5.0
(
=
−⋅
=
−⋅
y
wg e
f
t
t l
mmt f t l wg ye 2.30310)420/2351414(2)/23514(2 =+⋅⋅⋅=+⋅⋅⋅=
mm A
t ht
At h
At
A
z G
fg wg p fg
wg
w
fg
f
6.403222
0 =
⎟ ⎠
⎞⎜⎝
⎛ ++⋅+⎟⎟ ⎠
⎞⎜⎜⎝
⎛ +⋅+⋅
=hwg = 870-20
= 850
t wg = 10
l e = 303.2
b fg = 300
t = 14
z 0
t fg = 20
hc
ht
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Elastic moment of inertia (gross section):
Effective girder web according to NS3472:
Elastic buckling stress
Web slenderness:
Effective compression web height, see Figure 9-10:
Figure 9-10Effective Girder Section
Effective girder cross section properties
Reduction in web height:Δh = hc -hce = 466.4 – 430.8 = 35.6 mm
Effective cross section area: Ae = AG -Δh ·t wg = 18745.1 – 35.6·10.0 = 18389.1 mm2
Distance to neutral axis from bottom of girder flange:
Effective elastic moment of inertia:
( ) 492
0
222
222 10407.222212
1mm z At h
t At
h A
t At Ah At A I G fg wg p fg
wg
w
fg
f pwg w fg f G ⋅=⋅−⎟ ⎠
⎞⎜⎝
⎛ ++⋅+⎟⎟ ⎠
⎞⎜⎜⎝
⎛ +⋅+⎟⎟
⎠
⎞⎜⎜⎝
⎛ ⋅+⋅+⋅+⋅⋅=
( ) ( ) MPa
h
t E f wg
wg
e 9.627850
10
3.0112
101.29.23112
9.23
2
2
52
2
2
2
=⎟ ⎠ ⎞⎜
⎝ ⎛ ⋅
−⋅⋅⋅⋅=⎟
⎟ ⎠ ⎞
⎜⎜⎝ ⎛ ⋅
−⋅⋅⋅= π
ν
π
818.09.627
0.420===
e
y
p f
f λ
⎪⎩
⎪
⎨
⎧
>⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟
⎠
⎞⎜⎜⎝
⎛
⋅−⋅
≤
= 724.05
11
724.0
p
p p
c
pc
ce if h
if h
h λ λ λ
λ
mmhce 8.430818.05
1
1818.0
2.341
=⎥⎦
⎤
⎢⎣
⎡
⎟ ⎠
⎞
⎜⎝
⎛
⋅−⋅=
½ hce
hwg = 870-20
= 850
t wg = 10
l e = 303.2
b fg = 300
t = 14
e
z 0
hc
ht
½ hce
t fg = 20
Δh
ht
mm A
t hhh
t h z A
z e
fg t
ce
wg G
e 1.3991.18389
206.3832
8.4306.35106.356.4031.18745
20
0 =⎟ ⎠
⎞⎜⎝
⎛ +++
⋅−⋅=
⎟ ⎠
⎞⎜⎝
⎛ ++
+Δ⋅Δ−⋅
=
2
03
2121 ⎟
⎠ ⎞⎜
⎝ ⎛ −++⋅⋅Δ−⋅Δ⋅−= e
ct fg wg wg GGe z hht t ht h I I
49
2
39 10387.21.3992
4.4666.38320106.35106.35
12
110407.2 mm I Ge ⋅=⎟
⎠
⎞⎜⎝
⎛ −++⋅⋅−⋅⋅−⋅=
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Effective elastic section modulus:
Plastic section modulus:
Plastic section modulus if A p > Aw1 + Aw2 + Af :
Plastic section modulus if A p + Aw1 > Aw2 + Af :
Plastic section modulus if A p + Aw1 < Aw2 + Af :
369
0
10923.41.3991485020
10387.2mm
z t ht
I W
ewg fg
Ge
eo ⋅=−++
⋅=
−++=
369
0
10982.51.399
10387.2mm
z
I W
e
Ge
eu ⋅=⋅
==
3610923.4),min( mmW W W eueoe ⋅==
Web areas:
Eccentricities (see figure):
2
1 0.2154102
8.430
2mmt
h A wg
ce
w =⋅=⋅=
½ hce
Aw1½ hce
h t
e1
e3
Aw2
2
2 0.5990106.3832
8.430
2 mmt h
h
A wg t
ce
w =⋅⎟ ⎠
⎞
⎜⎝
⎛
+=⋅⎟ ⎠
⎞
⎜⎝
⎛
+=
mmt
A A A Ae
wg
pww f 9.494
102
1.42450.59900.21546000
2
21
1 =⋅
−++=
⋅
−++=
mmt
A A A Ae
wg
pww f 5.279
102
1.42450.59900.21456000
2
21
3 =⋅
−+−=
⋅
−+−=
⎪⎩
⎪
⎨
⎧
+≤
+>+
=t
c
t c
t c
hh
eif e
hh
eif hh
e
2
22
233
23
2
2 mme 5.2792 =
36
211 10719.82
2
422mm
hh
h Ah
Aht
At
AW t
ce
wg w
ce
wwg
fg
f p p ⋅=
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛ +
−⋅+⋅+⎟⎟ ⎠
⎞⎜⎜⎝
⎛ +⋅+⋅=
( )36
12
2
12
1
112 10392.62
2
2
2
22 mme
hh
h At
eh
t
e
eht Ae
t
AW
t
ce
wg wwg
ce
wg wg fg f p p ⋅=⎟⎟⎟
⎟
⎠
⎞
⎜⎜⎜
⎜
⎝
⎛
−
+
−⋅+⋅
⎟ ⎠
⎞⎜⎝
⎛ −
+⋅+−+⋅+⎟ ⎠
⎞⎜⎝
⎛ +⋅=
36
2231 10259.42222
mmet h
h Aehht
AW f ce
t f
ce
p p ⋅=⎟⎟ ⎠
⎞⎜⎜⎝
⎛ −++⋅+⎟
⎠
⎞⎜⎝
⎛ +Δ++⋅=
36
2
22
22132 10812.1
2
2
24mmt
ehh
t e
ehh
AW wg
t
ce
wg
ce
w p ⋅=⋅
⎟ ⎠
⎞⎜⎝
⎛ −+
+⋅+⎟ ⎠
⎞⎜⎝
⎛ +Δ+⋅=
36
213 10070.6 mmW W W p p p ⋅=+=
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Plastic section modulus:
Ratio between plastic and elastic section modulus:
9.5.3 Mass
Mass from permanent loads and possible live loads (to be eval- uated in each case):
Total mass:
9.5.4 Natural period
Linear Stiffness, Ref. Table 6-2 in Section 6.10:
Natural period assuming uniformly distributed mass ( K lm,u istaken from Table 6-2):
Ratio of pulse load period versus natural period:
36
213
212
211
10070.6 mm
A A A Aif W
A A A Aif W
A A A Aif W
W
f ww p p
f ww p p
f ww p p
p ⋅=⎪⎩
⎪⎨
⎧
+<+
+>+
++>
=
23.1=e
p
W W
Mass from plate:
Mass from stiffener, see figure:
Mass from girder:
m
kg l t w p 7.3517850200.314 =⋅⋅=⋅⋅= ρ
hws = 240-29
= 211
t ws = 10
b fs = 39
t fs = 29
23241293910211 mmt bt h A fs fswsws s =⋅+⋅=⋅+⋅=
m
kg
s
l Aw s s 8.101800
3200785010
32416 =⋅⋅=⋅⋅= ρ
m
kg Aw G g 1.1477850
10
1.187456
=⋅=⋅= ρ
m
kg l
g
pw P
PL 1.3263200.3807.9
1010 3
=⋅⋅
=⋅=
m
kg wwwww PL g s p 7.38631.32631.1478.1017.351 =+++=++⋅+=
m
N
mm
N
L
I E k Ge
l
85
3
95
310114.110114.1
12000
10387.2101.2384384⋅=⋅=
⋅⋅⋅⋅=
⋅⋅=
sec113.010114.1
0.127.386377.02
77.022
8
, =⋅
⋅⋅⋅⋅=
⋅⋅⋅⋅=
⋅⋅⋅= π π π
l l
uulm
k
Lw
k
M K T
33.1113.0
15.0==
T
t d
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9.5.5 Ductility ratio
The maximum lateral deformation prior to buckling can be cal-culated according to equation 3.19 in sub-section 3.10.2:
where;d c is characteristic dimension for local buckling, i.e.2·(t +½hce+dh+e3) = 2·(14+½·430.8+35.6+279.5)= 1089mm
c1 is 2 for clamped beams
κ L is the smaller the distance from load to adjacent joint (0.5).Here set to 0.5· L, i.e. 6000
,and c is non-dimensional spring stiffness, ref Section 3.7;
k node is axial stiffness of the node with the considered member removed, here assumed infinitely.
Calculation of cross sectional slenderness factor, ref. Section3.10, i.e. the maximum of the following:
Plate flange:
Bottom flange:
Web (bending):
Based on these input parameters, the maximum plastic defor-mation is calculated to:
The maximum elastic deformation is found from:
Ductility ratio:
9.5.6 Maximum blast pressure capacity
From Figure 9-11, the dynamic load factor is found:
With reference to Figure 9-11, k 3 was set to 0, which ensuresconservative results.
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟ ⎠
⎞⎜⎜⎝
⎛ ⋅
⋅
⋅⋅−−⋅
⋅=
2
3
1
1411
2
1
c
y f
f c
p
d
L
c
f c
cd
w κ
β
994.01066241
106624
1
22
=⎟⎟ ⎠
⎞⎜⎜⎝
⎛
+=⎟
⎟ ⎠
⎞⎜⎜⎝
⎛
+=
c
cc f
106624120001.18389420
1.40610873.7244 292
1
=⋅⋅
⋅⋅⋅⋅
=⋅⋅
⋅⋅⋅
= l A f
wk c
ce y
c
9
520
10873.7
1.18745101.22
1
101
1
1
2
11
1⋅=
⋅⋅⋅+
⋅
=
⋅⋅+
=
Gnode A E k
k
1.3961.18389
10070.62.12.1 6
=⋅⋅
=⋅
=e
p
c A
W w
9.86420/235
14/2.3033
/235
/3 ===
y
e
f
t l β
2.60420/235
20/3003
/235
/3 ===
y
fg fg
f
t b β
9.90420/235
10/8508.0
/235
/8.0 ===
y
wg wg
f
t h β
mmw p 37.331089
6000
9.902
420994.01411
994.02
10892
3 =⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛ ⎟ ⎠
⎞⎜⎝
⎛ ⋅⋅
⋅⋅−−⋅
⋅=
mm I E
LW f w
Ge
e y
e 56.1810387.2101.232
1200010923.4420
32 95
262
=⋅⋅⋅⋅
⋅⋅⋅=
⋅⋅
⋅⋅=
Maximum elastic deformation:
p
L
12
2 L p
W f M e ye
⋅=
⋅=
I E
L pwe ⋅
⋅⋅=
4
384
1
I E
LW f
I E
L M
I E
L L pw
e y
ee ⋅⋅
⋅⋅=
⋅⋅⋅=
⋅⋅
⋅⋅
⋅=3232
112
12384
12222
80.156.18
37.33===
e
p
w
wμ
99.0)( ==l
m
F
R DLF μ
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Figure 9-11Dynamic Response of a SDOF system due to a triangular pulse load profile (rise time = 0.50td)
Maximum resistance for a fixed supported beam, see Figure9-12:
Figure 9-12Moment diagram (elastic and fully plastic)
Resistance utilised in order to take the permanent and live load:
The maximum blast pressure capacity is obtained from the fol-lowing two equations:
0.1
1
10
100
0.1 1 10
td/T
y m a x / y e l
= 1.1
Rel/Fmax= 0.8
= 1.0
= 0.9
= 1.2= 1.5
=0.1
= 0.7
= 0.6= 0.5Rel/Fmax=0.05 = 0.3
yel y
R
Rel
F
Fmax
td0.50td
k1
k3 = 0.5k1 =0.2k1 =0.1k1
k3 = 0
k3 = 0.1k1
k3 = 0.2k1
k3 = 0.5k1
μ = 1.80
t d /T = 1.33
kN N L
f W
L
M R
y p p
m 2.3399102.339912000
42010070.61616163
6
=⋅=⋅⋅⋅
=⋅⋅
=⋅
=
Plastic
pP
Elastic
M = pL2/24
p
R m = pPL = 16MP/L
M = pL2/12MP = pPL2/16
MP = pPL2/16
LL
( ) ( ) kN N Ll p p L g w R l p g 3.593103.593122.31051012807.91.147 33
0 =⋅=⋅⋅⋅++⋅⋅=⋅⋅++⋅⋅=
Ll p F and DLF
R R F l
ml ⋅⋅=
−= max
0
)(μ