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Recommended Practice DNV-RP-C204, October 2010 – Changes



Recommended Practice DNV-RP-C204, October 2010Contents –

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 .........................................133.6 Force-deformation relationships for denting of

tubular 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........................................................173.10.1 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 ...................................................................194.2 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



Recommended Practice DNV-RP-C204, October 2010

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/mm 2)

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 resistanceR 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/s 2

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 deformationk Q Shear stiffness in linear domain for beam

Temperature reduction of effective yield stress for maximum temperature in connection

Plate length, beam length

m Distributed mass

ms Ship mass

m i 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, timetd 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




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 formatThe 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 loadsThe 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 criteriaExamples 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/m 3

ρw Density of sea water, 1025 kg/m 3

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 effectγf = partial factor for loadsR = Characteristic resistanceγM = Material factor

dd R S ≤

f k γS

M

k

γR




2.5 Analysis considerationsThe 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, safeand 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 GeneralThe 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 principlesWith 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 theship 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.

Strengthdesign

Shared-energydesign

Ductiledesign

Relative strength - installation/ship

ship

installation E n e r g y

d i s s i p a

t i o n




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 dissipationThe 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. v i = 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 energyThe 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 displacement 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

2ssss

amam

1

vv

1

)va(m21

E

+++

−

+=

2ssss )va(m

21

E +=

J

zm1

v

v1

)a(m21E

2s

2

s

i

sss

+

−+=

(3.4)

Collision response Model

dw s dw i

R iR s

Ship Installation

E s,sE s,i

∫∫ +=+= maxi,maxs, w

0 ii

w

0 ssis,ss,s dwR dwR EEE




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 3-4Recommended-deformation curve for beam, bow and stern impact

Figure 3-5Force -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 sideD = 10 m

= 1.5 m

Stern endD = 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 sideD = 10 m

= 1.5 m

Stern endD = 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-




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 arevalid 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

68

10

12

14

16

18

C o n

t a c

t d i m e n s

i o n

[ m ]

Forcesuperstructure

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 ]

ForceEnergy

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 3.00.35 1.65 6.40.20 1.15 5.4

Table 3-2 Local concentrated collision force -evenly distributedover a rectangular area. Stern end impact

Contact area Force (MN)a (m) b (m)0.6 0.3 5.60.9 0.5 7.52.0 1.1 10

Table 3-3 Energy dissipation in bow versus brace resistance

Contact location Energy dissipation in bow

if brace resistance R 0> 3 MN > 6 MN > 8 MN > 10 MN

Above bulb 1 MJ 4 MJ 7 MJ 11 MJFirst deck 0 MJ 2 MJ 4 MJ 17 MJFirst deck - oblique brace 0 MJ 2 MJ 4 MJ 17 MJBetween forcastle/firstdeck

1 MJ 5 MJ 10 MJ 15 MJ

Arbitrary location 0 MJ 2 MJ 4 MJ 11 MJ

a

b

b

a




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 of tubular membersThe 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 w d / 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 GeneralThe 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 flexibilityRelatively 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 member 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 32

Dtf 0.51.5y ≥

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 )210.50

b/D =

Rd

Sd

Rd

Sd

Rd

Sd

Rd

Sd

2

1

2

yc

cd

1c

N N

0.60k

0.6 N N

0.20.2 N N

21.0k

0.2 N N

1.0k

DB

3.5

1.925c

D

B1.222c

tD

4t

f R

Dw

kcR R 2

≤=

<<

−−=

≤=

+=

+=

=

=

(3.7)

plastic collapse resistance in bending for the 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+=

lP1Mc4

R 0 =

cwc

ww

1

=

lAf kw4c

cy

2c1=




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 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 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 membersThe 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−+= 0.1lim

≤w

w

R R * = 0.1lim

≥w

w

*0R

lP2P1P*

0

2M2M4MR

++=

∑ ≤=i

PiPj,Pj MMM

(3.15)

lim 2w wπ =

lim 1.2w w=

−=

=

−=

D2w

1arccosθ

tDf M

sinθ2

1

2

θcos

M

M

d

2yP

P

red




Figure 3-11Reduction of moment capacity due to local dent

3.8 Strength of connectionsProvided 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 structureThe 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 GeneralThe 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 bucklingTubular 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-sections

c1 = 2 for clamped ends= 1 for pinned ends

c = non-dimensional spring stiffness, refer Section 3.7.2.

κ ≤0.5 = the smaller distance from location of collisionload to adjacent jointIf 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= 2h w for stiffened plating (for simplicity)

31

2

c1

yf

dκ

c

f 14cβ

≤

l

yf 235tD

β =

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 dκ

βc

3.5f

dw

=

l




For flanges subjected to compression;

For webs subjected to bending

bf = flange widthtf = flange thicknesshw = web heighttw = web thickness

3.10.3 Tensile fractureThe 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 meansof 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 of collision load to adjacent joint

W = elastic section modulusWP = plastic section modulusεcr = critical strain for rupture (see Table 3-4 for

recommended values)

= yield strain

f y = yield strengthf cr = strength corresponding to εcr

dc = D diameter of tubular beams= 2h w twice the web height for stiffened

plates (s e·t > A s)= 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 of cross-section

−+= 1

cεc4c

12cc

dw

1

cr f w

f

1

c

2

ccr

y

Plplp

1w d

κ ε

ε

WW

14c31

1cc1

c

−+

−=

l

1HWW1

εε

HWW

1εε

c

Py

cr

Py

cr

lp

+

−

−

=

2

f c1

cc

+

=

−−

==ycr

ycr p

εε

f f

E1

E

EH

l l

E

f ε y

y =




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 stiffenersFor 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 stiffenersIn 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 bulkheadsCalculation 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-selsFor strength design the side or stern shall resist crushing forceof 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 global 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 Objects4.1 GeneralThe 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 steelgrades

Steel grade εcr H S 235 20 % 0.0022S 355 15 % 0.0034S 460 10 % 0.0034

(3.32)

cr wc

εcdw =

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 =




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 velocityThe 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 = v o 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 the

considered motionm = mass of objectA p = projected cross-sectional area of the objectV = 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 mv21

E =

(in water) (4.2)

(4.3)

( ) 2kin vam

21

E +=

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/v t]

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/v t]

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/v t]

In water

In air

D i s t a n c e

[ s / s c ]

(4.4)

=

terminal velocity for theobject (drag force and

buoyancy force balance thegravity force)

∫=Δ dt

0F(t)dtvm

Δvvv 0 −=

pdw

wt ACρ

V)ρ2g(mv −=

= characteristic distance)

mVρ

2g(1

)ma(1v

ACρam

sw

2t

pdw −

+=+=c




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 energyThe 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 natureof 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 ≤τcr 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/girdersIn 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 structuresThe 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 objectsFor stability of cross-sections and tensile fracture, refer Sec-tion 3.10.

5. Fire

5.1 GeneralThe 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 waterItem Mass

[kN]Terminal velocity

[m/s]Drill collar Winch,Riser pump

28250100

23-24

BOP annular preventer 50 16Mud pump 330 7

(4.5)

(4.6)

: stiffness of plateenclosed by hinge circle

∫∫ +=+= max,maxo, w

0 ii

w

0 oois,os,s dwR dwR EEEi

dw o dw i

R iR o

Object Installation

E s,oE s,i

2i

2

sp m

m0.481

2k R

E

+=

( )

+ +−+

=2

2

2

yc1

2r d6.256c

r d51

tπf 21

k

= mass of plate enclosed by hinge circle

m = mass of dropped objectρ p = mass density of steel plated = smaller diameter at threaded end of drill

collar r = smaller distance from the point of impact to

the 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




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 members,

— 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 methodsshall 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 methodsStructural 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 modellingIn 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 imperfectionsTo 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 factorThe 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 bucklingIf 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 beamsIf 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.


5.7.1 GeneralThe ductility of beams and connections increase at elevatedtemperatures compared to normal conditions. Little informa-

α π 0

y*

zi

E

f 1e =l

αAI

W

E

f 1α

zi

E

f 1W

We py

0

y*

p

π π ==

l

l




Figure 6-1Failure modes for two-way stiffened panel

Table 6-1 Analysis models

Failure mode Simplified analysis model Resistance models Comment

Elastic-plastic deformation of plate SDOF Section 6.9Stiffener plastic

– plate elasticSDOF Stiffener: Section 6.10.1-2.

Plate: Section 6.9.1Elastic, 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 w max exceeds the elastic limit w el




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 = N P)

m = distributed mass

M i = concentrated massq = explosion loadFi = concentrated load (e.g.

support reactions)xi = position of concentrated

mass/load

(6.6)

= load-mass transformation factor for uniform mass

= load-mass transformation factor for concentrated mass

= mass transformation factor for uni-form mass

=mass transformation factor for concentrated mass

( ) ( ) ( )tyxt,xw φ=

( )tf yk m =+ y&&

( ) ∑∫ +=i

2ii

2 φMdxxmφml

( )∫ ∑+=l i

iiφFdxxq(t)φ)t(f

( )∫=l

dxxEIφk 2xx,

0k =

( )∫=l

dxx Nφk 2x,

( )ii xxφφ ==

F(t)K(y)yy)MK M(K ccm,uum, =++ &&ll

l

lK

K K um,

um, =

l

lK

K K um,

um, =

u

2

um, M

dx(x)m

K ∫

= l

c

i

2

cm, M

M

K ∑

=

i

= load transformation factor for uniformly distributed load

= load transformation factor for concentrated 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 ∫

= ll

F

FK i

∑=

ii

l

∫=l

mdxM u

c ii

M M= ∑

∫=l

qdxF

ii

F F= ∑

lk k

k e =

e

ccm,uum,

k

MK MK 2

k

m2T

ll +== π π




Figure 6-2Maximum response a SDOF system to a triangular pressure pulse with zero rise time. F max / R el = 2

Design charts for systems with bi- or tri-linear resistancecurves subjected to a triangular pressure pulse with 0.5 t d risetime is given in Figure 6.3. Curves for different rise times aregiven in Ch.8, Commentary Figure 8-15 to Figure 8-17.

Baker's method The governing equations (6.1) and (6.2) for the maximumresponse in the impulsive domain and the quasi-static domainmay also be used along with response charts for maximum dis-

placement for different F max /R el ratios to produce pressure-impulse (F max , I) diagrams - iso-damage curves - provided thatthe maximum pressure is known.The advantage of using iso-damage diagrams is that "back-ward" calculations are possible:The diagram is established on the basis of the resistance curve.The information may be used to screen explosion pressure his-tories and eliminate those that obviously lie in the admissible

domain. This will reduce the need for large complex simula-tion of explosion scenarios.

6.5 Dynamic response charts for SDOF systemTransformation factors for elastic–plastic-membrane deforma-tion of beams and one-way slabs with different boundary con-ditions are given in Table 6-2.

Maximum displacement for a SDOF system exposed to a tri-angular pressure pulse with rise time of 0.5t d is displayed inFigure 6.3. Maximum displacement for a SDOF systemexposed to different pressure pulses are given in Ch.8, Com-mentary Figure 8-15 to Figure 8-17.

The characteristic response of the system is based upon theresistance in the linear range, k = k 1, where the equivalent stiff-ness is determined from the elastic solution to the actual sys-tem.

0,1

1

10

100

0,1 1 10td/T

Impulsive asymptote, k 3=0.2k 1

Elastic-perfectly plastic, k 3=0

Static asymptote, k 3=0.2k 1

k 3=0.2k 1

k 3=0.1k 1

td/T for system

ymax /yel

for system

td

F(t)

0,1

1

10

100

0,1 1 10td/T

Impulsive asymptote, k 3=0.2k 1

Elastic-perfectly plastic, k 3=0

Static asymptote, k 3=0.2k 1

k 3=0.2k 1

k 3=0.1k 1

td/T for system

ymax /yel

for system

td

F(t)




Figure 6-3Dynamic response of a SDOF system to a triangular load (rise time = 0.50 t d)

6.6 MDOF analysisSDOF 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 momentThe 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/F max = 0.

= 1.0

= 0.9

= 1.2= 1.5

=0.1= 0.7

= 0.6= 0.5R el/Fmax =0.05 = 0.3

yel y

R

R el

F

Fmax

td0.50t d

k1

k3 = 0.5k 1 =0.2k 1 =0.1k 1

k3 = 0k3 = 0.1k 1

k3 = 0.2k 1

k3 = 0.5k 1

Elastic-perfectly plastic

elasto-plastic

Moment

Curvature




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 strainingof the plate is accounted for by the 2 nd 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 hingesIn 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 restraintEquivalent 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/h w), 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

3α92α3α

w1r r 2

2

c

≤

−−++=

( ) 1w1w31

α3α2α1w2

r r

2c

>

−−−+=

22

2y

c α

t6f r

tw

2wl

==

22

2y

c α

t12f r

tw

wl

==

−

+= l l l ss

3s

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

321

w

w

(6.9)

(6.10)

+

+

+==

w

2g

s'1

ccm,uum,

AA

GE

1L

r πc1

k

MK MK 2

k

m2T

ll

π π

L

GAck ,

k 1

k 1

k

1 wQQ

Q1'1

=+=




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 clamped

L = 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-

trated mass

Uni- formmass

Concen-trated mass

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


resistance Rel

Linear stiffness

k 1

Equiva-lent lin-

ear stiffness

k e

Dynamic reaction

V Concen-

trated mass

Uniformmass

Con-cen-

trated mass

Uniformmass

Elastic 0.53 0.41 0.77

Elasto- plastic

bending0.64 0.50 0.78


Plasticmem- brane

0.50 0.33 0.66

F=pL

L

8 M

L p 384

5 3

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 48

3

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 L

0 525 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 384

3

EI

L F R 14.036.0 +

( )8 M M

L

ps Pm+ 384

5 3

EI

L 0 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




Where:

q = explosion load per unit length= ps for stiffeners= p for girders

m1, m2 and m 3 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- plasticbending

0.64 0.50 0.78


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


resistance Rel

Linear stiffness

k 1

Equiva-lent lin-

ear stiffness

k e

Dynamic reaction

V Concen-

trated mass

Uniformmass

Con-cen-

trated mass

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/29 ps M

L 3

260 EI L

13

212m

L

EI

0.48 0.02 R F +

( )6 ps Pm M M

L

+3

56.4 EI L

0.52 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 185

3

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

163 M L

Ps 1073

EI

L23

160m

L

EI

V R F 1 0 25 0 07= +. .V R F 2 0 54 014= +. .

( )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

L33

122m 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 Lel

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




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 = 2.0can

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 for the member with uniform load.

= member length

2EAk

1

k

1

node

l+=

21for 1 N N

MM

α

p p

<<=

+ α

l

py10

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 beamsc1 = 1 = for pinned beamsWP = plastic section modulus for the cross sec-

tion of the beamW p = zgAg = plastic section modulus for stiffened

plate for s et > A sA = A s + st = total area of stiffener and plate flangeAe = A s + s et = effective cross-sectional area of stiffener

and plate flange,zg = distance from plate flange to stiffener

centre of gravity.As = stiffener areas = stiffener spacingse = effective width of plate flange see Sec-

tion 6.10.4

c1wcw

w =

A

αWw p

c =

lAf kw4c

cy

2c1=

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 & membraneMembrane only

k k

F (explosion load)

w

R / R

0α = 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


k k

F (explosion load)

w

R / R

0




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

0.10.20.51

0

w

c = ∞


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

0.10.20.51

0

w

c = ∞


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

0.50.2 0.1

0

w

c = ∞


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

0.50.2 0.1

0

w

c = ∞


k k

F (explosion load)

w

R / R

0

(6.13) pinnedclamped ζ)R (1ζR R −+= (6.14)11

M8

R ζ0

p

actual0 ≤−=≤

l




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 forceintensity during a collision between the stern corner of a supplyvessel and a stiffened column. In the beginning the contact isconcentrated at the extreme end of the corner, but as the corner deforms it undergoes inversion and the contact ceases in thecentral 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 tototal 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 finiteelement analysis of stern/column impacts.

The information given in 3.5.2 may be used to performstrength design. If strength design is not achieved numericalanalyses have shown that the column is likely to undergosevere 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 strainenergy.

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. Afundamental 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 localdeformation. 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 compliedwith.

Figure 8-8 indicates the level of the various contact locations.

Figure 8-4 shows the minimum thickness as a function of bracediameter and resistance level in order to achieve sufficientresistance 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 wellas 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 andlength diameter ratio that results from Equation (8.1). Thethickness can generally be smaller than the values shown, andstill 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 diameterssubstantially larger than 1.0 m, because the resistance of the

bow will increase. For brace diameters smaller than 1.0 m, therequirement 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 forceintensity

Deformed stern corner

(8.1)2

21

D 10.14

t c D ≤

l




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 =20L/D =25L/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 0.35x/ l

S t r a

i nε

Hardening parameter H = 0.005

Maximum strainεcr /εY

= 50= 40= 20

No hardening

P

l

x




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 m i/m < 0.75.

The formula neglect the local energy dissipation which can beadded as E loc = 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 isapplied 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 responseEquation (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




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 analogyThe displacement at the end of the initial, linear resistancedomain y el will generally not coincide with the displacement atfirst yield. Typically, y el represents the displacement at the ini-tiation of a plastic collapse mechanism. Hence, y el 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 y max /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 thedisplacement of the component. From the diagram it is seenthat several combinations of pulses characterised by F max 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 ratioSince 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 t d/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 y max /yelastic = 10Elastic-perfectly plastic resistance

(8.3)

10y

y

y el

max

el

allow ==

el

max

R F

R F =

T

t

FR 2

1

TR

tF21

RT

I d

max

elel

dmax

==

( )μ μ plastic

melastic

maveragem

k k k ll

l

1−+=




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, s e, of simply supported or clamped stiffeners/girders may be taken from Figure 8-18.

Figure 8-18Effective 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 impactThe 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= 0.5R el/F max =0.05 = 0.3

= 1.1= 1.0

= 0.9

R el/Fmax = 0.

= 1.2= 1.5

yel y

R

R el

F

Fmax

td0.30t d

k1

k3 = 0.5k 1 =0.2k 1 =0.1k 1k3 = 0k3 = 0.1k 1

k3 = 0.2k 1

k3 = 0.5k 1

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 = 4n = 3

Uniform distribution or

nF

= L

nF

= 0.6L




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 (c 1 = 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 theadjacent 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 w crit = 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, d c = D = 0.762 m. For steel gradeS 355 a strain hardening coefficient of H = 0.0034 is used, refer Table 3-3. c 1 = 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/W P = π /4. Thisyields w crit = 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 mml = 23.3 m

m MN K node /95511

7361

21

=

+=

−

m MN EA

/12343.23

0286.0762.0101.222 5

== π l

MN/m881234

19511 =+=

K

18.03.230286.0355

762.08822Af

Kw4cc

y

2c1 ===

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




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 predominatesOn 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 2 nd 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 w el = 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 c 1= 2. The area of the stiffener A s= 1.88·10 -2 m2 and the distanceto the centroid is z g = 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 > A s. 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 W P = A s 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 w el = 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!).

21412 2 t

t t

AW

w P c === α

∞=c

m MN s Et EA

k /8400122

=== 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 = 1.0Plate c = 0.3Strip 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

wcα W P

A------------ α z g 1.2 0.109 0.13 m= = = =

MN/m230L

384EIk

3==




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 m 1 = 1:

and w el = 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, h w = 1.5 m, web thickness, t w =13 mm, top flange breadth, b top = 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/m 2 and mounted equipment with mass 1.8·10 5 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/s 2, density of steel 7.86 103 kg/m 3.

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, W P = 2.51 10-2 m3,total cross-sectional area 0.048 m 2. 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 (m 3 = 1) is.

The plastic bending resistance is

and w el* = 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 w el = 14.1 mm is available in the equivalent elas-tic range. The limiting deformation for rupture calculated in9.4.3 is w max = 95 mm, yielding ductility ratio μ = w/ max / w el= 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 = 0.56and

k lmaverage,c = (0.83 + (6.7 − 1) 1.0) / 6.7 = 0.975,respectively. The eigenperiod becomes

and hence t d/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 F max = 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 d c = 2 (1.5 − 0.315) = 2.37 m. The criticallocation is at the clamped side, whereby κ =1/3. Clamped endyields c 1 = 2 for the fracture check. With H = 0.0034 and c =0, there is obtained w/d c = 0.069 and w = 0.095 m.


9.4.1 PlatingRupture 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, d c = 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, w crit ,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 d c = 2h w =0.36 m. This yields λ/dc = 5.56, only. This critical deformation

becomes wcrit

= 0.1dc

= 36 mm, almost independent of thespring 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 = 1.0c = 0.5

c = 0.2c = 0.1

MN/m274L

122EIk

3==

Table 9-1 Ductility limit as a function of the spring stiffnessc ∞ 1.0 0.3 0

wcrit [mm] 35 51 59 76

*8

5.95 MN Pmel

M R

L= =

T 2p0.56 2.9 10 4 0.975 1.8 10 5 +

274 10 6----------------------------------------------------------------------------------- 0.166s= =

l




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 d c = 2 (1.5 − 0.315) = 2.37 m. The critical

location at the clamped side, whereby κ =1/3. Clamped endyields c 1 = 2 for the fracture check. With H = 0.0034 and c =0, there is obtained w/d c = 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 MPaStrain rate factor: γε = 1.0Effective yield strength: f y = f y· γε = 420 MPaModulus of elasticity: E = 2.1·10 5 MPaMaterial density: ρ = 7850 kg/m 3

Poisson’s ratio: ν = 0.3Max. plastic strain: 1.0% (maximum allowable, corre-

spond to cross section class 3 or 4, seesub-section 9.5.2)

Permanent loads: p P = 10.0 kN/m2Live loads: p L = 5.0 kN/m2Explosion pulse

period:td = 0.15 sec (triangular load

with a rise time =0.50·t d)

t = 14

Bulkhead

Girder:

800 (typ.)

10

Stiffener: Hp240

Girder: TG870x300x10x20

Bulkhead

12000

Stiffener:

870

300

20

10 240

39

29

3200(typ.)




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 propertiesReduction in web height:Δh = hc -hce = 466.4 – 430.8 = 35.6 mmEffective cross section area:

Ae = AG -Δh ·t wg = 18745.1 – 35.6·10.0 = 18389.1 mm 2 Distance to neutral axis from bottom of girder flange:

Effective elastic moment of inertia:

( ) 4920

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 =−

+++

++

+++=

( ) ( ) MPaht E f

wg

wg e 9.627

85010

3.0112101.29.23

1129.23

2

2

522

2

2 =

−=

−= π

ν π

818.09.6270.420 ===

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 8.430818.051

1818.02.341

=

−=

½ hce

hwg = 870-20

= 850

t wg = 10

l e = 303.2

b fg = 300

t = 14

e

z 0

hc

ht

½ h ce

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 =

+++−

=

+++ΔΔ−

=

2

03

2121

−++Δ−Δ−= e

ct fg wg wg GGe z hht t ht h I I

492

39 10387.21.3992

4.4666.38320106.35106.35

121

10407.2 mm I Ge =

−++−−=




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

Geeo =

−++=

−++=

369

0

10982.51.39910387.2

mm z

I W

e

Geeu ===

3610923.4),min( mmW W W eueoe ==

Web areas:

Eccentricities (see figure):

21 0.215410

28.430

2mmt

h A wg

cew ===

½ h ce

A w 1½ h ce

h t

e1

e3

A w 2

2

2 0.5990106.38328.430

2 mmt h

h

A wg t

ce

w =

+=

+=

mmt

A A A Ae

wg

pww f 9.494

1021.42450.59900.21546000

221

1 =−++=−++

=

mmt

A A A Ae

wg

pww f 5.279

1021.42450.59900.21456000

221

3 =−+−=−+−

=

+≤

+>+

=t

c

t c

t c

hh

eif e

hh

eif hh

e2

222

33

23

2

2 mme 5.2792 =

36211 10719.8

22

422mm

hh

h Ah

Aht

At

AW t

ce

wg wce

wwg fg

f p p =

+−++

++=

( )36

12

2

121

112 10392.622

2

2

22 mme

hh

h At

eh

t e

eht Aet

AW

t ce

wg wwg

ce

wg wg fg f p p =

−

+

−+

−

++−++

+=

362231 10259.4

2222mme

t hh Aeh

ht AW f ce

t f ce

p p =

−+++

+Δ++=

36

2

222

2132 10812.12

2

24mmt

ehh

t e

ehh

AW wg

t ce

wg ce

w p =

−+

++

+Δ+=

36213 10070.6 mmW W W p p p =+=




Plastic section modulus:

Ratio between plastic and elastic section modulus:

9.5.3 Mass

Mass from permanent loads and possible live loads (to be evaluated in each case):

Total mass:

9.5.4 Natural periodLinear 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:

mkg

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 =+=+=

mkg

sl

Aw s s 8.1018003200

7850103241

6 === ρ

mkg

Aw G g 1.147785010

1.187456

=== ρ

mkg

l g

pw P

PL 1.3263200.3807.91010 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

33.1113.015.0 ==

T

t d




9.5.5 Ductility ratioThe 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 +½h ce+dh+ e3) = 2·(14+½·430.8+35.6+279.5)= 1089mmc1 is 2 for clamped beamsκ 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 capacityFrom 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

31

1411

21

c

y f

f c

p

d L

c

f c

cd

w κ β

994.01066241

106624

1

22

=

+

=

+

=c

cc f

106624120001.183894201.40610873.7244 292

1

=== l A f

wk cc

e y

c

9

520

10873.7

1.18745101.221

1011

1

211

1 =+

=+

=

Gnode A E k

k

1.3961.1838910070.62.12.1 6

===e

pc 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 37.3310896000

9.902420994.014

11994.02

10892

3=

−−=

mm I E

LW f w

Ge

e ye 56.18

10387.2101.232

1200010923.442032 95

262

===

Maximum elastic deformation:

p

L

12

2 L p

W f M e ye

=

=

I E L p

we =4

3841

I E

LW f

I E L

M I E L L p

w e yee ===

3232112

123841

2222

80.156.1837.33 ===

e

p

w

wμ

99.0)( ==l

m

F

R DLF μ

Dnv Dal Spec.

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