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RECOMMENDED PRACTICE DNV-RP-C204

DESIGN AGAINST ACCIDENTAL LOADS OCTOBER 2010

DET NORSKE VERITAS

FOREWORD DET NORSKE VERITAS (DNV) is an autonomous and independent foundation with the objectives of safeguarding life, property and the environment, at sea and onshore. DNV undertakes classification, certification, and other verification and consultancy services relating to quality of ships, offshore units and installations, and onshore industries worldwide, and carries out research in relation to these functions. DNV service documents consist of amongst other the following types of documents: — Service Specifications. Procedual requirements. — Standards. Technical requirements. — Recommended Practices. Guidance. The Standards and Recommended Practices are offered within the following areas: A) Qualification, Quality and Safety Methodology B) Materials Technology C) Structures D) Systems E) Special Facilities F) Pipelines and Risers G) Asset Operation H) Marine Operations J) Cleaner Energy

O) Subsea Systems

The electronic pdf version of this document found through http://www.dnv.com is the officially binding version © Det Norske Veritas Any comments may be sent by e-mail to [email protected] For subscription orders or information about subscription terms, please use [email protected] Computer Typesetting (Adobe Frame Maker) by Det Norske Veritas

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Recommended Practice DNV-RP-C204, October 2010 Changes – Page 3

CHANGES •

General

•

As of October 2010 all DNV service documents are primarily published electronically. In order to ensure a practical transition from the “print” scheme to the “electronic” scheme, all documents having incorporated amendments and corrections more recent than the date of the latest printed issue, have been given the date October 2010.

Main changes

Since the previous edition (November 2004), this document has been amended, most recently in April 2005. All changes have been incorporated and a new date (October 2010) has been given as explained under “General”.

An overview of DNV service documents, their update status and historical “amendments and corrections” may be found through http://www.dnv.com/resources/rules_standards/.

DET NORSKE VERITAS

Recommended Practice DNV-RP-C204, October 2010 Page 4 – Changes

DET NORSKE VERITAS

Recommended Practice DNV-RP-C204, October 2010 Contents – Page 5

CONTENTS 1. 1.1 1.2 1.3 1.4

GENERAL .............................................................. 7 Introduction .............................................................7 Application ...............................................................7 Objectives .................................................................7 Normative references ..............................................7

1.4.1 1.4.2

DNV Offshore Standards (OS)........................................... 7 DNV Recommended Practices (RP)................................... 7

1.5 1.6

Definitions ................................................................7 Symbols.....................................................................8

2. 2.1 2.2 2.3 2.4 2.5

DESIGN PHILOSOPHY ....................................... 9 General .....................................................................9 Safety format............................................................9 Accidental loads .......................................................9 Acceptance criteria..................................................9 Analysis considerations .........................................10

3. 3.1 3.2 3.3

SHIP COLLISIONS............................................. 10 General ...................................................................10 Design principles....................................................10 Collision mechanics ...............................................11

3.3.1 3.3.2

Strain energy dissipation................................................... 11 Reaction force to deck ...................................................... 11

3.4 3.5

Dissipation of strain energy ..................................11 Ship collision forces ...............................................11

3.5.1 3.5.2 3.5.3

3.6 3.7

3.7.1 3.7.2

Recommended force-deformation relationships............... 11 Force contact area for strength design of large diameter columns............................................................................. 13 Energy dissipation is ship bow ......................................... 13

Force-deformation relationships for denting of tubular members ...................................................14 Force-deformation relationships for beams........14

3.7.4

General.............................................................................. 14 Plastic force-deformation relationships including elastic, axial flexibility.................................................................. 14 Support capacity smaller than plastic bending moment of the beam............................................................................ 16 Bending capacity of dented tubular members .................. 16

3.8 3.9 3.10

Strength of connections.........................................17 Strength of adjacent structure .............................17 Ductility limits........................................................17

3.7.3

3.10.1 3.10.2 3.10.3 3.10.4

3.11

General.............................................................................. 17 Local buckling ................................................................. 17 Tensile fracture ................................................................. 18 Tensile fracture in yield hinges......................................... 18

Resistance of large diameter, stiffened columns.19

3.11.1 3.11.2 3.11.3 3.11.4

General.............................................................................. 19 Longitudinal stiffeners...................................................... 19 Ring stiffeners................................................................... 19 Decks and bulkheads ........................................................ 19

3.12 3.13

Energy dissipation in floating production vessels......................................................................19 Global integrity during impact ............................19

4. 4.1 4.2 4.3 4.4

DROPPED OBJECTS ......................................... 19 General ...................................................................19 Impact velocity.......................................................20 Dissipation of strain energy ..................................21 Resistance/energy dissipation ...............................21

4.4.1 4.4.2 4.4.3

4.5

Stiffened plates subjected to drill collar impact ............... 21 Stiffeners/girders .............................................................. 21 Dropped object ................................................................. 21

Limits for energy dissipation ............................... 21

4.5.1 4.5.2

Pipes on plated structures ................................................. 21 Blunt objects ..................................................................... 21

5. 5.1 5.2 5.3 5.4 5.5 5.6 5.7

FIRE ...................................................................... 21 General................................................................... 21 General calculation methods................................ 22 Material modelling................................................ 22 Equivalent imperfections...................................... 22 Empirical correction factor.................................. 22 Local cross sectional buckling.............................. 22 Ductility limits ....................................................... 22

5.8

Capacity of connections ........................................ 23

6. 6.1 6.2 6.3

EXPLOSIONS ...................................................... 23 General................................................................... 23 Classification of response ..................................... 23 Recommended analysis models for stiffened panels...................................................................... 23 SDOF system analogy ........................................... 25 Dynamic response charts for SDOF system ....... 26 MDOF analysis...................................................... 27 Classification of resistance properties ................ 27

5.7.1 5.7.2 5.7.3

6.4 6.5 6.6 6.7

6.7.1

6.8 6.9 6.9.1 6.9.2 6.9.3

6.10 6.10.1 6.10.2 6.10.3

General.............................................................................. 22 Beams in bending ............................................................. 23 Beams in tension............................................................... 23

Cross-sectional behaviour................................................. 27

Idealisation of resistance curves .......................... 28 Resistance curves and transformation factors for plates ................................................................ 28 Elastic - rigid plastic relationships.................................... 28 Axial restraint ................................................................... 29 Tensile fracture of yield hinges ........................................ 29

Resistance curves and transformation factors for beams................................................................ 29

6.10.4 6.10.5 6.10.6 6.10.7

Beams with no- or full axial restraint ............................... 29 Beams with partial end restraint. ...................................... 32 Beams with partial end restraint - support capacity smaller than plastic bending moment of member............. 34 Effective flange................................................................. 34 Strength of adjacent structure ........................................... 34 Strength of connections .................................................... 34 Ductility limits.................................................................. 34

7.

REFERENCES..................................................... 35

8.

COMMENTARY ................................................. 35

9. 9.1

EXAMPLES ......................................................... 43 Design against ship collisions ............................... 43

9.1.1

9.2

9.2.1 9.2.2 9.2.3

9.3

9.3.1 9.3.2 9.3.3

Jacket subjected to supply vessel impact.......................... 43

Design against explosions ..................................... 44

Geometry .......................................................................... 44 Calculation of dynamic response of plate: ....................... 44 Calculation of dynamic response of stiffened plate.......... 44

Resistance curves and transformation factors .. 44

Plates................................................................................. 44 Calculation of resistance curve for stiffened plate ........... 45 Calculation of resistance curve for girder......................... 46

DET NORSKE VERITAS

Recommended Practice DNV-RP-C204, October 2010 Page 6 – Contents

9.4

Ductility limits ..................................................... 46

9.4.1 9.4.2 9.4.3

Plating ...............................................................................46 Stiffener: ...........................................................................46 Girder: ...............................................................................47

9.5

Design against explosions - girder ....................... 47

9.5.1 9.5.2 9.5.3 9.5.4 9.5.5 9.5.6

Geometry, material and loads ...........................................47 Cross sectional of properties for the girder.......................48 Mass ..................................................................................51 Natural period ...................................................................51 Ductility ratio ....................................................................52 Maximum blast pressure capacity.....................................52

DET NORSKE VERITAS

Recommended Practice DNV-RP-C204, October 2010 Page 7

1. General

DNV-OS-C301

1.1 Introduction This Recommended Practice deals with design to maintain the load-bearing function of the structures during accidental events. The overall goal of the design against accidental loads is to achieve a system where the main safety functions of the installation are not impaired. The Recommended Practice has been developed for general world-wide application. Governmental legislation may include requirements in excess of the provisions of this Recommended Practice depending on type, location and intended service of the unit/installation. The Design Accidental Loads and associated performance criteria are given in DNV-OS-A101. The Accidental Loads in this standard are prescriptive loads. This Recommended Practice may also be used in cases where the Design Accidental Loads are determined by a formal safety assessment (see DNV-OS-A101, Appendix C) or Quantified Risk Assessment (QRA). The following main subjects are covered: — — — — —

Design philosophy Ship Collisions Dropped Objects Fire Explosions.

The Recommended Practice is applicable to all types of floating and fixed offshore structures made of steel. The methods described are relevant for both substructures and topside structures. The document is limited to load-carrying structures and does not cover pressurised equipment.

1.3 Objectives The objective with this Recommended Practice is to provide recommendations for design of structures exposed to accidental events.

1.4 Normative references The following standards include requirements which, through reference in the text constitute provisions of this Recommended Practice. Latest issue of the references shall be used unless otherwise agreed. Other recognised standards may be used provided it can be demonstrated that these meet or exceed the requirements of the standards referenced below. Any deviations, exceptions and modifications to the codes and standards shall be documented and agreed between the supplier, purchaser and verifier, as applicable. 1.4.1 DNV Offshore Standards (OS) The latest revision of the following documents applies:

DNV-OS-C102 DNV-OS-C103 DNV-OS-C104 DNV-OS-C105 DNV-OS-C106

1.4.2 DNV Recommended Practices (RP) The latest revision of the following documents applies: DNV-RP-C201 DNV-RP-C202

Buckling Strength of Plated Structures Buckling Strength of Shells

1.5 Definitions Load-bearing structure: That part of the facility whose main function is to transfer loads. Accidental Event: An undesired incident or condition which, in combination with other conditions (e.g.: weather conditions, failure of safety barrier, etc.), determines the accidental effects. Accidental Effect: The result of an accidental event, expressed in terms of heat flux, impact force and energy, acceleration, etc. which is the basis for the safety evaluations. Design Accidental Event (DAE): An accidental event, which results in effects that, the platform should be designed to sustain. Acceptance criteria: Functional requirements, which are concerned with the platforms' resistance to accidental effects. This should be in accordance with the authority's definition of acceptable safety levels.

1.2 Application

DNV-OS-A101 DNV-OS-C101

Stability and Watertight Integrity of Offshore Units

Safety Principles and Arrangements Design of Offshore Steel Structures, General (LRFD Method) Structural Design of Offshore Ships Structural Design of Column Stabilised Units (LRFD) Structural Design of Self-Elevating Units (LRFD) Structural Design of TLPs (LRFD) Structural Design of Deep Draught Floating Units (LRFD)

Active protection: Operational loads and mechanical equipment which are brought into operation when an accident is threatening or after the accident has occurred, in order to limit the probability of the accident and the effects thereof, respectively. Some examples are safety valves, shut down systems, water drenching systems, working procedures, drills for coping with accidents, etc. Passive protection: Protection against damage by means of distance, location, strength and durability of structural elements, insulation, etc. Event control: Implementation of measures for reducing the probability and consequence of accidental events, such as changes and improvements in equipment, working procedures, active protection devices, arrangement of the platform, personnel training, etc. Indirect design: Implementation of measures for improving structural ductility and resistance without numerical calculations and determination of specific accidental effects. Direct design: Determination of structural resistance, dimensions, etc. on basis of specific design accidental effects. Load: Any action causing load effect in the structure. Characteristic load: Reference value of a load to be used in determination of load effects when using the partial coefficient method or the allowable stress method. Load effect: Effect of a single load or combination of loads on the structure, such as stress, stress resultant (internal force and moment), deformation, displacement, motion, etc. Resistance: Capability of a structure or part of a structure to resist load effect. Characteristic resistance: The nominal capacity that may be used for determination of design resistance of a structure or structural element. The characteristic value of resistance is to be based on a defined percentile of the test results. Design life: The time period from commencement of construction until condemnation of the structure. Limit state: A state where a criterion governing the load-carrying ability or use of the structure is reached.

DET NORSKE VERITAS

Recommended Practice DNV-RP-C204, October 2010 Page 8

1.6 Symbols

clp

Plastic zone length factor

A

Cross-sectional area

cs

Shear factor for vibration eigenperiod

Ae

Effective area of stiffener and effective plate flange

cQ

Shear stiffness factor

As

Area of stiffener

cw

Displacement factor for strain calculation

Ap

Projected cross-sectional area

d

Smaller diameter of threaded end of drill collar

Aw

Shear area of stiffener/girder

dc

Characteristic dimension for strain calculation

B

Width of contact area

f

Generalised load

CD

Hydrodynamic drag coefficient

fu

Ultimate material tensile strength

D

Diameter of circular sections, plate stiffness

fy

Characteristic yield strength

E

g

Acceleration of gravity, 9.81 m/s2

Ep

Young's Modulus of elasticity, (for steel 2.1⋅105 N/mm2) Plastic modulus

hw

Web height for stiffener/girder

Ekin

Kinetic energy

i

Radius of gyration

Es

Strain energy

k

Stiffness, characteristic stiffness, plate stiffness, factor

F

Lateral load, total load

G

Shear modulus

H

Non-dimensional plastic stiffness

I

Moment of inertia, impulse

J

Mass moment of inertia

Kl

Load transformation factor

Km

Mass transformation factor

Klm

k ke

kl k

' 1

kQ

Generalised stiffness Equivalent stiffness Bending stiffness in linear domain for beam Stiffness in linear domain including shear deformation Shear stiffness in linear domain for beam

k y,θ

Temperature reduction of effective yield stress for maximum temperature in connection Plate length, beam length

Load-mass transformation factor

l m

L

Girder length

ms

Ship mass

M

Total mass, cross-sectional moment

mi

Installation mass

MP

Plastic bending moment resistance

meq

Equivalent mass

NP

Plastic axial resistance

m

Generalised mass

Sd

Design load effect

p

Explosion pressure

T

Fundamental period of vibration

r

Radius of deformed area, resistance

N

Axial force

rc

Plastic collapse resistance in bending for plate

NSd

Design axial compressive force

rg

Radius of gyration

NRd

Design axial compressive capacity

s

Distance, stiffener spacing

NP

Axial resistance of cross section

sc

Characteristic distance

R

Resistance

se

Effective width of plate

RD

Design resistance

t

Thickness, time

R0

Plastic collapse resistance in bending

td

Duration of explosion

V

Volume, displacement

tf

Flange thickness

WP

Plastic section modulus

tw

Web thickness

W

Elastic section modulus

vs

Velocity of ship

a

Added mass

vi

Velocity of installation

as

Added mass for ship

vt

Terminal velocity

ai

Added mass for installation

w

Deformation, displacement

b

Width of collision contact zone

wc

Characteristic deformation

bf

Flange width

wd

dent depth

c

Factor

w

Non-dimensional deformation

cf

Axial flexibility factor

x

Axial coordinate

Distributed mass

DET NORSKE VERITAS

Recommended Practice DNV-RP-C204, October 2010 Page 9

y

Generalised displacement, displacement amplitude

2.2 Safety format

yel

Generalised displacement at elastic limit

z

Distance from pivot point to collision point

The requirements to structures exposed for accidental loads are given in DNV-OS-C101 Section 7.

zplast 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

φ

Displacement shape function

λ μ

Reduced slenderness ratio

ν

Poisson's ratio, 0.3

θ

Angle

The structure should be checked in two steps: — First the structure will be checked for the loads to which it is exposed due to the accidental event — Secondly in case the structural capacity towards ordinary loads is reduced as a result of the accident then the strength of 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 Accidental Limit States (ALS). The requirement may be written as

Sd ≤ R d

Ductility ratio

(2.1)

where: Sd

=

kg/m3

ρ

Density of steel, 7860

ρw

Density of sea water, 1025 kg/m3

Rd

=

τ

Shear stress

τcr

Critical shear stress for plate plugging

ξ

Interpolation factor

ψ

Plate stiffness parameter

Sk γf R γM

= = = =

Sk γ f Design load effect Rk γM

Design resistance Characteristic load effect partial factor for loads Characteristic resistance Material factor

For check of Accidental limit states (ALS) the load and material factor should be taken as 1.0.

2. Design Philosophy

The failure criterion needs to be seen in conjunction with the assumptions made in the safety evaluations.

2.1 General

The limit states may need to be alternatively formulated to be on the form of energy formulation, as acceptable deformation, or as usual on force or moment.

The overall goal for the design of the structure against accidental loads is to prevent an incident to develop into an accident disproportional to the original cause. This means that the main safety functions should not be impaired by failure in the structure due to the design accidental loads. With the main safety functions is understood: — usability of escapeways, — integrity of shelter areas, — global load bearing capacity

The design against accidental loads may be done by direct calculation of the effects imposed on the structure, or indirectly, by design of the structure as tolerable to accidents. Examples of the latter are compartmentation of floating units which provides sufficient integrity to survive certain collision scenarios without further calculations. The inherent uncertainty of the frequency and magnitude of the accidental loads, as well as the approximate nature of the methods for determination of accidental load effects, shall be recognised. It is therefore essential to apply sound engineering judgement and pragmatic evaluations in the design. — — — —

Ship collision Dropped objects Fire Explosion

The accidental loads are either prescriptive values or defined in a Formal Safety Assessment. Prescriptive values may be given by authorities, the owner or found in DNV Offshore Standard DNV-OS-A101. Usually the simplification that accidental loads need not to be combined with environmental loads is valid.

In this section the design procedure that is intended to fulfil this goal is presented.

Typical accidental events are:

2.3 Accidental loads

For check of the residual strength in cases where the accident lead to reduced load carrying capacity in the structure the check should be made with the characteristic environmental loads determined as the most probable annual maximum value.

2.4 Acceptance criteria Examples of failure criteria are: — Critical deformation criteria defined by integrity of passive fire protection. To be considered for walls resisting explosion pressure and shall serve as fire barrier after the explosion. — Critical deflection for structures to avoid damage to process equipment (Riser, gas pipe, etc). To be considered for structures or part of structures exposed to impact loads as ship collision, dropped object etc. — Critical deformation to avoid leakage of compartments. To be considered in case of impact against floating structures where the acceptable collision damage is defined by the minimum number of undamaged compartments to remain stable.

DET NORSKE VERITAS

Recommended Practice DNV-RP-C204, October 2010 Page 10

The mechanical response to accidental loads is generally concerned with energy dissipation, involving large deformations and strains far beyond the elastic range. Hence, load effects (stresses forces, moments etc.) obtained from elastic analysis and used in ultimate limit state (ULS) checks on component level are generally not applicable, and plastic methods of analysis should be used. Plastic analysis is most conveniently based upon the kinematical approach, taking into account the effect of the strengthening (membrane tension) or softening (compression) caused by finite deformations, where applicable. The requirements in this RP are generally derived from plastic methods of analysis, including the effect of finite deformations. Plastic methods of analysis are valid for materials that can undergo considerable straining and during this process exhibit considerable strain hardening. If the material is ductile as such, i.e. it can be strained significantly, but has little strain hardening, 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 members undergoing large, plastic rotations is compact cross-sections; typically type I cross-sections (refer DNV-OS-C101, Table A1). The methods may also be utilised for type II sections provided that the detrimental effect of local buckling is taken into account. Note that for members subjected to significant tensile straining, the tendency for local buckling may be overridden by membrane tension for large deformations. The straining, and hence the amount of energy dissipation, is limited by fracture. This key parameter is associated with considerable uncertainty, with respect to both physical occurrence as well as modelling in theoretical analysis. If good and validated models for prediction of fracture are not available, safe and conservative assumptions for ductility limits should be adopted. If non-linear, dynamic finite elements analysis is applied, it shall be verified that all behavioural effects and local failure modes (e.g. strain rate, local buckling, joint overloading, and joint fracture) are accounted for implicitly by the modelling adopted, or else subjected to explicit evaluation.

3. Ship Collisions 3.1 General The requirements and methods given in this section have historically been developed for jackets. They are generally valid also for jack-up type platforms, provided that the increased importance of global inertia effects are accounted for. Columnstabilised platforms and floating production and storage vessels (FPSOs) consist typically plane or curved, stiffened panels, for which methods for assessment of energy dissipation in braced platforms (jackets and jack-ups) sometimes are not relevant. Procedures especially dedicated to assessment of energy dissipation in stiffened plating are, however, also given based on equivalent beam-column models. The ship collision load is characterised by a kinetic energy, governed by the mass of the ship, including hydrodynamic added 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. The remainder of the kinetic energy has to be dissipated as strain energy in the installation and, possibly, in the vessel. Generally this involves large plastic strains and significant structural damage to the installation, the ship or both. The strain energy dissipation is estimated from force-deformation relationships

for the installation and the ship, where the deformations in the installation shall comply with ductility and stability requirements. The load bearing function of the installation shall remain intact with the damages imposed by the ship collision load. In addition, damaged condition should be checked if relevant, see Section 2.2. The structural effects from ship collision may either be determined by non-linear dynamic finite element analyses or by energy considerations combined with simple elastic-plastic methods. If non-linear dynamic finite element analysis is applied all effects described in the following paragraphs shall either be implicitly covered by the modelling adopted or subjected to special considerations, whenever relevant. Often the integrity of the installation can be verified by means of simple calculation models. If simple calculation models are used the part of the collision energy that needs to be dissipated as strain energy can be calculated by means of the principles of conservation of momentum and conservation of energy, refer Section 3.3. It is convenient to consider the strain energy dissipation in the installation 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 contact with the ship. Elastic strain energy can in most cases be disregarded, but elastic axial flexibility may have a substantial effect on the load-deformation relationships for components/ sub-structures. Elastic energy may contribute significantly on a global level.

3.2 Design principles With respect to the distribution of strain energy dissipation there may be distinguished between, see Figure 3-1: — strength design — ductility design — shared-energy design Energy dissipation

2.5 Analysis considerations

Ductile design

Shared-energy design

Strength design ship

installation Relative strength - installation/ship

Figure 3-1 Energy dissipation for strength, ductile and shared-energy design

Strength design implies that the installation is strong enough to resist the collision force with minor deformation, so that the ship is forced to deform and dissipate the major part of the energy. Ductility design implies that the installation undergoes large, plastic deformations and dissipates the major part of the collision energy.

DET NORSKE VERITAS

Recommended Practice DNV-RP-C204, October 2010 Page 11

Shared energy design implies that both the installation and ship contribute significantly to the energy dissipation. From calculation point of view strength design or ductility design is favourable. In this case the response of the «soft» structure can be calculated on the basis of simple considerations of the geometry of the «rigid» structure. In shared energy design both the magnitude and distribution of the collision force depends upon the deformation of both structures. This interaction makes the analysis more complex. In most cases ductility or shared energy design is used. However, strength design may in some cases be achievable with little increase in steel weight.

3.3 Collision mechanics 3.3.1 Strain energy dissipation The collision energy to be dissipated as strain energy may depending on the type of installation and the purpose of the analysis - be taken as: Compliant installations 2

⎛ vi ⎞ ⎜1 − ⎟ ⎜ v ⎟ 1 2 ⎝ s ⎠ E s = (ms + a s )vs ms + a s 2 1+ mi + a i

Model

Collision response

Figure 3-2 Model for assessment of reaction force to deck

3.4 Dissipation of strain energy The structural response of the ship and installation can formally be represented as load-deformation relationships as illustrated in Figure 3-3. The strain energy dissipated by the ship and installation equals the total area under the load-deformation curves.

Rs

(3.1)

Ri

Fixed installations 1 2 E s = (m s + a s )v s 2

(3.2)

Articulated columns

dws 2

⎛ v ⎞ ⎜1 − i ⎟ ⎜ v s ⎟⎠ 1 ⎝ E s = (m s + a s ) 2 m z2 1+ s J

ms as vs mi ai vi J z

(3.3)

= = = = = = =

ship mass ship added mass impact speed mass of installation added mass of installation velocity of installation mass moment of inertia of installation (including added mass) with respect to effective pivot point = distance from pivot point to point of contact

In most cases the velocity of the installation can be disregarded, i.e. vi = 0. The installation can be assumed compliant if the duration of impact is small compared to the fundamental period of vibration of the installation. If the duration of impact is comparatively long, the installation can be assumed fixed. Floating platforms (semi-submersibles, TLP’s, production vessels) can normally be considered as compliant. Jack-ups may be classified as fixed or compliant. Jacket structures can normally 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 maximum force between the collision zone and the deck for bottom supported installations may be obtained by considering the platform compliant for the assessment of total strain energy dissipation and assume the platform fixed at deck level when the collision response is evaluated.

Es,i

Es,s Ship

Installation

dwi

Figure 3-3 Dissipation of strain energy in ship and platform

E s = E s,s + E s,i = ∫

w s, max 0

R s dw s + ∫

w i, max 0

R i dw i

(3.4)

As the load level is not known a priori an incremental procedure is generally needed. The load-deformation relationships for the ship and the installation are often established independently of each other assuming the other object infinitely rigid. This method may have, however, severe limitations; both structures will dissipate some energy regardless of the relative strength. Often the stronger of the ship and platform will experience less damage and the softer more damage than what is predicted with the approach described above. As the softer structure deforms the impact force is distributed over a larger contact area. Accordingly, the resistance of the strong structure increases. This may be interpreted as an "upward" shift of the resistance curve for the stronger structure (refer Figure 3-3 ). Care should be exercised that the load-deformation curves calculated are representative for the true, interactive nature of the contact 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 with stern roller. The curves for broad side and stern end impacts are based upon

DET NORSKE VERITAS

Recommended Practice DNV-RP-C204, October 2010 Page 12

collision events, e.g. impact against tubular braces.

penetration of an infinitely rigid, vertical cylinder with a given diameter and may be used for impacts against jacket legs (D = 1.5 m) and large diameter columns (D = 10 m).

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.

The curve for stern corner impact is based upon penetration of an infinitely rigid cylinder and may be used for large diameter column impacts.

For bow impacts against jacket braces, reference is made to Section 3.5.3.

In lieu of more accurate calculations the curves in Figure 3-4 may be used for square-rounded columns.

For supply vessels and merchant vessels in the range of 25000 tons displacement, the force deformation relationships given in Figure 3-5 may be used for impacts against jacket legs with diameter 1.5 m – 2.5 m.

The curve for bow impact is based upon collision with an infinitely rigid, plane wall and may be used for large diameter column impacts, but should not be used for significantly different

50 D

Broad side D = 10 m = 1.5 m

Impact force (MN)

40 30 20

Stern corner

Stern end D = 10 m = 1.5 m

D

10 Bo Bow

0 0

1

2 Indentation (m)

3

4

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

80

40 Design curve - plane wall

30

Energy Contact force

40

with bulb

20

20

10 no bulb

0

0 0

1

2

3

Deformation [m] Figure 3-5 Force -deformation relationship for bow with and without bulb (2-5.000 dwt)

DET NORSKE VERITAS

4

5

Force [MN]

Energy [MJ]

60

Recommended Practice DNV-RP-C204, October 2010 Page 13

70

Bulb force

60

10

Force [MN]

50

8

b

40

6

30

a

20

4

b

2

10

a

0 0

1

2

3

4

Contact dimension [m]

12

0

5

3.5.2 Force contact area for strength design of large diameter columns.

6

Deformation [m] 70

16

Force [MN]

14

50

a

12

40

10

30

8

b

20

6

b Force superstructure

10 0 0

1

2

4 2

a

3

Contact dimension [m]

18

60

0

4

5

Deformation [m]

800

Force [MN]

120 100

600 500

80

400

60

300

40

200

20 0

100 0 0

1

2

3 4 5 6 Deformation [m]

7

8

Contact dimensions [m]

a b

6

b

2

a

0 0

1

2

3 4 5 Deformation [m]

6

7

10

If strength design is not aimed for - and in lieu of more accurate assessment (e.g. nonlinear finite element analysis) - all strain energy has to be assumed dissipated by the column, corresponding to indentation by an infinitely rigid stern corner. For typical supply vessels bows and bows of merchant vessels of similar size (i.e. 2-5000 tons displacement), energy dissipation in ship bow may be taken into account provided that the collapse resistance in bending for the brace, R0, see Section 3.7 is according to the values given in Table 3-3. The figures are valid for normal bows without ice strengthening and for brace diameters < 1.25 m. The values should be used as step functions, i.e. interpolation for intermediate resistance levels is not allowed. If contact location is not governed by operation conditions, size of ship and platform etc., the values for arbitrary contact location shall be used. (see also Ch.8, Comm. 3.5.3).

10

4

1.1

3.5.3 Energy dissipation is ship bow

12

8

Table 3-1 Local concentrated collision force -evenly distributed over a rectangular area. Stern corner impact Contact area Force (MN) a (m) b (m) b 0.35 0.65 3.0 0.35 1.65 6.4 a 0.20 1.15 5.4

2.0

700

Force Energy

Energy [MJ]

140

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, large diameter columns have to resist local concentrations (subsets) of the collision force, given for stern corner impact in Table 31 and stern end impact in Table 3-2.

Table 3-2 Local concentrated collision force -evenly distributed over a rectangular area. Stern end impact Contact area Force (MN) a (m) b (m) b 0.6 0.3 5.6 a 0.9 0.5 7.5

Figure 3-6 Force -deformation relationship for tanker bow impact (~ 125.000 dwt)

160

Force-deformation relationships for tanker bow impact are given in Figure 3-6 for the bulbous part and the superstructure, respectively, and for the bulb of a VLCC in Figure 3-7. The curves may be used provided that the impacted structure (e.g. stern of floating production vessels) does not undergo substantial deformation i.e. strength design requirements are complied with. If this condition is not met interaction between the bow and the impacted structure shall be taken into consideration. Non-linear finite element methods or simplified plastic analysis techniques of members subjected to axial crushing shall be employed, see Ch.7 /3/, /4/.

8

Figure 3-7 Force -deformation relationship and contact area for the bulbous bow of a VLCC (~ 340.000 dwt)

Table 3-3 Energy dissipation in bow versus brace resistance Energy dissipation in bow Contact location if brace resistance R0 > 3 MN > 6 MN > 8 MN > 10 MN Above bulb 1 MJ 4 MJ 7 MJ 11 MJ First deck 0 MJ 2 MJ 4 MJ 17 MJ First deck - oblique brace 0 MJ 2 MJ 4 MJ 17 MJ Between forcastle/first 1 MJ 5 MJ 10 MJ 15 MJ deck Arbitrary location 0 MJ 2 MJ 4 MJ 11 MJ

In addition, the brace cross-section must satisfy the following

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Recommended Practice DNV-RP-C204, October 2010 Page 14

compactness requirement f y t 1.5 D 0.5 ≥

2 ⋅ factor 3

(3.5)

where factor is the required resistance in [MN] given in Table 3-3. See Section 3.6 for notation. If the brace is designed to comply with these provisions, special care should be exercised that the joints and adjacent structure is strong enough to support the reactions from the brace.

3.6 Force-deformation relationships for denting of tubular members The contribution from local denting to energy dissipation is small for brace members in typical jackets and should be neglected. The resistance to indentation of unstiffened tubes may be taken from Figure 3-8. Alternatively, the resistance may be calculated from Equation (3.6): 20 18 16

b/D =

R/(kRc)

14 12 10 8

2 1 0.5 0

4 2 0 0.1

0.2

0.3

0.4

0.5

wd/D

⎞ ⎟⎟ ⎠

l 1 1 = + k k node 2EA

t2 4

(3.6)

k = 1.0

N Sd ≤ 0.2 N Rd

⎛N ⎞ k = 1.0 − 2⎜⎜ Sd − 0.2 ⎟⎟ ⎝ N Rd ⎠

0.2
0.05. The width of contact area is in theory equal to the height of the vertical, plane section of the ship side that is assumed to be in contact with the tubular member. For large widths, and depending on the relative rigidity of the cross-section and the ship side, it may be unrealistic to assume that the tube is subjected to flattening over the entire contact area. In lieu of more accurate calculations it is proposed that the width of contact area be taken equal to the diameter of the hit cross-section (i.e. B/D = 1).

knode = axial stiffness of the node with the considered member removed. This may be determined by introducing 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.

N Sd < 0.6 N Rd 0.6 ≤

N Sd N Rd

The following notation applies: R0 =

w=

NSd NRd B wd

= = = =

(3.7)

design axial compressive force design axial compressive resistance width of contactarea dent depth

c=

4c1M P l

w c1 wc

4c 1 kw c f y Al

c1 = 2

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plastic collapse resistance in bending for the member, for the case that contact point is at midspan non-dimensional deformation

2

non-dimensional spring stiffness for clamped beams

Recommended Practice DNV-RP-C204, October 2010 Page 15

c1 = 1

member the force-deformation relationship may be interpolated from the curves for pinned ends and clamped ends:

for pinned beams

wc =

D 2

characteristic deformation for tubular beams

wc =

1 .2 WP A

characteristic deformation for stiffened plating

where

0≤ξ=

WP

=

plastic section modulus

l

=

member length

R = ζR clamped + (1 − ζ) R pinned

(3.8)

actual

For non-central collisions the force-deformation relationship may be taken as the mean value of the force-deformation curves for central collision with member half length equal to the smaller and the larger portion of the member length, respectively. For members where the plastic moment capacity of adjacent members is smaller than the moment capacity of the impacted member the force-deformation relationship may be interpolated from the curves for pinned ends and clamped ends: For non-central collisions the force-deformation relationship may be taken as the mean value of the force-deformation curves for central collision with member half length equal to the smaller and the larger portion of the member length, respectively. For members where the plastic moment capacity of adjacent members is smaller than the moment capacity of the impacted

R0 R0

R0 −1 ≤ 1 MP 4 l

actual

actual

(3.9)

resistance by bending action of beam account= Plastic ing for actual bending resistance of adjacent members

=

4M P + 2M P1 + 2M P2 l

(3.10)

M Pj = ∑ M Pj,i ≤ M P

(3.11)

i

i = adjacent member no i j = end number {1,2} MPj,i = Plastic bending resistance for member number i at end j. Elastic, rotational flexibility of the node is normally of moderate significance.

6,5 6 5,5 5 4,5 4

R/R0

Bending & membrane Membrane only

0.2 0,3

3,5

F

0.1

(collision load)

0.5

3 2,5

c =∞

2

k

1

w

0.05

1,5 1 0,5 0 0

0,5

1

1,5

2

2,5

3

3,5

w

Deformation Figure 3-9 Force-deformation relationship for tubular beam with axial flexibility

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k

4

Recommended Practice DNV-RP-C204, October 2010 Page 16

5 4,5 4

Bending & membrane Membrane only

3,5

R/R0

3

F

0.1

2,5

0.2

0.5

c =∞

2

k

k

w

1

1,5

(collision load)

0

1 0,5 0 0

0,5

1

1,5

Deformation

2

2,5

3

3,5

4

w

Figure 3-10 Force-deformation relationship for stiffened plate with axial flexibility

3.7.3 Support capacity smaller than plastic bending moment of the beam

For beams where the plastic moment capacity of adjacent members is smaller than the moment capacity of the impacted beam, the force-deformation relationship, R*, may be derived from the resistance curve, R, for beams where the plastic moment capacity of adjacent members is larger than the moment capacity of the impacted beam (Section 3.7.2), using the expression:

R = R + (R 0 − R ) *

* 0

w w lim

,

w w lim

(3.12)

w

R* = R

≤ 1 .0

w lim

≥ 1.0

where R0 = Plastic bending resistance with clamped ends (c1 = 2) – moment capacity of adjacent members larger than the plastic bending moment of the beam * R 0 = Plastic bending resistance - moment capacity of adjacent members at one or both ends smaller than the plastic bending moment of the beam

i = j = MPj,i= wlim =

adjacent member no i end number {1,2} Plastic bending resistance for member no. i 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 = ∞, w lim = π w for 2 tubular beams and wlim = 1.2 w for stiffened plate

3.7.4 Bending capacity of dented tubular members The reduction in plastic moment capacity due to local denting shall be considered for members in compression or moderate tension, but can be neglected for members entering the fully plastic membrane state. Conservatively, the flat part of the dented section according to the model shown in Figure 3-11 may be assumed non-effective. This gives: M red θ 1 = cos − sinθ MP 2 2 M P = f y D2 t ⎛ 2w d ⎞ θ = arccos⎜1 − ⎟ D ⎠ ⎝

wd = dent depth as defined in Figure 3-11.

4M P + 2M P1 + 2M P2 R *0 = l

(3.13)

M Pj = ∑ M Pj,i ≤ M P

(3.14)

i

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(3.15)

Recommended Practice DNV-RP-C204, October 2010 Page 17

1

wd

0,8

Mred/MP

D 0,6 0,4 0,2 0 0

0,2

0,4

0,6

0,8

1

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 maintain moment capacity during significant plastic rotation it is recommended that cross-sections be proportioned to section type I requirements, defined in DNV-OS-C101. Initiation of local buckling does, however, not necessarily imply that the capacity with respect to energy dissipation is exhausted, particularly for type I and type II cross-sections. The degradation of the cross-sectional resistance in the postbuckling range may be taken into account provided that such information is available, refer Ch.8, Comm. 3.10.1. For members undergoing membrane stretching a lower bound to the post-buckling load-carrying capacity may be obtained by using the load-deformation curve for pure membrane action. 3.10.2 Local buckling Tubular cross-sections: Buckling does not need to be considered for a beam with axial restraints if the following condition is fulfilled:

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

3.8 Strength of connections Provided that large plastic strains can develop in the impacted member, the strength of the connections that the member frames into should be checked. The resistance of connections should be taken from ULS requirements in relevant standards. For braces reaching the fully plastic tension state, the connection shall be checked for a load equal to the axial capacity of the member. The design axial stress shall be assumed equal to the ultimate tensile strength of the material. If the axial force in a tension member becomes equal to the axial capacity of the connection, the connection has to undergo gross deformations. The energy dissipation will be limited and rupture should be considered at a given deformation. A safe approach is to assume failure (disconnection of the member) once the axial force in the member reaches the axial capacity of the connection. If the capacity of the connection is exceeded in compression and bending, this does not necessarily mean failure of the member. The post-collapse strength of the connection may be taken into account provided that such information is available.

3.9 Strength of adjacent structure The strength of structural members adjacent to the impacted member/sub-structure must be checked to see whether they can provide the support required by the assumed collapse mechanism. If the adjacent structure fails, the collapse mechanism must be modified accordingly. Since, the physical behaviour becomes more complex with mechanisms consisting of an increasing number of members it is recommended to consider a design which involves as few members as possible for each collision scenario.

3.10 Ductility limits 3.10.1 General The maximum energy that the impacted member can dissipate will – ultimately - be limited by local buckling on the compressive side or fracture on the tensile side of cross-sections undergoing finite rotation. If the member is restrained against inward axial displacement, any local buckling must take place before the tensile strain due to membrane elongation overrides the effect of rotation induced compressive strain. If local buckling does not take place, fracture is assumed to

⎛ 14c f f y β≤⎜ ⎜ c1 ⎝

1

⎛ κl ⎞ ⎜ ⎟ ⎜d ⎟ ⎝ c⎠

2

⎞3 ⎟ ⎟ ⎠

(3.16)

where β=

Dt 235 f y

(3.17)

axial flexibility factor ⎛ c ⎞⎟ cf = ⎜ ⎜1 + c ⎟ ⎠ ⎝

dc

= = = = =

c1 c

2

(3.18)

characteristic dimension D for circular cross-sections 2 for clamped ends 1 for pinned ends non-dimensional spring stiffness, refer Section 3.7.2.

κ l ≤ 0.5 l = the smaller distance from location of collision load to adjacent joint If this condition is not met, buckling may be assumed to occur when the lateral deformation exceeds ⎛ 14c f f y w 1 ⎜ = 1− 1− d c 2c f ⎜⎜ c 1β 3 ⎝

2 ⎞ ⎛ κl ⎞ ⎟ ⎜ ⎟ ⎟ ⎜d ⎟ ⎝ c⎠ ⎟ ⎠

(3.19)

For small axial restraint (c < 0.05) the critical deformation may be taken as w 3.5f y = dc c 1β 3

⎛ κl ⎞ ⎜ ⎟ ⎜d ⎟ ⎝ c⎠

2

(3.20)

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 used with dc =

characteristic dimension for local buckling, equal to twice the distance from the plastic neutral axis in bending to the extreme fibre of the cross-section = h height of cross-section for symmetric I –profiles = 2hw for stiffened plating (for simplicity)

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Recommended Practice DNV-RP-C204, October 2010 Page 18

For flanges subjected to compression; β = 2.5

bf t f

bf t f

β=3

type I cross-sections

235 f y 235 f y

type II and type III cross-sections

(3.21) (3.22)

For webs subjected to bending β = 0.7 β = 0.8

bf tf hw tw

= = = =

hw tw

type I cross-sections

235 f y hw tw 235 f y

type I and type III cross-sections

(3.23) (3.24)

3.10.4 Tensile fracture in yield hinges When the force deformation relationships for beams given in Section 3.7.2 are used rupture may be assumed to occur when the deformation exceeds a value given by

⎞ 4c c ε c ⎛ w = 1 ⎜ 1 + w f cr − 1⎟ ⎜ ⎟ d c 2c f ⎝ c1 ⎠ where the following factors are defined; Displacement factor

1 cw = c1

⎛ ⎛ 1 ⎞ ⎛ ⎜ c lp ⎜1 − c lp ⎟ + 4⎜1 − W ⎜ ⎝ 3 ⎠ ⎜ W P ⎝ ⎝

— — — —

⎛ c ⎞⎟ cf = ⎜ ⎜1+ c ⎟ ⎝ ⎠

material toughness presence of defects strain rate presence of strain concentrations

The critical strain for plastic deformations of sections containing defects need to be determined based on fracture mechanics methods. Welds normally contain defects and welded joints are likely to achieve lower toughness than the parent material. For these reasons structures that need to undergo large plastic deformations should be designed in such a way that the plastic straining takes place outside the weld. In ordinary full penetration welds, the overmatching weld material will ensure that minimal plastic straining occurs in the welded joints even in cases with yielding of the gross cross section of the member. In such situations, the critical strain will be in the parent material 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 strains as such. Therefore, strain levels should be assessed by means of adequate analytic models of the strain distributions in the plastic zones or by non-linear finite element analysis with a sufficiently detailed mesh in the plastic zones. (For information 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 average strain in axially loaded plate material may be used in conjunction with nonlinear finite element analysis or simple plastic analysis t l

where: t

l

= plate thickness = length of plastic zone. Minimum 5t

⎞ ⎛ κl ⎞ ⎟⎜ ⎟ ⎟⎜d ⎟ ⎠⎝ c ⎠

⎛ ε cr ⎞ W ⎜ − 1⎟ H ⎜ εy ⎟ WP ⎝ ⎠ c lp = ⎛ ε cr ⎞ W ⎜ − 1⎟ H +1 ⎜ εy ⎟ WP ⎝ ⎠

axial flexibility factor

ε cr = 0.02 + 0.65

⎞ εy ⎟⎟ ⎠ ε cr

2

(3.27)

plastic zone length factor

flange width flange thickness web height web thickness

3.10.3 Tensile fracture The degree of plastic deformation or critical strain at fracture will show a significant scatter and depends upon the following factors:

— — — — —

(3.26)

(3.25)

(3.28)

2

(3.29)

non-dimensional plastic stiffness H=

Ep E

=

c1

1 ⎛⎜ f cr − f y ⎞⎟ E ⎜⎝ ε cr − ε y ⎟⎠

c

= = =

κl

≤

W WP εcr

= = =

εy =

fy E

fy fcr

(3.30)

2 for clamped ends 1 for pinned ends non-dimensional spring stiffness, refer Section 3.7.2 0.5 l the smaller distance from location of collision load to adjacent joint elastic section modulus plastic section modulus critical strain for rupture (see Table 3-4 for recommended values)

=

yield strain

= =

yield strength strength corresponding to εcr

The characteristic dimension shall be taken as: dc

= D = 2hw = h = 2 (h − zplast)

diameter of tubular beams twice the web height for stiffened plates (se·t > As) height of cross-section for symmetric I-profiles for unsymmetrical I-profiles

zplast = smaller distance from flange to plastic neutral axis of cross-section For small axial restraint (c < 0.05) the critical deformation may

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Recommended Practice DNV-RP-C204, October 2010 Page 19

be taken as w = c w ε cr dc

(3.31)

The critical strain εcr and corresponding strength fcr should be selected so that idealised bi-linear stress-strain relation gives reasonable results, see Ch.8, Commentary. For typical steel material grades the following values are proposed: Table 3-4 Proposed values for εcr and H for different steel grades Steel grade εcr H S 235 20 % 0.0022 S 355 15 % 0.0034 S 460 10 % 0.0034

3.11 Resistance of large diameter, stiffened columns 3.11.1 General Impact on a ring stiffener as well as midway between ring stiffeners shall be considered. Plastic methods of analysis are generally applicable. 3.11.2 Longitudinal stiffeners For ductile design the resistance of longitudinal stiffeners in the beam mode of deformation can be calculated using the procedure described for stiffened plating, Section 3.7. For strength design against stern corner impact, the plastic bending moment capacity of the longitudinal stiffeners has to comply with the requirement given in Figure 3-12, on the assumption that the entire load given in Table 3-1 is taken by one stiffener.

Plastic bending capacity (MNm)

3

2

1

0 1

2

3

(3.32)

where wc =

WP Ae

3.12 Energy dissipation in floating production vessels For strength design the side or stern shall resist crushing force of the bow of the off-take tanker. In lieu of more accurate calculations the force-deformation curve given in Section 3.5.2 may 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 procedure described in Section 3.7.2. (See Ch.8, Comm. 3.12 on resistance 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, because the collision load is typically an order of magnitude smaller than the resultant design wave force. Linear analysis often suffices to check that global integrity is maintained. The installation should be checked for the maximum collision force. For installations responding predominantly statically the maximum collision force occurs at maximum deformation. For structures responding predominantly impulsively the maximum collision force occurs at small global deformation of the platform. An upper bound to the collision force is to assume that the installation is fixed with respect to global displacement. (e.g. jack-up fixed with respect to deck displacement).

4.1 General

3.11.3 Ring stiffeners In lieu of more accurate analysis the plastic collapse load of a ring-stiffener can be estimated from:

wc D

3.11.4 Decks and bulkheads Calculation of energy dissipation in decks and bulkheads has to be based upon recognised methods for plastic analysis of deep, axial crushing. It shall be documented that the collapse mechanisms assumed yield a realistic representation of the true deformation field.

4. Dropped Objects

Figure 3-12 Required bending capacity of longitudinal stiffeners

4 2M P

Effective flange of shell plating: Use effective flange of stiffened plates, see Chapter 6. For ductile design it can be assumed that the resistance of the ring stiffener is constant and equal to the plastic collapse load, provided that requirements for stability of cross-sections are complied with, refer Section 3.10.2.

4

Distance between ring stiffeners (m)

F0 =

effective shell flange WP = plastic section modulus of ring stiffener including effective shell flange Ae = area of ring stiffener including effective shell flange

=characteristic deformation of ring stiffener

D = column radius MP = plastic bending resistance of ring-stiffener including

The dropped object load is characterised by a kinetic energy, governed by the mass of the object, including any hydrodynamic added mass, and the velocity of the object at the instantof impact. In most cases the major part of the kinetic energy has to be dissipated as strain energy in the impacted component and, possibly, in the dropped object. Generally, this involves large plastic strains and significant structural damage to the impacted component. The strain energy dissipation is estimated from force-deformation relationships for the component and the object, where the deformations in the component shall comply with ductility and stability requirements. The load bearing function of the installation shall remain intact with the damages imposed by the dropped object load. In addition, damaged condition should be checked if relevant, see Section 2.2. Dropped objects are rarely critical to the global integrity of the

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Recommended Practice DNV-RP-C204, October 2010 Page 20

installation and will mostly cause local damages. The major threat to global integrity is probably puncturing of buoyancy tanks, which could impair the hydrostatic stability of floating installations. Puncturing of a single tank is normally covered by the general requirements to compartmentation and watertight integrity given in DNV-OS-C301. The structural effects from dropped objects may either be determined by non-linear dynamic finite element analyses or by energy considerations combined with simple elastic-plastic methods as given in Sections 4.2 - 4.5. If non-linear dynamic finite element analysis is applied all effects described in the following paragraphs shall either be implicitly covered by the modelling adopted or subjected to special considerations, whenever relevant.

4.2 Impact velocity The kinetic energy of a falling object is given by: E kin =

1 mv 2 2

(in air)

and E kin =

a

1 (m + a )v 2 (in water) 2

(4.2)

= hydrodynamic added mass for considered motion

For impacts in air the velocity is given by (4.3)

v = 2gs

s v

= travelled distance from drop point = vo at sea surface

For objects falling rectilinearly in water the velocity depends upon the reduction of speed during impact with water and the falling distance relative to the characteristic distance for the object.

(4.1)

-3

In air

-2

Velocity [v/vt]

-1 0

0,5

1

1,5

2

2,5

3

3,5

4

s

Distance [s/sc]

0 1 2

In water

3 4 5 6 7

Figure 4-1 Velocity profile for objects falling in water

The loss of momentum during impact with water is given by mΔv = ∫ d F(t)dt t

(4.4)

0

F(t) = force during impact with sea surface After the impact with water the object proceeds with the speed v = v 0 − Δv Assuming that the hydrodynamic resistance during fall in water is of drag type the velocity in water can be taken from Figure 4-1 where

2g(m − ρ w V) vt = ρ w Cd A p

terminal velocity for the = object (drag force and buoyancy force balance the gravity force)

m+a sc = = ρ w Cd A p

a ) m = characteristic distance ρ V 2g(1 − w ) m v t 2 (1 +

ρw = density of sea water Cd = hydrodynamic drag coefficient for the object in the considered motion m = mass of object Ap = projected cross-sectional area of the object V = object displacement The major uncertainty is associated with calculating the loss of momentum during impact with sea surface, given by Equation (4.4). However, if the travelled distance is such that the velocity is close to the terminal velocity, the impact with sea surface is of little significance. Typical terminal velocities for some typical objects are given

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Recommended Practice DNV-RP-C204, October 2010 Page 21

fy = characteristic yield strength

in Table 4-1. Table 4-1 Terminal velocities for objects falling in water Item Mass Terminal velocity [kN] [m/s] Drill collar 28 23-24 Winch, 250 Riser pump 100 BOP annular preventer 50 16 Mud pump 330 7

c = −e

⎛ d ⎞ − 2.5⎜⎜ 1− ⎟⎟ ⎝ 2r ⎠

R = πdtτ = contact force for τ ≤τ cr refer Section 4.5.1 for τ cr

Rectilinear motion is likely for blunt objects and objects which do not rotate about their longitudinal axis. Bar-like objects (e.g. pipes) which do not rotate about their longitudinal axis may execute lateral, damped oscillatory motions as illustrated in Figure 4-1.

m i = ρ p πr 2 t

= mass of plate enclosed by hinge circle

m ρp d

= mass of dropped object = mass density of steel plate = smaller diameter at threaded end of drill collar = smaller distance from the point of impact to the plate boundary defined by adjacent stiffeners/girders, refer Figure 4-3.

r

4.3 Dissipation of strain energy The structural response of the dropped object and the impacted component can formally be represented as load-deformation relationships as illustrated in Figure 4-2. The part of the impact energy dissipated as strain energy equals the total area under the load-deformation curves. E s = E s,o + E s,i = ∫

w o, max 0

R o dw o + ∫

w i , max 0

(4.5)

R i dw i

As the load level is not known a priori an incremental approach is generally required. Often the object can be assumed to be infinitely rigid (e.g. axial impact from pipes and massive objects) so that all energy is to be dissipated by the impacted component.

Ro

Ri

dwo

Object

Installation

r

r

r

Figure 4-3 Definition of distance to plate boundary

4.4.2 Stiffeners/girders In lieu of more accurate calculations stiffeners and girders subjected to impact with blunt objects may be analysed with resistance models given in Section 6.10. 4.4.3 Dropped object Calculation of energy dissipation in deformable dropped objects shall be based upon recognised methods for plastic analysis. It shall be documented that the collapse mechanisms assumed yield a realistic representation of the true deformation field.

Es,i

Es,o

For validity range of design formula reference is given to Ch.8, Comm. 4.4.1.

4.5 Limits for energy dissipation

dwi

Figure 4-2 Dissipation of strain energy in dropped object and installation

4.5.1 Pipes on plated structures The maximum shear stress for plugging of plates due to drill collar impacts may be taken as

If the object is assumed to be deformable, the interactive nature of the deformation of the two structures should be recognised.

t⎞ ⎛ τ cr = f u ⎜ 0.42 + 0.41 ⎟ d ⎝ ⎠

4.4 Resistance/energy dissipation

f u = ultimate material tensile strength

4.4.1 Stiffened plates subjected to drill collar impact The energy dissipated in the plating subjected to drill collar impact is given by

4.5.2 Blunt objects For stability of cross-sections and tensile fracture, refer Section 3.10.

E sp

m ⎞ R2 ⎛ = ⎜1 + 0.48 i ⎟⎟ ⎜ 2k ⎝ m ⎠

(4.7)

2

(4.6)

5.1 General

where:

⎛ ⎜ 1 + 5 d − 6c 2 + 6.25⎛⎜ d ⎞⎟ ⎜ 1 r ⎝ 2r ⎠ k = πf y t ⎜ 2 2 (1 + c) ⎜ ⎜ ⎝

5. Fire

2

⎞ ⎟ ⎟ : stiffness of plate ⎟ enclosed by hinge circle ⎟ ⎟ ⎠

The characteristic fire structural load is temperature rise in exposed members. The temporal and spatial variation of temperature depends on the fire intensity, whether or not the structural members are fully or partly engulfed by the flame and to what extent the members are insulated. Structural steel expands at elevated temperatures and internal stresses are developed in redundant structures. These stresses

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are most often of moderate significance with respect to global integrity. The heating causes also progressive loss of strength and stiffness and is, in redundant structures, accompanied by redistribution of forces from members with low strength to members that retain their load bearing capacity. A substantial loss of load-bearing capacity of individual members and subassemblies may take place, but the load bearing function of the installation 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

tions compressive members shall be modelled with an initial, sinusoidal imperfection with amplitude given by Elastic-perfectly plastic material model, refer Figure 6-4 : e* 1 = l π

Elasto-plastic material models, refer Figure 6-4 :

e* W 1 = W π l p

— individual members — subassemblies — entire system

fy i α E z0

fy i α = E z0

1

f y Wp

π

E

AI

α

α

The assessment of fire load effect and mechanical response shall be based on either — simple calculation methods applied to individual members, — general calculation methods, or a combination. Simple calculation methods may give overly conservative results. General calculation methods are methods in which engineering principles are applied in a realistic manner to specific applications. Assessment of individual members by means of simple calculation methods should be based upon the provisions given in Ch.7 /2/ Eurocode 3 Part 1.2. /2/ . Assessment by means of general calculation methods shall satisfy the provisions given in Ch.7 /2/ Eurocode 3 Part1.2, Section 4.3. In addition, the requirements given in this section for mechanical response analysis with nonlinear finite element methods shall be complied with. Assessment of ultimate strength is not needed if the maximum steel temperature is below 400°C, but deformation criteria may have to be checked for impairment of main safety functions.

5.2 General calculation methods Structural analysis methods for non-linear, ultimate strength assessment may be classified as — stress-strain based methods — stress-resultants based (yield/plastic hinge) methods Stress-strain based methods are methods where non-linear material behaviour is accounted for on fibre level. Stress-resultants based methods are methods where non-linear material behaviour is accounted for on stress-resultants level based upon closed form solutions/interaction equation for cross-sectional forces and moments.

5.3 Material modelling In stress-strain based analysis temperature dependent stressstrain relationships given in Ch.7 /2/ Eurocode 3, Part 1.2, Section 3.2 may be used. For stress resultants based design the temperature reduction of the elastic modulus may be taken as kE,θ according to Ch.7 / 2/ Eurocode 3. The yield stress may be taken equal to the effective yield stress, fy,θ. The temperature reduction of the effective yield stress may be taken as ky,θ. Provided that the above requirements are complied with creep does need explicit consideration.

5.4 Equivalent imperfections To account for the effect of residual stresses and lateral distor-

= 0.5 for fire exposed members according to column curve c, Ch.7 /2/ Eurocode 3 i = radius of gyration z0 = distance from neutral axis to extreme fibre of crosssection WP = plastic section modulus W = elastic section modulus A = cross-sectional area I = moment of inertia e* = amplitude of initial distortion l = member length The initial out-of-straightness should be applied on each physical member. If the member is modelled by several finite elements the initial out-of-straightness should be applied as displaced nodes. The initial out-of-straightness shall be applied in the same direction as the deformations caused by the temperature gradients.

5.5 Empirical correction factor The empirical correction factor of 1.2 should be accounted for in calculating the critical strength in compression and bending for design according to Ch.7 /2/ Eurocode 3, refer Ch.8, Comm. A.5.5.

5.6 Local cross sectional buckling If shell modelling is used, it shall be verified that the software and the modelling is capable of predicting local buckling with sufficient accuracy. If necessary, local shell imperfections have to be introduced in a similar manner to the approach adopted for lateral distortion of beams If beam modelling is used local cross-sectional buckling shall be given explicit consideration. In lieu of more accurate analysis cross-sections subjected to plastic deformations shall satisfy compactness requirements given 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 considerations shall be performed. The load-bearing capacity will be reduced significantly after the onset of buckling, but may still be significant. A conservative approach is to remove the member from further analysis. Compactness requirements for type I and type I cross-sections may be disregarded provided that the member is capable of developing significant membrane forces.

5.7 Ductility limits 5.7.1 General The ductility of beams and connections increase at elevated temperatures compared to normal conditions. Little informa-

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tion exists. 5.7.2 Beams in bending In lieu of more accurate analysis requirements given in Section 3.10 shall be complied with. 5.7.3 Beams in tension In lieu of more accurate analysis an average elongation of 3% of the member length with a reasonably uniform temperature can be assumed. Local temperature peaks may localise plastic strains. It is considered to be to the conservative side to use critical strains for steel under normal temperatures. See Section 3.10 and 3.10.4.

5.8 Capacity of connections

The load bearing function of the installation shall remain intact with the damages imposed by the explosion loads. In addition, damaged condition should be checked if relevant, see Section 2.2.

6.2 Classification of response The response of structural components can conveniently be classified into three categories according to the duration of the explosion pressure pulse, td, relative to the fundamental period of vibration of the component, T: Impulsive domain Dynamic domain Quasi-static domain

td/T < 0.3 0.3 < td/T < 3 3 < td/T

In lieu of more accurate calculations the capacity of the connection can be taken as: Rθ = ky,θ R0 where

Impulsive domain: The response is governed by the impulse defined by

R0 = capacity of connection at normal temperature ky,θ = temperature reduction of effective yield stress for maximum temperature in connection

Hence, the structure may resist a very high peak pressure provided that the duration is sufficiently small. The maximum deformation, wmax, of the component can be calculated iteratively from the equation

6. Explosions

I = 2m eq ∫

I = ∫ F(t )dt td

(6.1)

0

w max

0

6.1 General Explosion loads are characterised by temporal and spatial pressure distribution. The most important temporal parameters are rise time, maximum pressure and pulse duration. For components and sub-structures the explosion pressure shall normally be considered uniformly distributed. On global level the spatial distribution is normally non-uniform both with respect to pressure and duration. The response to explosion loads may either be determined by non-linear dynamic finite element analysis or by simple calculation models based on Single Degree Of Freedom (SDOF) analogies and elastic-plastic methods of analysis. If non-linear dynamic finite element analysis is applied all effects described in the following paragraphs shall either be implicitly covered by the modelling adopted or subjected to special considerations, whenever relevant. In the simple calculation models the component is transformed to a single spring-mass system exposed to an equivalent load pulse by means of suitable shape functions for the displacements in the elastic and elastic-plastic range. The shape functions allow calculation of the characteristic resistance curve and equivalent mass in the elastic and elastic-plastic range as well as the fundamental period of vibration for the SDOF system in the elastic range. Provided that the temporal variation of the pressure can be assumed to be triangular, the maximum displacement of the component can be calculated from design charts for the SDOF system as a function of pressure duration versus fundamental period of vibration and equivalent load amplitude versus maximum resistance in the elastic range. The maximum displacement must comply with ductility and stability requirements for the component.

R (w )dw

(6.2)

where R(w)= force-deformation relationship for the component meq = equivalent mass for the component. Quasi-static-domain: The response is governed by the peak pressure and the rise time of the pressure relative to the fundamental period of vibration. If the rise time is small the maximum deformation of the component can be solved iteratively from the equation: w max =

1 Fmax

∫

w max

0

R (w )dw

(6.3)

If the rise time is large the maximum deformation can be solved from the static condition Fmax = R(w max )

(6.4)

Dynamic domain: The response has to be solved from numerical integration of the dynamic equations of equilibrium.

6.3 Recommended analysis models for stiffened panels Various failure modes for a stiffened panel are illustrated in Figure 6-1. Suggested analysis model and reference to applicable resistance functions are listed in Table 6.1. Application of a Single Degree of Freedom (SDOF) model in the analysis of stiffeners/girders with effective flange is implicitly based on the assumption that dynamic interaction between the plate flange and the profile can be neglected.

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Figure 6-1 Failure modes for two-way stiffened panel Table 6-1 Analysis models Failure mode Elastic-plastic deformation of plate Stiffener plastic – plate elastic Stiffener plastic – plate plastic Girder plastic – stiffener and plating elastic Girder plastic – stiffener elastic – plate plastic Girder and stiffener plastic – plate elastic

Simplified analysis model Resistance models SDOF Section 6.9 SDOF Stiffener: Section 6.10.1-2. Plate: Section 6.9.1 SDOF Stiffener: Section 6.10.1-2. Plate: Section 6.9 SDOF Girder: Section 6.10.1-2 Plate: Section 6.9 SDOF Girder: Section 6.10.1-2 Plate: Section 6.9

MDOF

Comment

Elastic, effective flange of plate Effective width of plate at mid span. Elastic, effective flange of plate at ends. Elastic, effective flange of plate with concentrated loads (stiffener reactions). Stiffener mass included. Effective width of plate at girder mid span and ends. Stiffener mass included

Girder and stiffener: Dynamic reactions of stiffeners → loading for girders Section 6.10.1-2 Plate: Section 6.9 Girder and stiffener plastic MDOF Girder and stiffener: Dynamic reactions of stiffeners → loading for girders – plate plastic Section 6.10.1-2 Plate: Section 6.9 By girder/stiffener plastic is understood that the maximum displacement wmax exceeds the elastic limit wel

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Recommended Practice DNV-RP-C204, October 2010 Page 25

6.4 SDOF system analogy Biggs method: For many practical design problems it is convenient to assume that the structure - exposed to the dynamic pressure pulse - ultimately attains a deformed configuration comparable to the static deformation pattern. Using the static deformation pattern as displacement shape function, i.e.

m&y& + ky = f (t )

(6.5)

φ(x) y(t) m = ∫ mφ(x ) dx + ∑ M i φ i 2

2

= displacement shape function = displacement amplitude = generalized mass

i

f ( t ) = ∫ q(t)φ(x )dx + ∑ Fi φ i

= generalized load

k = ∫ EIφ , xx (x ) dx

= generalized elastic bending stiffness

2

l

k=0

k = ∫ Nφ , x (x ) dx 2

l

m Mi q Fi xi

= generalized plastic bending stiffness (fully developed mechanism) = generalized membrane stiffness (fully plastic: N = NP) = distributed mass = concentrated mass = explosion load = concentrated load (e.g. support reactions) = position of concentrated mass/load

φ i = φ (x = x i ) The equilibrium equation can alternatively be expressed as: (K l m, u M u + K l m,c M c )&y& + K(y)y = F(t)

(6.6)

where

K m,u =

K m,u

= load-mass transformation factor for uniform mass

Kl K m,u

transformation factor = load-mass for concentrated mass

Kl

∫ mϕ (x) l

Mu

∑M ϕ i

K m,c =

M

i

Mc

2

i

= load transformation factor for concentrated load

i

F

= total uniformly, distributed mass

dx

=

c

∑M

i

= total concentrated mass

i

F = ∫ qdx

load in case of uniformly = total distributed load

l

F=

∑F

load in case of concentrated = total load

k kl

= equivalent stiffness

i

i

ke =

T = 2π

= mass transformation factor for uniform mass mass transformation factor for = concentrated mass

m k

= 2π

K l m, u M u + K l m,c M c ke

(6.7)

The response, y(t), is - in addition to the load history - entirely governed by the total mass, load-mass factor and the characteristic stiffness. For a linear system, the load mass factor and the characteristic stiffness are constant k = k1. The response is then alternatively governed by the eigenperiod and the characteristic stiffness. For a non-linear system, the load-mass factor and the characteristic stiffness depend on the response (deformations). Nonlinear systems may often conveniently be approximated by equivalent bi-linear or tri-linear systems, see Section 6.8. In such cases the response can be expressed in terms of (see Figure 6-6 for definitions): k1

= characteristic stiffness in the initial, linear resistance domain yel = displacement at the end of the initial, linear resistance domain T = eigenperiod in the initial, linear resistance domain and, if relevant, k3

2

∑i F ϕ

The natural period of vibration for the equivalent system in the linear resistance domain is given by

i

l

K l m,u =

F

load transformation factor for = uniformly distributed load

l

the dynamic equations of equilibrium can be transformed to an equivalent single degree of freedom system:

K l m,u =

Kl =

l

M u = ∫ mdx

w (x , t ) = φ(x )y(t )

l

Kl =

∫ qϕ (x)dx

= normalised characteristic resistance in the third linear resistance domain.

Characteristic stiffness is given explicitly or can be derived from 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 equation (6.6). For standard load histories and standard resistance curves maximum displacements can be presented in design charts. Figure 6-2 shows the normalised maximum displacement of a SDOF system with a bi- (k3 = 0) or tri-linear (k3 > 0) resistance function, exposed to a triangular pressure pulse with zero rise time. When the duration of the pressure pulse relative to the eigenperiod in the initial, linear resistance range is known, the maximum displacement can be determined directly from the diagram as illustrated in Figure 6-2.

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Recommended Practice DNV-RP-C204, October 2010 Page 26

100 Static asymptote, k 3=0.2k 1

10

Impulsive asymptote, k 3=0.2k 1

k 3=0.1k 1

k 3=0.2k 1

ymax/yel for system

Elastic-perfectly plastic, k 3=0

1

td/T for system

F(t)

td

0,1 0,1

1 td/T

10

Figure 6-2 Maximum response a SDOF system to a triangular pressure pulse with zero rise time. Fmax / Rel = 2

Design charts for systems with bi- or tri-linear resistance curves subjected to a triangular pressure pulse with 0.5 td rise time is given in Figure 6.3. Curves for different rise times are given in Ch.8, Commentary Figure 8-15 to Figure 8-17. Baker's method The governing equations (6.1) and (6.2) for the maximum response in the impulsive domain and the quasi-static domain may also be used along with response charts for maximum displacement for different Fmax/Rel ratios to produce pressureimpulse (Fmax, I) diagrams - iso-damage curves - provided that the maximum pressure is known. The advantage of using iso-damage diagrams is that "backward" calculations are possible: The diagram is established on the basis of the resistance curve. The information may be used to screen explosion pressure histories and eliminate those that obviously lie in the admissible

domain. This will reduce the need for large complex simulation of explosion scenarios.

6.5 Dynamic response charts for SDOF system Transformation factors for elastic–plastic-membrane deformation of beams and one-way slabs with different boundary conditions are given in Table 6-2. Maximum displacement for a SDOF system exposed to a triangular pressure pulse with rise time of 0.5td is displayed in Figure 6.3. Maximum displacement for a SDOF system exposed to different pressure pulses are given in Ch.8, Commentary Figure 8-15 to Figure 8-17. The characteristic response of the system is based upon the resistance in the linear range, k = k1, where the equivalent stiffness is determined from the elastic solution to the actual system.

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Rel/Fmax=0.05

100

=0.1

= 0.3

= 0.5

= 0.6 = 0.7

Rel/Fmax= 0.8

ymax/yel

10

= 0.9

1

k3 = 0 k 3 = 0.1k1 k 3 = 0.2k 1 k 3 = 0.5k 1

F

R

Fmax

Rel

k3 = 0.5k1 =0.2k1

=0.1k1

= 1.0 = 1.1 = 1.2 = 1.5

k1 0.50td

td

yel

y

0.1 0.1

1

10

td/T Figure 6-3 Dynamic response of a SDOF system to a triangular load (rise time = 0.50 td)

6.6 MDOF analysis SDOF analysis of built-up structures (e.g. stiffeners supported by girders) is admissible if — the fundamental periods of elastic vibration are sufficiently separated — the response of the component with the smallest eigenperiod 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 a multi degree of freedom analysis is required with simultaneous time integration of the coupled system.

6.7 Classification of resistance properties 6.7.1 Cross-sectional behaviour

Moment

Elasto-plastic : The effect of partial yielding on bending moment is accounted for Elastic-perfectly plastic: Linear elastic up to fully plastic bending moment The simple models described herein assume elastic-perfectly plastic material behaviour. Note: Even if the analysis is based upon elastic-perfectly plastic behaviour, the material has to exhibit strain hardening in practice for the theory to be valid. The effect of strain hardening on the plastic, cross-sectional resistances may be accounted 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 equivalent yield stress equal to the mean of the lower yield stress and the ultimate stress. In the clauses for the ductility limits the effect of strain hardening is accounted for.

Elastic-perfectly plastic elasto-plastic Curvature

Figure 6-4 Bending moment-curvature relationships

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Recommended Practice DNV-RP-C204, October 2010 Page 28

Component behaviour R

R k1

k2 k1

w

k2

R

Elastic-plastic (determinate)

k2

k3

k1

k1 w

Elastic

R

w

w

Elastic-plastic (indeterminate)

Elastic-plastic with membrane

Figure 6-5 Resistance curves

Elastic: Elastic material, small deformations Elastic-plastic (determinate): Elastic-perfectly plastic material. Statically determinate system. Bending mechanism fully developed with occurrence of first plastic hinge(s)/yield lines. No axial restraint. Elastic-plastic (indeterminate): Elastic perfectly plastic material. Statically indeterminate system. Bending mechanism develops with sequential formation of plastic hinges/yield lines. No axial restraint. For simplified analysis this system may be modelled as an elastic-plastic (determinate) system with equivalent initial stiffness. In lieu of more accurate analysis the equivalent stiffness should be determined such that the area under the resistance curve is preserved. Elastic-plastic with membrane: Elastic-perfectly plastic material. Statically determinate (or indeterminate). Ends restrained against axial displacement. Increase in load-carrying capacity caused by development of membrane forces.

6.8 Idealisation of resistance curves The resistance curves in 6.7 are idealised. For simplified SDOF analysis the resistance characteristics of a real non-linear system may be approximately modelled. An example with a tri-linear approximation is illustrated in Figure 6-6. The stiffness in the k3 phase may have some contribution from strain hardening, but in most cases the predominant effect is development of membrane forces when member ends are restrained form inward displacement.

Elastic

k3

6.9.1 Elastic - rigid plastic relationships In lieu of more accurate calculations rigid plastic theory combined with elastic theory may be used. In the elastic range the stiffness and fundamental period of vibration of a clamped plate under uniform lateral pressure can be expressed as:

r = k1w k1 = ψ T=

= resistance-displacement relationship for plate centre D s4

= plate stiffness

ρ ts 4 D

2π η

(

t3

12 1 − ν 2

= natural period of vibration

)

= plate bending stiffness

The factors ψ and η are given in Figure 6-8.

k1 wel

Elastic-plastic with membrane

6.9 Resistance curves and transformation factors for plates

k 2=0

Rel

Rigid-plastic

Figure 6-7 Construction of elastic -plastic resistance curve

D=E

R

=

+

800

40

700

35

600

ψ

w

500 400

25

20 η

ψ

300

Figure 6-6 Representation of a non-linear resistance by an equivalent tri-linear system

30

η

15

200

10

100

5

0

In lieu of more accurate analysis the resistance curve of elasticplastic systems may be composed by an elastic resistance and a rigid-plastic resistance as illustrated in Figure 6-7.

0 1

1.5

Figure 6-8 Coefficients ψ and η.

DET NORSKE VERITAS

2

l/s

2.5

3

Recommended Practice DNV-RP-C204, October 2010 Page 29

In the plastic range the resistance (r) of plates with edges fully restrained against inward displacement and subjected to uniform pressure can be taken as: 2 ⎛ α + (3 − 2α ) r = 1 + w ⎜⎜ rc ⎝ 9 − 3α

2

⎞ ⎟ ⎟ ⎠

w ≤1 (6.8)

⎛ α(2 − α ) ⎛ 1 ⎞⎞ r ⎜ ⎟⎟ = 2 w ⎜1 + − 1 2 ⎜ ⎟⎟ ⎜ rc 3 − α ⎝ 3w ⎠⎠ ⎝ Pinned ends: 6f t 2 w w=2 rc = 2 y 2 t l α

w >1

Clamped ends: 12f t 2 w w= rc = 2 y2 t l α 2 ⎛ ⎞ s⎜ s⎟ ⎛s⎞ α = ⎜ 3+⎜ ⎟ − ⎟ l⎜ l⎟ ⎝l⎠ ⎝ ⎠

l (>s) s t rc w

= plate aspect parameter

= = = =

plate length plate width plate thickness plastic resistance in bending for plates with no axial restraint = non-dimensional displacement parameter 6

l/s = 100

Resistance [r/rc]

5 3 2 1

3

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 methods of analysis are generally applicable. Special considerations shall be given to the effect of:

2 1 0 0

0.5

1

1.5

Relative displacement

2

2.5

6.9.3 Tensile fracture of yield hinges In lieu of more accurate calculations the procedure described in Section 3.10.4 may be used for a beam with rectangular cross-section (plate thickness x unit width) and length equal to stiffener spacing.

6.10 Resistance curves and transformation factors for beams

5 4

The effect of flexibility may be taken into account in an approximate way by means of plate strip theory and the procedure described in Section 3.7.2. The relative reduction of the plate’s plastic resistance, with respect to the values given in Equation (6.8), is taken equal to the relative reduction of the resistance for a beam with rectangular cross-section (plate thickness x unit width) and length equal to stiffener spacing, using the diagram for α = 2 (Figure 6-12). The elastic straining of the plate is accounted for by the 2nd term in Equation (6.8). In lieu of more accurate calculation, the effect of pull-in, given by the first term in Equation (6.8) may be estimated by removing the plate and apply a uniformly distributed unit in-plane force normal to the plate edges. The axial stiffness should be taken as the inverse of the maximum in-plane displacement of the long edge. In lieu of more accurate calculation, it should be conservatively assumed that no membrane effects exist for a plate located close to an unsupported boundary, i.e. the resistance should be taken as constant and equal to the resistance in bending, r = rc over the allowable displacement range. In lieu of more accurate calculations, it is suggested to assess the relative reduction of the resistance for a uniformly loaded plate located some distance from an unsupported boundary with c = 1.0. If membrane forces are taken into account it must be verified that the adjacent structure is strong enough to anchor the fully plastic membrane tension forces.

3

w

Figure 6-9 Plastic load-carrying capacities of plates as a function of lateral displacement

6.9.2 Axial restraint

In Equation (6.8) the beneficial effect of membrane stiffening is represented by the term containing the non-dimensional displacement parameter w . Great caution should be exercised when assuming the presence of the membrane effect, because the membrane forces must be anchored in the adjacent structure. For plates located in the middle of a continuous plate field, the boundaries have often considerable restraint against pull-in. If the plate is located close to the boundary, the edges are often not sufficiently stiffened to prevent pull-in of edges. Unlike stiffeners no simple method with a clear physical interpretation exists to quantify the effect of flexibility on the resistance of plates under uniform pressure. In the formulations used in this RP the flexibility may be split into two contributions

— — — — — —

Elastic flexibility of member/adjacent structure Local deformation of cross-section Local buckling Strength of connections Strength of adjacent structure Fracture

6.10.1 Beams with no- or full axial restraint Equivalent springs and transformation factors for load and mass for various idealised elasto-plastic systems are shown in Table 6-2. For more than two concentrated loads, equal in magnitude and spacing, use values for uniform loading. Shear deformation may have a significant impact on the elastic flexibility and eigenperiod of beams and girders with a short span/web height ratio (L/hw), notably for clamped ends. The eigenperiod and stiffness in the linear domain including shear deformation may be calculated as: T = 2π

m k

= 2π

Kl m,u Mu + Kl m,c Mc k1'

2

⎛ πrg ⎞ ⎛ E A ⎞ ⎟ ⎜1+ ⎟ (6.9) 1+ ⎜⎜ cs ⎟ ⎟ ⎜ ⎝ L ⎠ ⎝ G Aw ⎠

and

1) Pull-in of edges

1 1 1 = + , ' k k k1 1 Q

2) Elastic straining of the plate

where DET NORSKE VERITAS

k Q = cQ

GA w L

(6.10)

Recommended Practice DNV-RP-C204, October 2010 Page 30

cs

= 1.0 for both ends simply supported = 1.25 for one end clamped and one end simply supported = 1.5 for both ends clamped L = length of beam/girder E = elastic modulus G = shear modulus A = total cross-sectional area of beam/girder Aw = shear area of beam/girder kQ = shear stiffness for beam/girder k1 = bending stiffness of beam/girder in the linear domain according to Table 6-2 rg = radius of gyration

Mps = plastic bending capacity of beam at support Mpm = plastic bending capacity of beam at midspan and regardless of rotational boundary conditions the following values 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 increasingly inaccurate for loads with short duration and/or high magnitudes.

Table 6-2 Transformation factors for beams with various boundary and load conditions Mass factor Load-mass factor Load case Km Klm Load Maximum Resistance Factor ConcenUniConcen- Uniform resistance domain Kl Rel trated form trated mass mass mass mass

F L/2

L/2

F/2

Load case

V

0.50

0.78

8Mp

384 EI

L

5 L3

Plastic bending

0.50

0.33

0.66

8Mp

Plastic membrane

0.50

0.33

0.66

Elastic

1.0

1.0

0.49

1.0

0.49

4Mp

48 EI

L

L3

Plastic bending

1.0

1.0

0.33

1.0

0.33

4Mp

Plastic membrane

1.0

1.0

0.33

1.0

0.33

4NP L

Elastic

0.87

0.76

0.52

0.87

0.60

6Mp

56.4 EI

L

L3

Plastic bending

1.0

1.0

0.56

1.0

0.56

6Mp

Plastic membrane

1.0

1.0

0.56

1.0

0.56

Load-mass factor Klm Concen- Uniform trated mass mass

0

L

0 6N P L

Elastic

0.53

0.41

0.77

Elastoplastic bending

0.64

0.50

0.78

8 M ps + M Pm

Plastic bending

0.50

0.33

0.66

8 M ps + M Pm

Plastic membrane

0.50

0.33

0.66

0.38 Rel + 012 . F

0.78 R − 0.28 F 0.75Rel − 0.25F

2 N P ymax L

L

Resistance domain

0.39 R + 011 . F

2 N P ymax L

4NP L

Mass factor Km Load Factor Concen- Uniform trated Kl mass mass

DET NORSKE VERITAS

0

L

0.525R − 0.025F 0.52 Rel − 0.02 F 3 N P ymax L

Maximum resistance Rel

EquivaLinear lent lin- Dynamic reaction stiffness ear k1 stiffness V ke

12 M ps

384 EI

L

L3

F=pL L

k1

0.64

F/2

L/3 L/3 L/3

Dynamic reaction

Elastic

F=pL

L

Linear stiffness

(

)

L

(

L

)

384 EI 5 L3

0 4NP L

0.36 R + 0.14 F

0.39 Rel + 011 . F

0.38 Rel + 012 . F 2 N p ymax L

Recommended Practice DNV-RP-C204, October 2010 Page 31

Load case

Resistance domain

F L/2

L/2

F/2

F/2

L/3 L/3 L/3

F=pL V1

V2

L

F V1

V2 L/2

F/2

L/2

F/2

V1

V2 L/3 L/3 L/3

Mass factor Km Load Factor Concen- Uniform trated Kl mass mass

Load-mass factor Klm Concen- Uniform trated mass mass

Maximum resistance Rel

EquivaLinear lent lin- Dynamic reaction stiffness ear k1 stiffness V ke

(

192 EI

)

Elastic

1.0

1.0

0.37

1.0

0.37

4 M ps + M Pm

Plastic bending

1.0

1.0

0.33

1.0

0.33

4 M ps + M Pm

Plastic membrane

1.0

1.0

0.33

1.0

0.33

Elastic

080

0.64

0.41

0.80

0.51

Elastoplastic bending

0.87

0.76

0.52

0.87

0.60

6 ( M ps + M Pm ) L

56.4EI L3

Plastic bending

1.0

1.0

0.56

1.0

0.56

6 ( M ps + M Pm )

0

Plastic membrane

1.0

1.0

0.56

1.0

0.56

Elastic

0.58

0.45

0.78

L

(

)

L

9 M ps L

L

8 M ps

0.50

0.78

Plastic bending

0.50

0.33

0.66

4 M ps + 2 M Pm

Plastic membrane

0.50

0.33

Elastic

1.0

1.0

(

)

(

)

L

16 M Ps 3L

1.0

1.0

0.49

1.0

0.49

2 M ps + 2 M Pm

Plastic bending

1.0

1.0

0.33

1.0

0.33

2 M ps + 2 M Pm

Plastic membrane

1.0

1.0

0.33

1.0

0.33

Elastic

0.81

0.67

0.45

0.83

0.55

6 M Ps L

(

(

(

V1 = 0.26 R + 0.12 F V2 = 0.43R + 019 . F ⎛ 160 EI ⎞ ⎟ ⋅ m2 ⎜ 3 ⎝ L ⎠

4NP L

2 N P ymax L

107 EI

V1 = 0.25R + 0.07 F

)

0

V2 = 0.54 R + 014 . F

⎛ 160 EI ⎞ ⎜ ⎟ ⋅ m2 3 ⎝ L ⎠

L3

L3

0.87

0.60

2 M ps + 3 M Pm

Plastic bending

1.0

1.0

0.56

1.0

0.56

2 M ps + 3 M Pm

Plastic membrane

1.0

1.0

0.56

1.0

0.56

)

56 EI

)

0

L

(

L

1.5M ps

M ps + M pm

L3

6N P L

+ 0.25

1.5M ps

M ps + 2M pm 2M ps

M ps + 3M pm

+ 0.5 + 0.5

0.39 R + 011 . F ± M Ps L 0.38 R + 012 . F ± M Ps L

0

0.78 R − 0.28 F ± M Ps L 0.75R − 0.25F ± M Ps L 2 N P ymax L V1 = 017 . R + 017 . F V2 = 0.33R + 0.33F

132 EI

0.52

DET NORSKE VERITAS

0.52 Rel − 0.02 F

0.52 Rel − 0.02 F

4NP L

0.76

m3 =

⎛ 212 EI ⎞ ⎟ ⋅ m1 ⎜ 3 ⎝ L ⎠

3

48 EI

L

0.87

m2 =

384 EI

)

L

Elastoplastic bending

m1, m2 and m3 are factors for deriving the equivalent stiffness:

0.48R + 0.02 F

L3

Elastoplastic bending

m1 =

260EI L3

5L

L

0.66

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

2 N P ymax L

L

L

Where: q

0.75Rel − 0.25F

4NP L

3

0.64

0.43

⎛ 48EI ⎞ ⎜ 3 ⎟ ⋅ m1 ⎝ L ⎠

185EI

4 M ps + 2 M Pm

1.0

0

6N P L

Elastoplastic bending

0.43

0.71R − 0.21F

L3

⎛ 122 EI ⎞ ⎜ ⎟ ⋅ m3 3 ⎝ L ⎠

0.525R − 0.025F ± M Ps L 0.52 Rel − 0.02 F ± M Ps L 3 N P ymax L

Recommended Practice DNV-RP-C204, October 2010 Page 32

6.10.2 Beams with partial end restraint. Relatively small axial displacements have a significant influence on the development of tensile forces in members undergoing large lateral deformations. Equivalent elastic, axial stiffness may be defined as

l 1 1 = + k k node 2EA

α

⎞ ⎟ =1 ⎟ ⎠

for 1 < α < 2

(6.12)

In lieu of more accurate analysis α = 1.2 can be assumed for stiffened plates and H or I beams. For members with tubular section α = 1.5. For rectangular sections and plates α = 2.0 can be assumed.

R0 =

8c 1 f y Wp l

plastic collapse resistance in bending for = the member with uniform load.

w c1 w c

wc =

(6.11)

knode = axial stiffness of the node with the considered member removed. This may be determined by introducing unit loads in member axis direction at the end nodes with the member removed. Plastic force-deformation relationship for a beam under uniform pressure may be obtained from Figure 6-10, Figure 6-11 or Figure 6-12 if the plastic interaction between axial force and bending moment can be approximated by the following equation:

M ⎛⎜ N + M p ⎜⎝ N p

w=

c=

= non-dimensional deformation

αWp A

4c1 kw c f y Al

characteristic beam height for beams = described by plastic interaction equation (6.12).

2

c1 = 2 c1 = 1 WP

Wp = zgAg A = As + st Ae = As + set zg As s se

= non-dimensional spring stiffness

= for clamped beams = for pinned beams = plastic section modulus for the cross section of the beam = plastic section modulus for stiffened plate for set > As = total area of stiffener and plate flange = effective cross-sectional area of stiffener and plate flange, = distance from plate flange to stiffener centre of gravity. = stiffener area = stiffener spacing = effective width of plate flange see Section 6.10.4

= member length

l

6

5

α = 1.2

R/R 0

4

Bending & membrane Membrane only

F

3

c=∞

2

0.2

0.5

0.1

k

0

0 0,5

1

1,5

k

w

1

1

0

(explosion load)

2

2,5

Deformation

3

3,5

w

Figure 6-10 Plastic load-deformation relationship for beam with axial flexibility (α = 1.2)

DET NORSKE VERITAS

4

Recommended Practice DNV-RP-C204, October 2010 Page 33

7

6

α = 1.5

R/R0

5

Bending & membrane Membrane only

4

F

(explosion load)

3

c=∞

2

0.2

0.5

1

0.1

k

k

w 0

1

0 0

0,5

1

1,5

2

2,5

3

3,5

4

w

Deformation

Figure 6-11 Plastic load-deformation relationship for beam with axial flexibility (α = 1.5)

9 8

α=2

7

R/R0

6

Bending & membrane Membrane only

5

0.1

0.2

4

F

(explosion load)

0.5

c=∞

3

k

1

k

w

2

0 1 0 0

0,5

1

1,5

2

Deformation

2,5

3

3,5

4

w

Figure 6-12 Plastic load-deformation relationship for beam with axial flexibility (α = 2)

For members where the plastic moment capacity of adjacent members is smaller than the moment capacity of the exposed member the force-deformation relationship may be interpolated from the curves for pinned ends and clamped ends:

R = ζR clamped + (1 − ζ)R pinned

(6.13)

where

0≤ζ=

R0 8

DET NORSKE VERITAS

actual

Mp l

−1 ≤ 1

(6.14)

Recommended Practice DNV-RP-C204, October 2010 Page 34

R

= Collapse load in bending for beam accounting for actual bending resistance of adjacent members

actual 0

R 0actual =

8M p + 4M p1 + 4M p2 l

M Pj = ∑ M Pj,i ≤ M P

The effective width for elastic deformations may be used when the plate flange is on the tension side.

(6.16)

In most cases the flange will experience varying stress with parts in compression and parts in tension. It may be unduly conservative to use the effective width for the section with the largest compression stress to be valid for the whole member length. For continuous stiffeners it will be reasonable to use the mean value between effective width at the section with the largest compression stress and the full width. For simple supported stiffeners with compression in the plate it is judged to be reasonable to use the effective width at midspan for the total length of the stiffener.

i = adjacent member no i j = end number {1,2} MPj,i = Plastic bending moment for member no. i. Elastic, rotational flexibility of the node is normally of moderate significance 6.10.3 Beams with partial end restraint - support capacity smaller than plastic bending moment of member

For beams where the plastic moment capacity of adjacent members is smaller than the moment capacity of the impacted beam, the force-deformation relationship, R*, may be derived from the resistance curve, R, for beams where the plastic moment capacity of adjacent members is larger than the moment capacity of the impacted beam (Section 3.7.2), using the expression: w w lim

,

,

R* = R

w ≤ 1.0 w lim w w lim

(6.17)

≥ 1.0

R0 = Plastic bending resistance with clamped ends (c1 = 2) – moment capacity of adjacent members larger than the plastic bending moment of the beam R *0 = Plastic bending resistance - moment capacity of adjacent members at one or both ends smaller than the plastic bending moment of the member

4M P + 2M P1 + 2M P2 l

M Pj = ∑ M Pj,i ≤ M P

(6.18) (6.19)

i

i j MPj,i wlim

= = = =

adjacent member no i end number {1,2} Plastic bending resistance for member no. i. 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 = ∞,

wlim =

π

2

6.10.5 Strength of adjacent structure

The adjacent structure must be checked to see whether it can provide the support required by the assumed collapse mechanism for the member/sub-structure 6.10.6 Strength of connections

The capacity of connections can be taken from recognised codes. The connection shall be checked for the dynamic reaction force given in Table 6-2. For beams with axial restraint the connection should also be checked for lateral - and axial reaction in the membrane phase:

where

R *0 =

Determination of effective flange due to buckling can be made as for buckling of stiffened plates see DNV-RP-C201.

(6.15)

i

R * = R + (R 0 − R *0 )

Commentary.

— If the axial force in a tension member exceeds the axial capacity of the connection the member should be assumed disconnected. — If the capacity of the connection is exceeded in compression and bending, this does not necessarily mean failure of the member. The post-collapse strength of the connection may be taken into account provided that such information is available. 6.10.7 Ductility limits

Reference is made to Section 3.10. The local buckling criterion in Section 3.10.2 and tensile fracture criterion given in Section 3.10.3 may be used with: dc c

= characteristic dimension equal to twice the distance from the plastic neutral axis in bending to the extreme fibre of the cross-section = non-dimensional axial spring stiffness calculated in Section 6.10.2. y

max Alternatively, the ductility ratios μ = y el be used.

in Table 6-3 may

Table 6-3 Ductility ratios μ - beams with no axial restraint

w

Boundary conditions

for tubular beams and for stiffened plate 6.10.4 Effective flange

In order to analyse stiffened plate as a beam the effective width of the plate between stiffeners need to be determined. The effective width needs to be reduced due to buckling and/or shear lag. Shear lag effects may be neglected if the length is more than 2.5 times the width between stiffeners. For guidance see Ch.8,

Cantilevered Concentrated Distributed Pinned Concentrated Distributed Fixed Concentrated Distributed 1)

Cross-section type 1)

Load

Type I

Type II

Type III

6 7 6 12 6 4

4 5 4 8 4 3

2 2 2 3 2 2

Crossecton types are defined in DNV-OS-C101, Table A3, Appendix A

DET NORSKE VERITAS

Recommended Practice DNV-RP-C204, October 2010 Page 35

/1/ /2/

NORSOK Standard N-003 Action and Action Effect NS-ENV 1993-1 Eurocode 3: Design of Steel structures Part 1-2. General rules - Structural fire design /3/ Amdahl, J.: “Energy Absorption in Ship-Platform Impacts”, UR-83-34, Dept. Marine Structures, Norwegian Institute of Technology, Trondheim, 1983. /4/ SCI 1993: Interim Guidance Notes for the Design and Protection of Topside Structures against Explosion and Fire /5/ Amdahl, J.: “Mechanics of Ship-Ship Collisions- Basic Crushing Mechanics”. West Europene Graduate School of Marine Technology, WEGEMT , Copenhagen, 1995 /6/ Skallerud, B. and Amdahl, j.: “Nonlinear Analysis of Offshore Structures”, Research studies Press, UK 2002 /7/ Amdahl, J. and Johansen, A.: “High-Energy Ship Collision with Jacket Legs” ISOPE, Stavanger, 2001 /8/ Moan, T., Amdahl, J., Wang, X. and Spencer, J.: “ Risk Assessment of FPSOs, with Emphasis on Collisions”, SNAME Annual Meeting, Boston, 2002 /9/ Skallerud, B. and Amdahl, j.: “Nonlinear Analysis of Offshore Structures”, Research studies Press, UK 2002 /10/ Amdahl, J. and Johansen, A.: “High-Energy Ship Collision with Jacket Legs” ISOPE, Stavanger, 2001

8. Commentary Comm. 2.3 General The structural design is seen as having acceptable safety against accidental loads when the design accidental loads are less than the design resistance. This is similar to the check of the structure for ordinary loads but with the following differences: As ordinary loads are either permanent or occur frequent it will not be acceptable that the load lead to reduced load carrying capacity while the short duration and the low probability of accidental loads make this an acceptable assumptions. E.g. a blast wall need not be capable of resisting another explosion after a blast, but if the wall is used as a fire barrier it need to serve as such after the blast. Comm. 3.1 General For typical installations, the contribution to energy dissipation from elastic deformation of component/substructures in direct contact with the ship is very small and can normally be neglected. Consequently, plastic methods of analysis apply. However, elastic elongation of the hit member as well as axial flexibility of the nodes to which the member is connected, have a significant impact on the development of membrane forces in the member. This effect has to be taken into account in the analysis, which is otherwise based on plastic methods. The diagrams in Section 3.7.2 are based on such an approach. Depending on the structure size/configuration as well as the location of impact elastic strain energy over the entire structure may contribute significantly. Comm. 3.2 Design principles The transition from essentially strength behaviour to ductile response can be very abrupt and sensitive to minor changes in scantlings. E.g. integrated analyses of impact between the stern of a supply vessel and a large diameter column have shown that with moderate change of (ring - and longitudinal) stiffener size and/or spacing, the energy dissipation may shift from predominantly platform based to predominantly vessel based. Due attention should be paid to this sensitivity when the calculation procedure described in Section 3.5 is applied. Comm. 3.3 Collision mechanics

The added mass is due to the hydrodynamic pressure induced by the forced motion of water particles on the wet surface of the ship. By solving the velocity potential for the fluid on the body surface, the added mass is determined by means of 2-D (strip theory) or 3-D techniques. The added mass is frequency dependent, and thus varies with time during a collision, but a constant value is recommended for simple analysis. The fraction of collision energy to be dissipated as strain energy for shuttle tanker impact on FPSO stern is shown in Figure 8-1. Note the strong dependency of the mass ratio; the larger the mass of shuttle tanker, the lesser of the collision energy must be dissipated as strain energy. (However, provided that the speed of the shuttle tanker is constant, the absolute value of the strain energy increases) The relative size may differ considerably for the approach phase (shuttle tanker in ballast, FPSO fully loaded) and the departure phase (shuttle tanker fully loaded, FPSO in ballast). To illustrate this, possible values are listed in Table 8-1. In this example both the FPSO and shuttle tanker are large compared to typical North Sea conditions. The same added mass coefficient applies to both vessels. It is observed that the fraction of energy to be dissipated as strain energy varies between 0.33 (departure) and 0.71 (approach). This indicates that the approach phase may be particularly critical with respect to the consequences of collision. Table 8-1 Fraction of collision energy for strain energy dissipation Vessel size [dwt] Approach phase Departure phase Shuttle tanker 150.000 370.000 FPSO 320.000 160.000 Strain energy fraction 0.71 0.33

1,2 Strain energy fraction

7. References

1 0,8 0,6 0,4 0,2 0 0

1

2

3

4

Mass ratio [(ms + as)/(mi+ ai)] Figure 8-1 Fraction of collision energy to be dissipated as strain energy for shuttle tanker impact on an FPSO.

Comm. 3.5.1 Recommended force-deformation relationships

The force-deformation relationship for impacts from supply vessels/merchant vessels against jacket legs have been elaborated because of the need to consider high-energy impacts (collision energy ~50 MJ) for some installations in the North Sea. The likelihood of a central impact against a leg is obviously not very large, but has still been considered because loss of a leg could be critical for some platforms. Experience has shown, however, that many large North Sea jackets have sufficient strength to crush the bow. Reference is made to Amdahl and Johansen (2001).

DET NORSKE VERITAS

Recommended Practice DNV-RP-C204, October 2010 Page 36

The curve for bow impact in Figure 3-4 has been derived on the assumption of impacts against an infinitely rigid wall. Sometimes the curve has been used erroneously to assess the energy dissipation in bow-brace impacts. Experience from small-scale tests Ch.7, /3/ indicates that the bow undergoes very little deformation until the brace becomes strong 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 virtually undamaged. 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 function of impact location and brace diameter. These curves are reproduced in Figure 8-2. In order to fulfil a strength design requirement the brace should at least resist the load level indicated by the broken line (recommended design curve). For braces with a diameter to thickness ratio < 40 it should be sufficient to verify that the plastic collapse load in bending for the brace is larger than the required level. For larger diameter to thickness ratios local denting must probably be taken into account. Normally sized jacket braces are not strong enough to resist the likely bow forces given in Figure 8-2, and therefore it has to be assumed to absorb the entire strain energy. For the same reasons it has also to be assumed that the brace has to absorb all energy for stern and beam impact with supply vessels. Impact force [MN] Recommended design curve for brace impact 12 Between a stringer (D= 1.0 m) 8

On a stringer (D= 0.75 m) Between stringers (D= 0.75) m

4

0

0.5

1.0

1.5 2.0 Indentation [m]

Figure 8-2 Load-deformation curves for bow-bracing impact, Ch.7, /3/

Comm. 3.5.2 Force contact area for strength design of large diameter columns.

Figure 8-3 shows an example of the evolution of contact force intensity during a collision between the stern corner of a supply vessel and a stiffened column. In the beginning the contact is concentrated at the extreme end of the corner, but as the corner deforms it undergoes inversion and the contact ceases in the central part. The contact area is then, roughly speaking, bounded by two concentric circles, but the distribution is uneven. The force-deformation curves given in Figure 3-4 relate to total collision force for stern end - and stern corner impact , respectively. Table 3-1 and Table 3-2 give the anticipated maximum force intensities (local force and local contact areas, i.e. subsets of the total force and total area) at various stages of deformation. The basis for the design curves is integrated, non-linear finite element analysis of stern/column impacts. The information given in 3.5.2 may be used to perform strength design. If strength design is not achieved numerical analyses have shown that the column is likely to undergo severe deformations and absorb a major part of the strain energy. In lieu of more accurate calculations (e.g. non-linear FEM) it has to be assumed that the column absorbs all strain energy. Comm. 3.5.3 Energy dissipation is ship bow.

The requirements in this paragraph are based upon considerations of the relative resistance of a tubular brace to local denting and the bow to penetration of a tubular beam. A fundamental requirement for penetration of the brace into the bow is, first - the brace has sufficient resistance in bending, second - the cross-section does not undergo substantial local deformation. If the brace is subjected to local denting, i.e. undergoes flattening, the contact area with the bow increases and the bow inevitably gets increased resistance to indentation. The provisions ensure that both requirements are complied with. Figure 8-8 indicates the level of the various contact locations. Figure 8-4 shows the minimum thickness as a function of brace diameter and resistance level in order to achieve sufficient resistance to penetrate the ship bow without local denting. It may seem strange that the required thickness becomes smaller for increasing diameter, but the brace strength, globally as well as locally, decreases with decreasing diameter. Local denting in the bending phase can be disregarded provided that the following relationship holds true: D 1⎛l⎞ ≤ 0.14 2 ⎜ ⎟ t c1 ⎝ D ⎠

2

(8.1)

Figure 8-5 shows brace thickness as a function of diameter and length diameter ratio that results from Equation (8.1). The thickness can generally be smaller than the values shown, and still energy dissipation in the bow may be taken into account, but if Equation (8.1) is complied with denting does not need to be further considered.

Total collision force distributed over this Area with high force intensity Deformed stern corner

Figure 8-3 Distribution of contact force for stern corner/large diameter column impact

The requirements are based upon simulation with LS-DYNA for penetration of a tube with diameter 1.0 m. Great caution should therefore be exercised in extrapolation to diameters substantially larger than 1.0 m, because the resistance of the bow will increase. For brace diameters smaller than 1.0 m, the requirement is conservative.

DET NORSKE VERITAS

Recommended Practice DNV-RP-C204, October 2010 Page 37

Thickness [mm]

80

fy = 235 MPa, 6 MN

60

fy = 235 MPa, 3 MN

40

fy = 355 MPa, 6 MN fy = 355 MPa, 3 MN

20 0 0,6

0,8

1

1,2

1,4

phase. Between these extremities, simple linear interpolation is used. Comm. 3.10.1 General

If the degradation of bending capacity of the beam cross-section after buckling is known the load-carrying capacity may be interpolated from the curves with full bending capacity and no bending capacity according to the expression:

fy = 420 MPa, 6 MN

R ( w ) = RM P =1 ( w )ξ + RM P =0 ( w )(1 − ξ )

fy = 420 MPa, 3 MN

RM P =1 ( w ) = Collapse load with full bending contribution

RM P =0 ( w ) = Collapse load with no bending contribution

Diameter [m]

Figure 8-4 Required thickness versus grade and resistance level of brace to penetrate ship bow without local denting

Thickness [mm]

L/D =20 L/D =25 L/D =30

60 40 20

0,8

1 1,2 Diameter [m]

1,4

Figure 8-5 Brace thickness yielding little local deformation in the bending phase

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 moderate significance.

R M P , red R M P =1 ( w = 0)

= Plastic collapse load in bending with reduced cross-sectional capacities. This has to be updated along with the degradation of crosssectional bending capacity.

80

0

Comm. 3.10.4 Tensile fracture in yield hinges

The rupture criterion is calculated using conventional beam theory. A linear strain hardening model is adopted. For a cantilever beam subjected to a concentrated load at the end, the strain distribution along the beam can be determined from the bending moment variation. In Figure 8-7 the strain variation, ε = ε cr ε Y , is shown for a circular cross-section for a given hardening parameter. The extreme importance of strain hardening is evident; with no strain hardening the high strains are very localised close to the support (x = 0), with strain hardening 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 determined. If the beam response is affected by development of membrane forces it is assumed that the membrane strain follows the same relative distribution as the bending strain. By introducing the kinematic relationships for beam elongation, the maximum membrane strain can be calculated for a given displacement. 50

7

α = 1.5

6

45

c =0.5

40

5

Hardening parameter H = 0.005

35

Strain ε

R/R0 R*/R0

ξ=

R M P , red

100

0,6

(8.2)

4 R/R0

3 2

20

l

No hardening

10

R*/R0

0 0

25

P x

15

R *0 / R0

1

Maximum strain εcr/εY = 50 = 40 = 20

30

1

2

wlim

5

3

4

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x/l

Deformation w

Figure 8-6 Derivation of force-deformation relationship for beam with end moments 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 bending moment boundary condition is important in the bending phase, but has no influence on the resistance in the pure membrane

Figure 8-7 Axial variation of maximum strain for a cantilever beam with circular cross-section

Adding the bending strain and the membrane strain allows determination of the critical displacement as a function of the total critical strain. Figure 8-8 shows deformation at rupture for a fully clamped beam as a function of the axial flexibility factor c.

DET NORSKE VERITAS

Recommended Practice DNV-RP-C204, October 2010 Page 38

5

20%

4.5 4

NORSOK

15%

ABAQUS fine

2.5 2 1.5

l/D = 30

l/D = 20

c= 0 = 0.05 = 0.5 = 1000

c= = = =

0 0.05 0.5 1000

USFOS beam

10%

ABAQUS 5%

USFOS shell

0%

1

0.0

0.5

0.5

1.0

1.5

2.0

Displacement [m]

0 0

20

40

60

80

100

120

Figure 8-11 Strain versus displacement of clamped beam

εcr/εy

Figure 8-8 Maximum deformation for a tubular fully clamped beam (H=0.005)

The plastic stiffness factor H is determined from the stressstrain relationship for the material. The equivalent linear stiffness shall be determined such that the total area under the stress-strain curve up to the critical strain is preserved (The two portions of the shaded area shall be equal), refer Figure 8-9. It is un-conservative and not allowable to use a reduced effective yield stress and a plastic stiffness factor as illustrated in Figure 8-10.

fcr

HE E

HE

1600

fcr

It is observed that the strain estimated in ABAQUS analysis depend significantly on the mesh size evidencing the need for a mesh-size-dependent fracture strain criterion. The NORSOK criterion agrees fairly well with FEM calculations when a fine mesh is used. The criterion is conservative, as desired. The strain calculation in the USFOS beam element assumes a yield plateau followed by parabolic type hardening. Only the fine ABAQUS mesh captures the yield plateau effect. Comm. 3.12 Energy dissipation in floating production vessels

1600

w/D

3

Strain

3.5

E

εcr

εcr

1600

Figure 8-12 Design of an impact resistant stern – collision with a VLCC.

Figure 8-9 Determination of plastic stiffness

f HE

ε Figure 8-10 Erroneous determination of plastic stiffness

The accuracy of the calculation model for tensile fracture in yield hinges has been investigated by Amdahl and Skallerud (2002). The maximum strain as a function of lateral displacement (Equation (3.22)) for a tubular beam is compared with the maximum strain from finite element calculations in Figure 8-11. The beam is assumed to be clamped and fixed against inward axial displacement, l = 25 m, D = 1 m, t = 0.06 m, fy = 300 MPa, H = 0.00287 (i.e. ultimate stress fu = 390 MPa for at ultimate strain εu = 0.15). The mesh size for USFOS shell and ABAQUS is 0.25 ⋅ 0.39 m and for ABAQUS fine mesh 0.05 ⋅ 0.195 m. The element used in ABAQUS analyses is the S4R reduced integration element .

Calculation of energy dissipation in stringers, decks and bulkheads subjected to gross, axial crushing shall be based upon recognised methods for plastic analysis, e.g. Ch.7, /3/ and Ch.7, /4/. It shall be documented that the folding mechanisms assumed yield a realistic representation of the true deformation field. The force deformation relationships given in Figure 3-6 may be used to design a collision resistant stern of an FPSO. In order to be impact resistant, stringers and frames must be fairly closely spaced, typically in the range of 1.5 – 2 m. Given the relative dimensions of the girder system and the bulb crosssection, as illustrated in Figure 8-12, it is reasonable to apply the total collision force as uniformly distributed line loads on the stringers and frames. The integrity of the stringers and frames can then be checked in a FEM analysis. Moderate local yielding should be accepted. The stern structure must resist the collision force during all stages of the collision process. Normally, it suffices to analyse a few collision force and contact area situations. It is normally neither practical nor necessary to design the plating and stiffeners such that their response is elastic. Large plastic deformations can be accepted, but fracture of the plating should not occur (Note: provided that strength design is aimed for). In lieu of more accurate calculations, the contact force may be considered uniformly distributed over the plate field, and the resistance may be assessed using the provisions given for the resistance of plates and stiffeners to explosion loads.

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Even if the stiffeners are allowed to deform under extreme collision loads, they should be sufficiently robust to initiate crushing of the bulb. Engineering judgment must be applied, but it is recommended to design the stiffeners according to requirements for ships navigating in ice; DNV Ice Class POLAR. With respect to deformation resistance of stiffened plating, see next paragraph. The ductile resistance of stiffened plates may be analysed considering the side as an assembly of plate/stiffeners. The resistance of individual stiffeners with associated plate flange can be calculated with the methods given in Section 6.3 using relationships for a concentrated force, see example in Ch.8, Comm. 9.3. The resistance of the various stiffeners will be mobilised 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 stiffening has taken place. The initiation of fracture does not necessarily

30

30

Energy s upers tr. Energy bulb

25

imply that the resistance is totally lost, because fracture takes place in the top flange while the strain on the plate side is considerably smaller . The above procedure neglects the effect of membrane forces transverse 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 are given for supply vessels/merchant vessels, 18.000 tons chemical tanker and a 42.000 tons tanker and a shuttle tanker. The collision risk for all categories of vessels is discussed extensively. The consequences of a collision with a shuttle tanker servicing the FPSO are especially considered. Figure 8-13 shows the force-deformation relationship for supply vessel/merchant vessel colliding with the side of an FPSO. It is interesting to see that the force level for bow without bulb is smaller than the bow force-deformation curve given in Figure 3-4.

25

20

20

Forc e superstr. Forc e bulb

15

15

10

10

5

5

0

0 0

1 2 Bow Displacement [m]

Force [MN]

Energy [MJ]

Total force

3

Figure 8-13 Force-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 to 7 < 2 r/d < 41, t/d < 0.22 and mi/m < 0.75. The formula neglect the local energy dissipation which can be added as Eloc = R·0.2 t. In case of hit near the plate edges the energy dissipation will be low and may lead to unreasonable plate thickness. The failure criterion used for the formula is locking of the plate. In many cases locking may be acceptable as long as the falling object is stopped. 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 the calculations, no penetration of the drill collar will take place at any other hit location due to the collar of such dropped objects. Comm. 5.1 General

For redundant structures thermal expansion may cause buckling of members below 400°C. Forces due to thermal expansion are, however, purely internal and will be released once the member buckles. The net effect of thermal expansion is therefore often to create lateral distortions in heated members. In most cases these lateral distortions will have a moderate influence on the ultimate strength of the system. As thermal expansion is the source of considerable inconvenience in conjunction with numerical analysis it would tempting

to replace its effect by equivalent, initial lateral member distortions. There is however, not sufficient information to support such a procedure at present. Comm. 5.5 Empirical correction factor In Ch.7 /2/ Eurocode 3 an empirical reduction factor of 1.2 is applied in order to obtain better fit between test results and column curve c for fire exposed compressive members. In the design check this is performed by multiplying the design axial load by 1.2. In non-linear analysis such a procedure is impractical. In non-linear space frame, stress resultants based analysis the correction factor can be included by dividing the yield compressive load and the Euler buckling load by a factor of 1.2. (The influence of axial force on member’s stiffness is accounted for by the so-called Livesly’s stability multipliers, which are functions of the Euler buckling load.) In this way the reduction factor is applied consistently to both elastic and elasto-plastic buckling. The above correction factor comes in addition to the reduction caused by yield stress and elastic modulus degradation at elevated temperature if the reduced slenderness is larger than 0.2. Comm. 6.2 Classification of response Equation (6.2) is derived using the principle of conservation of momentum to determine the kinetic energy of the component at the end of the explosion pulse. The entire kinetic energy is

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then assumed dissipated as strain energy. Equation (6.3) is based on the assumption that the explosion pressure has remained at its peak value during the entire deformation and equates the external work with the total strain energy. In general, the explosion pressure is not balanced by resistance, giving rise to inertia forces. Eventually, these inertia forces will be dissipated as strain energy. Equation (6.4) is based on the assumption that the pressure increases slowly so that the static condition (pressure balanced by resistance) applies during the entire deformation. Comm. 6.4 SDOF system analogy The displacement at the end of the initial, linear resistance domain yel will generally not coincide with the displacement at first yield. Typically, yel represents the displacement at the initiation of a plastic collapse mechanism. Hence, yel is larger than the displacement at first yield for two reasons: i)

Change from elastic to plastic stress distribution over beam cross-section ii) Bending moment redistribution over the beam (redundant beams) as plastic hinges form

Impulsive asymptote

9

Pressure F/R

8 7

F Fmax = R R el and a normalised impulse 1 Fmax t d I = 2 = RT R el T

k lm

6 5

Iso-damage curve for ymax/yelastic = 10 Elastic-perfectly plastic resistance

3 2 1

average

Pressure asymptote

0 1

2

3

4

5

6

7

=

k lm

elastic

+ (μ − 1)k lm

μ

plastic

(8.3)

μ = ymax/yel ductility ratio Since μ is not known a priori iterative calculations may be necessary. Dynamic response charts for a SDOF system with triangular pressure pulses with rise time different from td/2 are given in Figure 8-15 to Figure 8-17.

4

0

t 1 ⋅ d R el T 2 Fmax

By plotting corresponding values of normalised impulse and normalised pressure the iso-damage curve given in Figure 8-14 is obtained. If the displacement shape function changes as a non-linear structure undergoes deformation the transformation factors change. In lieu of accurate analysis an average value of the combined load-mass transformation factor can be used:.

11 10

Figure 8-14 is derived from the dynamic response chart for a SDOF system subjected to a triangular load with zero rise time given in Figure 6-3. In the example it is assumed that from ductility considerations for the assumed mode of deformation a maximum displacement of ten times elastic limit is acceptable. Hence the line y allow y max = = 10 represents the upper limit for the y el y el displacement of the component. From the diagram it is seen that several combinations of pulses characterised by Fmax and td may produce this displacement limit. Each intersection with a response curve (e.g. k3 = 0) yields a normalized pressure

8

9

10

11

Impulse I/(RT)

Figure 8-14 Iso-damage curve for ymax/yel = 10. Triangular pressure

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Rel/Fmax=0.05

=0.1

= 0.3

100

= 0.5

= 0.6

= 0.7 Rel/Fmax= 0.8

= 0.9 10

= 1.0

ymax/yel

= 1.1 = 1.2 = 1.5

1

k3 = 0 k3 = 0.1k1 k3 = 0.2k 1 k3 = 0.5k 1

F

R

Fmax

Rel

k3 = 0.5k1 =0.2k1

=0.1k1

k1

td

yel

y

0.1 0.1

1

10

td/T Figure 8-15 Dynamic response of a SDOF system to a triangular load (rise time=0)

Rel/Fmax=0.05

=0.1

= 0.3

100

= 0.5

= 0.6

= 0.7

Rel/Fmax= 0.8

10

ymax/yel

= 0.9

= 1.0 = 1.1 = 1.2 = 1.5

1

k3 = 0 k3 = 0.1k1 k3 = 0.2k1 k3 = 0.5k1

F

R

Fmax

Rel

k3 = 0.5k1 =0.2k1

=0.1k1

k1 0.15td

td

yel

y

0.1 0.1

1

td/T Figure 8-16 Dynamic response of a SDOF system to a triangular load (rise time = 0.15td)

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Recommended Practice DNV-RP-C204, October 2010 Page 42

Rel/Fmax=0.05

100

=0.1

= 0.3

= 0.5

= 0.6

= 0.7

Rel/Fmax= 0.8

10

ymax/yel

= 0.9

= 1.0 = 1.1 = 1.2 = 1.5

1

k3 = 0 k 3 = 0.1k1 k 3 = 0.2k 1 k 3 = 0.5k 1

F

R

Fmax

Rel

k3 = 0.5k1 =0.2k1

=0.1k1

k1 0.30td

td

yel

y

0.1 0.1

1

10

td/T Figure 8-17 Dynamic response of a SDOF system to a triangular load (rise time = 0.30td)

Comm.6.7.1.1 Component behaviour For beams the characteristic linear stiffness given for the elasto-plastic resistance domain in Table 6-2 is derived from the equal area principle on the assumption that the support moment is equal to the plastic bending moment of the beam. Comm. 6.7.1.1 Component behaviour For deformations in the elastic range the effective width (shear lag effect) of the plate flange, se, of simply supported or clamped stiffeners/girders may be taken from Figure 8-18.

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 crosssectional shape factor 1.12 for beams and 1.5 for plates. Strain hardening and any membrane effect will increase the effective ductility ratio. The values are likely to be conservative.

9. Examples 9.1 Design against ship collisions 9.1.1 Jacket subjected to supply vessel impact The location of contact is at brace mid-span and the force acts parallel to global x-axis. The brace dimensions are 762 x 28.6 mm. From linear elastic analysis it is found that the stiffness of nodes 508 and 628 against displacement in the brace direction is 736 MN/m and 51 MN/m respectively, when the brace is removed. The unequal stiffness may be represented by two equal springs, each with stiffness:

1.2 1

Uniform distribution or

n> n= n= n=

s e/s

0.8

6 5 4 3

0.6 0.4 0.2

nFi

nFi

=L


= 0.6L

0 0

2

4

6

8

/s Figure 8-18 Effective flange for stiffeners and girders in the elastic range

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It must also be verified that the capacity of the joints is sufficient to support the force state in the brace both in the bending mode of deformation and in the membrane tension state. Figure 9-3 displays the simulated bending moment-axial force interaction history in the brace and shows that the membrane force becomes substantial, but doe not attain the fully plastic axial force. In lieu of accurate calculations, it should be assume that 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 by fracture due to excessive straining of the brace. Fracture criteria are given Section 3.10.3. Using the fracture criterion in Section 3.10.3 there is obtained wcrit = 2.2 m and a corresponding energy dissipation E = 6 MJ.

10 Impact force [MN]

508

762 x 28.6 mm l= 23.3 m

Figure 9-1 Jacket subjected to ship impact

1⎞ ⎛ 1 K node = 2⎜ + ⎟ 736 51 ⎝ ⎠

10 Energy dissipation

8

8

6

6 USFOS

4 2

4 2

Simple model

Energy dissipation [MJ]

628

−1

= 95 MN / m

0

0 0.0

The axial stiffness of the brace is given by

0.5

1.0

1.5

2.0

2.5

3.0

Displacement [m]

2 EA 2 ⋅ 2.1 ⋅ 10 ⋅ π ⋅ 0.762 ⋅ 0.0286 = = 1234 MN / m l 23.3 5

Figure 9-2 Load versus lateral deformation of the contact point

and is large compared to the stiffness of the node. This yields an effective stiffness of

Assuming clamped ends (c1 = 2) the non-dimensional spring stiffness comes out to be 4c 1 Kw c c= f y Al

2

2 Kd 2 ⋅ 88 ⋅ 0 .762 = = ≅ 0 .18 f y π t l 355 ⋅ π ⋅ 0 .0286 ⋅ 23 .3

The resulting end restraint is quite flexible. This is particularly due to low stiffness in node 628, in spite of the support by the adjacent braces. Hence, the build-up of tension force will be delayed compared to a full axial fixity. The collapse load in bending is calculated assuming clamped conditions at both ends. This is a good approximation at the lower end but slightly optimistic at the upper end.

R0 =

4c1M P 4 ⋅ 2 ⋅ 355 ⋅ (0.762 − 0.0286) 2 ⋅ 0.0286 = = 1.9 MN l 23.3

The load-deformation characteristics for the brace are obtained by interpolation of the curves given in Figure 3-7. The result is depicted in Figure 9-2. The response predicted by means of the nonlinear analysis program USFOS is also plotted. It appears that the simplified approach performs very well when axial flexibility is taken into account. The loss of stiffness predicted by USFOS at large displacements is due to initiation of failure of adjacent members at node 628. Collapse of these members takes place at a load level of 2.8 MN.

Normalised force N/NP

1.0

1 1 1 = + = 88 MN/m K 95 1234

0.8 0.6 0.4 0.2 0.0 -0.2 0.0

0.2 0.4 0.6 0.8 Normalised moment M/MP

1.0

Figure 9-3 Axial force-bending moment interaction in brace

Tensile fracture in jacket brace

Tensile fracture of the brace considered in is estimated. The characteristic dimension is, dc = D = 0.762 m. For steel grade S 355 a strain hardening coefficient of H = 0.0034 is used, refer Table 3-3. c1 = 2 (clamped ends are assumed), the collision occurs at mid span, hence κ = 0.5, and κ l /dc = 15.3. The nondimensional spring stiffness is c = 0.18 and W/WP = π /4. This yields wcrit = 2.2 m. Because of the large κ l /dc – ratio, the brace is capable of deforming almost three times its diameter.

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9.2 Design against explosions 9.2.1 Geometry The geometry of the structure is outlined in Figure 9-4. The plate, stiffeners and girders will be assessed. The main dimensions are:

t s l

= 10 mm = 500 mm = 2000 mm

Stiffener dimension Hp 180

Girder

t = 10

Stiffener Hp180

Figure 9-4 Geometry

9.2.2 Calculation of dynamic response of plate: The dynamic response of the plate considered in Section 9.3.1 is studied. The plate is subjected to a triangular pressure pulse with duration of 20 msecs. The peak pressure is fmax = 2.5 MPa. Assume that the resistance curve for c = 1.0 in Figure 97 applies. This yields rel/fmax = 0.3. The curve is redrawn below along with approximate relationships

Resistance [Mpa]

P late c = 1.0

S tatic

3

9.2.3 Calculation of dynamic response of stiffened plate The dynamic response of the stiffened plate considered in Section 9.3.2, subjected to a triangular explosion pulse with duration 20 msecs and peak pressure fmax = 2.5 MPa, is studied. The collapse resistance is R0 = Rel = 0.58 MN, and no membrane stiffening can be assumed, i.e. k3 = 0. As the plate/stiffener undergoes a phase with elasto-plastic bending, the resistance is approximated by a linear elastic-perfectly plastic model, with equivalent stiffness of 208 MN/m and wel = 2.8 mm. The critical deformation at rupture wcrit = 36 mm, hence the ductility ratio is μ = ymax/yel = 36/2.8 = 13. The total mass is 108 kg. The load-mass factor is ~ 0.77 and 0.66 in the elastic/elasto-plastic and plastic bending phase, respectively. Using Equation (8.3) the average load-mass facaverage tor becomes klm = ( 0.77 + (13 − 1) 0.66 ) /13 = 0.67 and the eigenperiod is: average

T = 2π

E q . linear T ri-lin ear

2

f crit =

1 0 0

10

20 30 D eform ation [m m ]

k lm

k1

M

= 3.7m sec s

This gives td/T = 5.4. By inspection of Figure 6-3 it is found that for μ = 13 and td/T =5.4 → Rel/Fmax ≅ 0.75 (in other words, because of limited pulse duration it is possible to “overload” the stiffener by 33% compared to the static collapse resistance in bending). The maximum peak pressure the stiffener can resist is:

5 4

eigenperiod is adjusted by Tmod = T 1 0.65 = 5.0 msecs to account for less stiffness. This yields td/Tmod = 4.0. From the response chart there is obtained ymax/yel ~ 0.7. This yields wmax = 0.7 · 47,3 = 33 mm. All these methods yield approximately the same result. The static approach is quite good, but there is a slight dynamic amplification > 1 in the present case. The plate must be checked with respect to rupture, see Section 9.4.1. It is noticed that if no membrane force can be taken into account, i.e. c = 0, then ymax/yel >> 100 and the plate will fail completely.

40

50

Alternative 1- static analysis: The eigenperiod of the plate according to Section 9.3.1 with η = 25 is T = 4.0 msecs. Hence td/T = 5. This is a fairly long duration and static behaviour can be assumed. The maximum deflection is determined directly from Figure 9-7, i.e. wmax = 27 mm. Alternative 2 - tri-linear resistance: By inspection of the dynamic response charts and the resistance curve for the plate it is noticed that none of the tri-linear curves apply very well. The best fit is obtained with k3 = 0.5 k1, but this underestimates the resistance for large deformations. From the response chart for td/T = 5 there is read ymax/yel ~ 4.8. This yields wmax = 4.8 · 6.15 = 30 mm. Alternative 3 – equivalent linear resistance: For large deformations the stiffness is fairly linear. Assume that the average stiffness is linear and equal to 65 % of the elastic stiffness, i.e. k = 0.65 · 123 = 80 MPa/m. In this case the rel can be set arbitrarily, but it should be ensured that the response is such that ymax/yel < 1.0, and it is practical to select a given rel/fmax ratio for which a response curve is provided. Hence assume rel/fmax = 1.5, which gives rel = 47.3 mm and then it follows r. The

1 Rel = 0.8 MPa. 0.75 sl

Consequently; the stiffener is not strong enough to resist the explosion pressure without rupture (see discussion in Section 9.3.2 as concerns rupture of stiffener). It is a fairly common experience that stiffeners are more likely to be critical with respect to explosion loads than the plating between stiffeners.

9.3 Resistance curves and transformation factors 9.3.1 Plates. Generation of elastic–plastic resistance curve is illustrated for a plate with the following particulars: Length, l = 2 m, width, s = 0.5 m, thickness, t = 10 mm, yield stress f y = 355 MPa. It is assumed that the plate is a part of a continuous plate field. Large deformations are expected so that the plate will yield along the boundaries. Then clamped boundaries are assumed. The rigid – plastic curve is given by Equation (8.3). The collapse resistance in bending is rc = 0.76 MPa. The resistance curve for fully fixed boundaries are indicated by the line “Plate c = inf” in Figure 9-6. Below, the curve will be adjusted for the effect of in-plane flexibility using the procedure described in Section 6.8.2. First, the resistance of a plate-strip is calculated, using information given in Section 6.9.2 with α = 2 (rectangular crosssection). Clamped boundaries with c1 = 2 are assumed also for

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αWP A

=

2 t 2 ⋅1 4 t = t ⋅1 2

The resistance curve for the plate strip is shown in Figure 9.6 for fully fixed boundaries c = ∞ , and for two values of the nondimensional spring stiffness, c = 1.0 and c = 0.3. It is observed that the difference between the plate strip and the plate solution is small for the present fairly large aspect ration, notably when the membrane effect predominates On the assumption that the plate experiences the same relative reduction of the resistance due to axial flexibility as does the plate strip, resistance curves for the plate with non-dimensional spring stiffness, c = 1.0, and c = 0.3 can be generated as shown in Figure 9-6. The next step is to assess the flexibility factor c: If the flexibility of the adjacent structure is neglected, accounting only for the 2nd term in Equation (6.11), there is obtained k=

2 EA 2 Et ⋅ 1 = = 8400 MN / m l s

This yields a non-dimensional spring stiffness, c = 0.95. Uniform stress field applied along boundary of removed plate

Resistance [Mpa]

wc =

5 4

Plate c = inf Plate c = 1.0 Plate c = 0.3 Strip c = inf Strip c = 1.0 Strip c = 0.3

3 2 1 0 0

10

20

30

40

50

Deformation [mm]

Figure 9-6 Derivation of rigid-plastic resistance curves for a plate

5 Resistance [Mpa]

the plate strip. The collapse resistance in bending for the plate strip is rc = 0.57 MPa. The characteristic beam height is.

4 3 2

Plate c = inf Plate c = 1.0 Plate c = 0.3

1 0 0

10

20

30

40

50

Deformation [mm]

Figure 9-7 Elastic-plastic resistance for a plate with various degrees of axial flexibility.

Inward displacement

9.3.2 Calculation of resistance curve for stiffened plate Figure 9-5 Approximate determination of flexibility by means of membrane analysis

In order to assess the influence of the flexibility of the adjacent structure, a membrane analysis is performed with the plate removed, see Figure 9-5. A constant stress of 100 MPa is applied perpendicular the boundaries. The maximum deformation obtained, at the mid-point of the long edges, is 0.25 mm. This yields an equivalent stiffness of knode = 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 be assessed 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 resistance in bending, rc, is added to the rigid-plastic solution. Using the information given in Section 6.9.1, ψ = 400, and k1 = 123 MPa/ m. The deformation corresponding to r = rc is wel = 6.15 mm. The resulting resistance curves are shown in Figure 9.7.

The plate considered in Section 9.3.1 is stiffened with HP 180x 8 stiffeners with yield stress fy = 355 MPa. The girder spacing is 2.0 m. It is assumed that the stiffener is continuous, so that yield hinges can form at the connections to the girder, hence c1 = 2. The area of the stiffener As= 1.88·10-2 m2 and the distance to the centroid is zg = 0.109 m. From Figure 8-18 it is found that the plate flange is approximately 80% for a uniformly distributed load when l /s = 0.6⋅2.0/0.5 = 2.4. The effective area of the plate flange is 0.8 s t = 4·10-3 m2 > As. Hence, it may be assumed that the plastic neutral axis for the effective section lies at the stiffener web toe. This yields the plastic section modulus WP = As zg = 2.05·10-3 m3 and collapse resistance in bending R0 =

8c1 f yWP l

= 0.58 MN

The characteristic beam height is. αW w c = -----------P- = α z g = 1.2 ⋅ 0.109 = 0.13 m A The moment of inertia for stiffener with effective plate flange is I = 2.28 10-5 m4. The initial elastic stiffness is taken from Table 6-2: 384EI k= = 230 MN/m L3 This yields a lateral “elastic” deformation of wel = 2.5 mm for R = R0. The resistance curve for the stiffener with associated plate flange is shown in Figure 9.8 for various degrees of axial flexibility (Note elastic part not included!).

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For uniformly loaded, clamped beams there will be an elastoplastic bending phase between the occurrence of first plastic hinge and final formation of final collapse mechanism. To account for this effect, the initial stiffness may be modified on the basis of equal area principle. The equivalent elastic stiffness is obtained from Table 6-2 with m1 = 1: k=

307EI = 184 MN/m L3

and wel = 3.2 mm for R = R0. It is noticed that the stiffener must undergo a substantial plastic deformation before membrane strengthening becomes significant according to the present model. Whether this is achievable depends on the ductility of the stiffener, refer Section 9.4.2. Recent investigations indicate that the model adopted for stiffened plate is considerably conservative, which may warrant a more accurate nonlinear finite element analysis if the stiffener response becomes critical. 2.0

4

α = 1.2

c = 1.0 c = 0.5 c = 0.2 c = 0.1

1.0

0.5

0.0 0

0.1

0.2

0.3

5

0.56 ⋅ 2.9 ⋅ 10 + 0.975 ⋅ 1.8 ⋅ 10 T = 2p ----------------------------------------------------------------------------------- = 0.166s 6 274 ⋅ 10

c = inf 1.5 R [MN]

The plastic bending resistance is 8M Pm Rel* = = 5.95 MN L and wel* = 21.8 mm. However, the functional loads amount to 1.8 + 0.29 = 2.09 MN (including steel weight), so 21.8·2.09/ 5.95 = 7.6 mm is already utilised and only Rel = 5.95-2.09 = 3.86 MN and wel = 14.1 mm is available in the equivalent elastic range. The limiting deformation for rupture calculated in 9.4.3 is wmax = 95 mm, yielding ductility ratio μ = w/max / wel = 95/14.1 = 6.7. When calculating the load-mass factor the change in transformation factor from the elastic to plastic regime may be accounted for, see Ch.8, Comm. 6.4. The factor for distributed mass and concentrated mass is klmaverage,u = (0.55 + (6.7 − 1) ⋅ 0.56) / 6.7 = 0.56 and average,c klm = (0.83 + (6.7 − 1) ⋅ 1.0) / 6.7 = 0.975, respectively. The eigenperiod becomes

0.4

0.5

Deformation w [m]

Figure 9-8 Resistance 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 to rupture, when the explosion load is triangular, with equal rise and decay time, and the duration is 0.33 s? The girder has the following dimensions: Length L = 12 m, web height, hw = 1.5 m, web thickness, tw = 13 mm, top flange breadth, btop = 0.45 m, top flange thickness ttop = 19 mm. The girder spacing is 2 m and the plate thickness is 10 mm. For simplicity it is assumed that the plate flange is fully effective. The girder has a distributed load of intensity 10 kN/m2 and mounted equipment with mass 1.8·105 kg. The equipment load acts equally at two points located L/3 from member ends. The girder is simply supported at one end and clamped at the other end. At the clamped end fully plastic bending moment of the girder can be assumed. There is no axial restraint. Yield stress fy = 355 MPa, acceleration of gravity g = 10 m/s2, density of steel 7.86⋅103 kg/m3. The following is obtained for the girder: Moment of inertia I = 1.84⋅10-2 m4, elastic section modulus, W = 1.96⋅10-2 m3, plastic section modulus, WP = 2.51⋅10-2 m3, total cross-sectional area 0.048 m2. The total distributed mass, including mass of girder is 0.29⋅10-5 kg, so the concentrated mass predominates. Hence, transformation factors for two concentrated loads in Table 6.2 are used. The equivalent stiffness in the elasto-plastic range (m3 = 1) is. 122EI k= = 274 MN/m L3

and hence td/T= 0.33/0.166 ~ 2. From Figure 6-3 there is read Rel/Fmax = 0.7 for coordinates (2,6.7). Hence, the girder can resist a dynamic load of Fmax = 3.86/0.7 = 5.5 MN, corresponding to a peak pressure of fmax= 0.23 MPa. Example girder: The neutral axis for the girder studied in Section 9.3.3 is located 0.315 m from the plate flange. This yields a characteristic dimension dc = 2 ⋅ (1.5 − 0.315) = 2.37 m. The critical location is at the clamped side, whereby κ =1/3. Clamped end yields c1 = 2 for the fracture check. With H = 0.0034 and c = 0, there is obtained w/dc = 0.069 and w = 0.095 m.

9.4 Ductility limits 9.4.1 Plating Rupture of the plating for the example considered in Section 9.2.2 may be estimated by means of the procedure given in Section 3.10.4, using the plate strip analogy. The characteristic dimension is, dc = t = 10 mm. For steel grade S 355 a strain hardening coefficient of H = 0.0034 is used, refer Table 3-4. κ = 0.5, c1 = 2 (clamped ends) and κ l /dc = 0.5 s/t = 25. This yields the following values for the critical deformation, wcrit, depending on the spring stiffness c, see Table 9.1 (Note: the elastic deformation rel = 6.15 mm is added to the values obtained). By inspection of Figure 9-7 it is noticed that the fully plastic membrane state according to this procedure is attained in all cases but c = 0. Table 9-1 Ductility limit as a function of the spring stiffness c ∞ 1.0 0.3 0 wcrit [mm] 35 51 59 76

9.4.2 Stiffener: Rupture is calculated for the stiffened plate considered in section 9.2.3 using the procedure given in Section 3.10.4. The steel 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. The characteristic dimension of the stiffened plate is dc = 2hw = 0.36 m. This yields λ/dc = 5.56, only. This critical deformation becomes wcrit = 0.1dc = 36 mm, almost independent of the spring stiffness c (Note: ductility ratio is μ = 36/2.2 = 16). This fairly small value is due to the low κλ/dc – ratio for the stiffener. The stiffener is far from entering the membrane stiffening phase, so that any discussion of the possibility for membrane forces to develop is irrelevant.

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Recommended Practice DNV-RP-C204, October 2010 Page 47

If the stiffener is free against rotation and/or has a longer span membrane effects may become important prior to rupture. Observe that rupture is calculated for the location subjected to the largest strains, i.e. at the stiffener top flange. Rupture in the top flange is not necessarily critical with respect to intactness to explosion loads, because the plate side experiences far less strains. It is likely that the plate will remain intact beyond the deformation limit corresponding to rupture in the top flange. A significant part of the contribution to resistance from the stiffener is lost, but the plating between girders may have a significant residual resistance after failure of stiffeners provided that the plate does not disintegrate. It is, however, difficult to provide validated, closed form solution for this situation. A stiffener subjected to pressure on the plate side may trip about the weld toe at mid span. In this case the assumptions used 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 is located 0.315 m from the plate flange. This yields a characteristic dimension dc = 2 ⋅ (1.5 − 0.315) = 2.37 m. The critical location at the clamped side, whereby κ =1/3. Clamped end yields c1 = 2 for the fracture check. With H = 0.0034 and c = 0, there is obtained w/dc = 0.069 and w = 0.095 m.

9.5 Design against explosions - girder 9.5.1 Geometry, material and loads

Plate thickness: t = 14 mm Stiffener dimension: HP240x10, simulated as an L-profile with dimension L240x39x10x29 Stiffener spacing: s = 800 mm l = 3200 mm Stiffener length: Girder dimension: T-girder with dimension: 870x300x10 x20 Girder length: L = 12000 mm The material properties are as follow: Yield strength: Strain rate factor: Effective yield strength: Modulus of elasticity: Material density: Poisson’s ratio: Max. plastic strain:

fy = 420 MPa γε = 1.0 fy = fy· γε = 420 MPa E = 2.1·105 MPa ρ = 7850 kg/m3 ν = 0.3 1.0% (maximum allowable, correspond to cross section class 3 or 4, see sub-section 9.5.2)

Permanent loads and live loads are as follow: Permanent loads: Live loads: Explosion pulse period:

The geometry of the structure is outlined in Figure 9-4. The main dimensions are:

pP = 10.0 kN/m2 pL = 5.0 kN/m2 td = 0.15 sec (triangular load with a rise time = 0.50·td)

Stiffener: 10

240

800 (typ.) 29

Bulkhead

39

Girder:

t = 14

12000

10

Stiffener: Hp240 870 Girder: TG870x300x10x20

Bulkhead

20

3200(typ.)

Figure 9-9 Geometry

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300

Recommended Practice DNV-RP-C204, October 2010 Page 48

9.5.2 Cross sectional of properties for the girder Effective plate flange according to DNV Classification Note 30.1 (July 1995), sub-section 3.4.3 and 3.5.4:

s t

β= ⋅

Cx =

1 .8

β

fy E

−

=

0 .8

β

2

800 420 ⋅ = 2.56 14 2.1E 5

=

1 .8 0.8 − = 0.58 2.56 2.56 2

2 2 ⎡ ⎡ 1 ⎞ ⎤ 1 ⎞ ⎤ ⎛ 3200 ⎞ ⎛ ⎛l ⎞ ⎛ l e = s ⋅ ⎢C x + 0.1 ⋅ ⎜ − 1⎟ ⋅ ⎜⎜1 + 2 ⎟⎟ ⎥ = 800 ⋅ ⎢0.58 + 0.1 ⋅ ⎜ − 1⎟ ⋅ ⎜ 1 + ⎟ ⎥ = 784.6 mm 2 ⎝ 800 ⎠ ⎝ 2.56 ⎠ ⎦⎥ ⎝ s ⎠ ⎝ β ⎠ ⎥⎦ ⎣⎢ ⎣⎢ Determination of cross section class, Ref. NS3472:2001, Sec- tion 12.1:

Web:

Bottom Flange:

(h wg / t wg ) 235 / f y (

=

(850 / 10) 235 / 420

0.5 ⋅ (b fg − t wg ) t fg 235 / f y

Plate Flange:

(

0.5 ⋅ (l e − t wg ) t 235 / f y

)

= 113.6

, i.e. class 3 (bending considered)

0.5 ⋅ (300 − 10) ) 20 = = 9 .7 235 / 420

, i.e. class 2 (bending & axial)

0.5 ⋅ (784.6 − 10) ) 14 = = 37.0 235 / 420

, i.e. class 4 (bending & axial)

)

(

(

In the following calculations, a plate flange width larger than

cross sectional class 3 will not be considered, i.e.:

l e = 2 ⋅ (14 ⋅ t ⋅ 235 / f y ) + t wg = 2 ⋅ (14 ⋅14 ⋅ 235 / 420 ) + 10 = 303.2 mm Gross sectional properties: Effective area of plate flange: Ap = le·t = 303.2·14 = 4245.1 mm2 Area of girder flange: Af = bfg·tfg = 300·20 = 6000.0 mm2 Total area of girder web: Aw = hwg·twg = 850·10 = 8500.0 mm2 Total area (gross section): AG = Ap+Af+Aw = 4245.1+6000+8500 = 18745.1 mm2 Distance to neutral axis (from bottom of girder flange): t fg ⎛ h wg ⎞ ⎛t ⎞ Af ⋅ + Aw ⋅ ⎜⎜ + t fg ⎟⎟ + A p ⋅ ⎜ + h wg + t fg ⎟ 2 2 2 ⎝ ⎠ ⎝ ⎠ z0 = = 403.6mm AG

Web height in tension: ht = z0-tfg = 403.6-20.0 = 383.6mm Web height in compression: hc = hwg-ht = 850.0-383.6 = 466.4mm

le = 303.2 t = 14 hc twg = 10 hwg = 870-20 = 850 z0 ht

tfg = 20 bfg = 300

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Recommended Practice DNV-RP-C204, October 2010 Page 49

Elastic moment of inertia (gross section): ⎛ t fg ⋅ ⎜⎜ ⎝ 2 Effective girder web according to NS3472: Elastic buckling stress

(

)

1 2 I G = ⋅ A f ⋅ t 2fg + Aw ⋅ h wg + Ap ⋅ t 2 + A f 12

f e = 23.9 ⋅

⎛ t wg π2 ⋅E ⋅⎜ 2 12 ⋅ (1 − ν ) ⎜⎝ hwg

2

⎞ ⎛h ⎟ + Aw ⋅ ⎜ wg + t fg ⎟ ⎜ 2 ⎝ ⎠

2

2 5 ⎞ ⎟ = 23.9 ⋅ π ⋅ 2.1 ⋅ 10 2 ⎟ 12 ⋅ 1 − 0.3 ⎠

(

)

2

2 ⎞ t ⎟ + A p ⋅ ⎛⎜ + h wg + t fg ⎞⎟ − AG ⋅ z 02 = 2.407 ⋅10 9 mm 4 ⎟ ⎝2 ⎠ ⎠

2

⎛ 10 ⎞ ⋅⎜ ⎟ = 627.9 MPa ⎝ 850 ⎠

Web slenderness:

λp =

fy fe

=

420.0 = 0.818 627.9

Effective compression web height, see Figure 9-10:

⎧ hc ⎪ hce = ⎨ ⎡ hc ⎪⎢ λ ⎩ ⎣⎢ p

⎛ 1 ⋅ ⎜1 − ⎜ 5⋅λ p ⎝

if

λ p ≤ 0.724

⎞⎤ ⎟⎥ if ⎟⎥ ⎠⎦

λ p > 0.724

⎡ 341.2 ⎛ 1 ⎞⎤ hce = ⎢ ⋅ ⎜1 − ⎟ ⎥ = 430.8mm ⎣ 0.818 ⎝ 5 ⋅ 0.818 ⎠ ⎦

le = 303.2 t = 14

½ hce

hc

Δh twg = 10

z0

hwg = 870-20 = 850

½ hce e

ht

ht

tfg = 20 bfg = 300

Figure 9-10 Effective Girder Section

Effective girder cross section properties Reduction in web height: Δh = hc -hce = 466.4 – 430.8 = 35.6 mm Effective cross section area: Ae = AG -Δh ·twg = 18745.1 – 35.6·10.0 = 18389.1 mm2 Distance to neutral axis from bottom of girder flange: ⎞ ⎛ Δh + hce 35.6 + 430.8 ⎞ AG ⋅ z 0 − Δh ⋅ t wg ⎜ + ht + t fg ⎟ 18745.1 ⋅ 403.6 − 35.6 ⋅10⎛⎜ + 383.6 + 20 ⎟ 2 2 ⎠ ⎝ ⎠ ⎝ z 0e = = = 399.1mm Ae 18389.1

Effective elastic moment of inertia: I Ge = I G −

h 1 ⎛ ⎞ ⋅ Δh 3 ⋅ t wg − Δh ⋅ t wg ⋅ ⎜ t fg + ht + c − z 0 e ⎟ 12 2 ⎝ ⎠

2

2

I Ge = 2.407 ⋅10 9 −

1 466.4 ⎛ ⎞ ⋅ 35.6 3 ⋅10 − 35.6 ⋅10 ⋅ ⎜ 20 + 383.6 + − 399.1⎟ = 2.387 ⋅10 9 mm 4 12 2 ⎝ ⎠

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Effective elastic section modulus: I Ge 2.387 ⋅10 9 W eo = = = 4.923 ⋅10 6 mm 3 t fg + h wg + t − z 0 e 20 + 850 + 14 − 399.1 W eu =

I Ge 2.387 ⋅10 9 = = 5.982 ⋅10 6 mm 3 z 0e 399.1

W e = min(W eo , W eu ) = 4.923 ⋅10 6 mm 3 Plastic section modulus: Web areas:

Aw1 =

Aw 2

hce 430.8 ⋅ t wg = ⋅10 = 2154.0mm 2 2 2

e1 ½ h ce

⎛h ⎞ ⎛ 430.8 ⎞ = ⎜ ce + ht ⎟ ⋅ t wg = ⎜ + 383.6 ⎟ ⋅10 = 5990.0mm 2 2 2 ⎝ ⎠ ⎝ ⎠

Eccentricities (see figure):

e1 =

e3 =

A f + Aw1 + Aw 2 − A p 2 ⋅ t wg A f − Aw1 + Aw 2 − A p 2 ⋅ t wg

⎧ hc 2 + ht if ⎪ e2 = ⎨ 2 ⎪ e3 if ⎩

6000 + 2154.0 + 5990.0 − 4245.1 = = 494.9mm 2 ⋅10 =

½ h ce

e3

6000 − 2145.0 + 5990.0 − 4245.1 = 279.5mm 2 ⋅10

Aw2 ht

hc 2 + ht 2 h e3 ≤ c 2 + ht 2 e3 >

e 2 = 279.5mm

Plastic section modulus if Ap > Aw1 + Aw2 + Af :

hce ⎛ + ht ⎜ ⎞ hce 2 ⎜ ⎟ + Aw1 ⋅ W p1 A h + ⋅ − w2 ⎟ ⎜ wg 4 2 ⎠ ⎜ ⎝ Plastic section modulus if Ap + Aw1 > Aw2 + Af : ⎛ t fg t = A p ⋅ + A f ⋅ ⎜⎜ + h wg 2 ⎝ 2

⎞ ⎟ ⎟ = 8.719 ⋅10 6 mm 3 ⎟ ⎟ ⎠

2

2

e ⎞ ⎛t W p 2 = A p ⋅ ⎜ + e1 ⎟ + A f ⋅ (t fg + h wg − e1 ) + 1 ⋅ t wg 2 2 ⎠ ⎝

⎛ hce ⎞ hce ⎛ ⎞ − e1 ⎟ ⎜ + ht ⎜ ⎟ ⎝ 2 ⎠ 2 + ⋅ t wg + Aw 2 ⋅ ⎜ h wg − − e1 ⎟ = 6.392 ⋅10 6 mm 3 ⎜ ⎟ 2 2 ⎜ ⎟ ⎝ ⎠

Plastic section modulus if Ap + Aw1 < Aw2 + Af : tf ⎛ ⎞ h ⎛t h ⎞ W p 31 = A p ⋅ ⎜ + ce + Δh + e 2 ⎟ + A f ⋅ ⎜⎜ ht + ce + − e 2 ⎟⎟ = 4.259 ⋅10 6 mm 3 2 2 ⎝2 2 ⎠ ⎝ ⎠ 2

2

⎛h ⎞ e W p 32 = Aw1 ⋅ ⎜ ce + Δh + e 2 ⎟ + 2 ⋅ t wg ⎝ 4 ⎠ 2

⎛ hce ⎞ + ht − e 2 ⎟ ⎜ 2 ⎝ ⎠ + ⋅ t wg = 1.812 ⋅10 6 mm 3 2

W p 3 = W p1 + W p 2 = 6.070 ⋅10 6 mm 3

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Aw1

Recommended Practice DNV-RP-C204, October 2010 Page 51

Plastic section modulus: ⎧W p1 if A p > Aw1 + Aw 2 + A f ⎪ W p = ⎨W p 2 if A p + Aw1 > Aw 2 + A f = 6.070 ⋅10 6 mm 3 ⎪W ⎩ p 3 if A p + Aw1 < Aw 2 + A f Ratio between plastic and elastic section modulus: Wp = 1.23 We 9.5.3 Mass Mass from plate:

w p = t ⋅ l ⋅ ρ = 14 ⋅ 3.200 ⋅ 7850 = 351.7

kg m

Mass from stiffener, see figure:

tws = 10

As = hws ⋅ t ws + b fs ⋅ t fs = 211 ⋅ 10 + 39 ⋅ 29 = 3241mm 2 ws = As ⋅ ρ ⋅

3200 l 3241 kg = ⋅ 7850 ⋅ = 101.8 800 s 10 6 m

tfs = 29

Mass from girder:

w g = AG ⋅ ρ =

kg 18745.1 ⋅ 7850 = 147.1 6 m 10

bfs = 39

Mass from permanent loads and possible live loads (to be evalp kg 10 ⋅ 10 3 w PL = P ⋅ l = ⋅ 3.200 = 3263.1 9.807 g m

uated in each case):

Total mass: w = w p + ⋅w s + w g + w PL = 351.7 + 101.8 + 147.1 + 3263.1 = 3863.7

kg m

9.5.4 Natural period Linear Stiffness, Ref. Table 6-2 in Section 6.10: kl =

384 ⋅ E ⋅ I Ge L3

=

384 ⋅ 2.1 ⋅10 5 ⋅ 2.387 ⋅10 9 N N = 1.114 ⋅10 5 = 1.114 ⋅10 8 3 mm m 12000

Natural period assuming uniformly distributed mass (Klm,u is taken from Table 6-2): T = 2 ⋅π ⋅

K lm ,u ⋅ M u kl

= 2 ⋅π ⋅

0.77 ⋅ w ⋅ L 0.77 ⋅ 3863.7 ⋅12.0 = 2 ⋅π ⋅ = 0.113 sec kl 1.114 ⋅10 8

Ratio of pulse load period versus natural period: td 0.15 = = 1.33 T 0.113

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hws = 240-29 = 211

Recommended Practice DNV-RP-C204, October 2010 Page 52

9.5.5 Ductility ratio The maximum lateral deformation prior to buckling can be calculated according to equation 3.19 in sub-section 3.10.2: 2 ⎛ wp 14 ⋅ c f ⋅ f y ⎛ κL ⎞ ⎞⎟ 1 ⎜ ⎟ ⎜ = ⋅ ⎜1 − 1 − ⋅ dc 2⋅c f ⎜ c1 ⋅ β 3 ⎜⎝ d c ⎟⎠ ⎟⎟ ⎝ ⎠

where; dc is characteristic dimension for local buckling, i.e. 2·(t+½hce+dh+e3) = 2·(14+½·430.8+35.6+279.5)= 1089mm c1 is 2 for clamped beams κL is the smaller the distance from load to adjacent joint (0.5). Here set to 0.5·L, i.e. 6000 2

2

⎛ c ⎞ ⎛ 106624 ⎞ ⎟ =⎜ ⎟ c f = ⎜⎜ ⎟ ⎜ 1 + 106624 ⎟ = 0.994 ⎝1+ c ⎠ ⎝ ⎠ ,and c is non-dimensional spring stiffness, ref Section 3.7; c=

k =

Web (bending):

β = 0.8

h wg / t wg

235 / f y

1 k node

1 + 2 ⋅ E ⋅ AG

=

we =

f y ⋅ W e ⋅ L2

32 ⋅ E ⋅ I Ge

=

Calculation of cross sectional slenderness factor, ref. Section 3.10, i.e. the maximum of the following: Plate flange: le / t 303.2 / 14 β =3 =3 = 86.9 235 / f y 235 / 420 Bottom flange:

β =3

b fg / t fg

235 / f y

=3

300 / 20 235 / 420

= 60.2

235 / 420

= 90.9

420 ⋅ 4.923 ⋅10 6 ⋅12000 2 = 18.56mm 32 ⋅ 2.1 ⋅10 5 ⋅ 2.387 ⋅10 9

p M e = f y ⋅ We =

1 = 7.873 ⋅10 9 1 1 + 1 ⋅10 20 2 ⋅ 2.1 ⋅10 5 ⋅18745.1

knode is axial stiffness of the node with the considered member removed, here assumed infinitely. 1.2 ⋅ W p 1.2 ⋅ 6.070 ⋅10 6 = = 396.1 wc = 18389.1 Ae

850 / 10

Based on these input parameters, the maximum plastic deformation is calculated to: 2 1089 ⎛⎜ 14 ⋅ 0.994 ⋅ 420 ⎛ 6000 ⎞ ⎞⎟ wp = ⋅ 1− 1− ⋅ ⎟ = 33.37 mm ⎜ 2 ⋅ 0.994 ⎜ 1089 ⎠ ⎟ 2 ⋅ 90.9 3 ⎝ ⎝ ⎠ The maximum elastic deformation is found from:

4 ⋅ c1 ⋅ k ⋅ wc2 4 ⋅ 2 ⋅ 7.873 ⋅10 9 ⋅ 406.12 = = 106624 f y ⋅ Ae ⋅ l 420 ⋅18389.1 ⋅12000

1

= 0 .8

p ⋅ L2 12

L Maximum elastic deformation: we =

1 p ⋅ L4 ⋅ 384 E ⋅ I

we =

f y ⋅ We ⋅ L2 1 p ⋅ L2 12 ⋅ L2 1 L2 ⋅ ⋅ = ⋅Me ⋅ = 384 12 E⋅I 32 32 ⋅ E ⋅ I E⋅I

Ductility ratio:

μ=

wp we

=

33.37 = 1.80 18.56

9.5.6 Maximum blast pressure capacity From Figure 9-11, the dynamic load factor is found: DLF ( μ ) =

Rm = 0.99 Fl

With reference to Figure 9-11, k3 was set to 0, which ensures conservative results.

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Recommended Practice DNV-RP-C204, October 2010 Page 53

Rel/Fmax=0.05

100

= 0.3

=0.1

= 0.5

= 0.6 = 0.7

Rel/Fmax= 0.8

ymax/yel

10

= 0.9

μ = 1.80

1

k3 = 0 k3 = 0.1k1 k3 = 0.2k1 k3 = 0.5k1

F

R

Fmax

Rel

k3 = 0.5k1 =0.2k1

=0.1k1

= 1.0 = 1.1 = 1.2 = 1.5

k1 0.50td

td

yel

y

0.1 0.1

1

td/T

10

td/T = 1.33

Figure 9-11 Dynamic Response of a SDOF system due to a triangular pulse load profile (rise time = 0.50td)

Maximum resistance for a fixed supported beam, see Figure 9-12: 16 ⋅ M p 16 ⋅ W p ⋅ f y 16 ⋅ 6.070 ⋅10 6 ⋅ 420 Rm = = = = 3399.2 ⋅10 3 N = 3399.2kN L L 12000 Elastic

Plastic

p

pP

L

L

M = pL2/12

MP = pPL2/16

M = pL2/24

MP = pPL2/16 Rm = pPL = 16MP/L

Figure 9-12 Moment diagram (elastic and fully plastic)

Resistance utilised in order to take the permanent and live load: R 0 = w g ⋅ g ⋅ L + ( p p + p l )⋅ l ⋅ L = 147.1 ⋅ 9.807 ⋅12 + (10 + 5) ⋅10 3 ⋅ 3.2 ⋅12 = 593.3 ⋅10 3 N = 593.3kN The maximum blast pressure capacity is obtained from the following two equations: Fl =

Rm − R0 DLF ( μ )

and

Fl = pmax ⋅ l ⋅ L

The maximum blast pressure capacity is then: pmax =

Rm − R0 1 (3399.2 − 593.3) ⋅ 103 1 ⋅ = ⋅ = 0.074 MPa = 0.74bar DLF ( μ ) l ⋅ L 0.99 3200 ⋅ 12000

Note that the maximum resistance (Rm) given above does not include a capacity check with respect to shear. The shear capacity can be determined from sub-section 12.4.4 in NS3472-2001. DET NORSKE VERITAS