An outline for a consistent structural analysis of Vortex-Induced vibration response of pipeline spans Collision analysis for MS DEXTRA

(1)

TECHNISCHE UNIVERSITEIT Laboratorium voor Scheepshydromechanlca Archief Mekelweg 2,2628 CD Deift Tel.: 015-786873 - Fwc 015-781838

AN OUTLINE FOR A CONSISTENT STRUCTURAL ANALYSIS

OF VORTEX-INDUCED VIBRATION RESPONSE OF PIPELINE

SPANS

BY

(2)

'i(Xj mcZJ

Ï3Q co

ac

(3)

N0

AN OUTLINE FOR A CONSISTENT STRUCTURAL

_{ANALYSIS OF}

VORTEX-INDUCED VIBRATION RESPONSE OF PIPELINE

_SPANS

INTRODUCT ION

It is the

_purpose

_of

_{the :present}

flote to _describe _a

non-linear,

_analytical

_procedure

which

can

be used to

determine

the static

_equilibrium

_form _of _a

prestressed

pipeline with a free span assuming a nonlinear

_{axial friction}

law.

As a perturbation

_to

_{this static}

equilibrium form

_{we shall}

present

a

procedure

for

_calculation

_of

free and _forced

vibrations

Figure 1. Pipeline with a free span on a rigid

_foundation.

In case of a pipeline on a rigid

foundation, see Fig.

1.

D.M. Richards

_{and A. Andjonicou}

have

_{presented a static}

as

well as a dynamic

analysis in the paper: "Seabed Irregularity

Effects

on the

_Buckling

_of

_Heated

Submarine

Pipelines"

presented

at

Holland

_Offshore

₈₆

"Advances

in

_Offshore

Technology" 25.-27.

_{November 1987.}

Thus,

_in

_{the case of}

a rigid

_foundation

_{a certain background}

exits,

_therefore

_in

the following

_{we shall}

concentrate on

cases where the

_{pipeline is embedded}

in sand or clay.

_See

(4)

Figure 2. Pipeline with a free span on an elastic-plastic

foundation

A PROCEDURE FOR CALCULATION OF THE STATIC EQUILIBRIUM FORM.

We shall assume that the undisturbed pipeliñe

_{is subjected to}

a compression load N

_{Which is composed of residual laying}

forces

_{forces caused by temperature differences and}

_forces

caused by internal pressure

_{p. The axial Stress in the free}

span part of the pipeline N cán differ 'from N0 due

_{to axial}

friction

_forces

between

the

pipeline

surface

and the

surrounding

soil and due to lateral

deflections

of the

pipe 1 inc.

In the static case the non-dimensional differential eqùation

govering the free span can be expressed

as

y'''(x) -X

(x) =

where- the effective compression load

N +

D2Ap

= -

XEI/L2

(5)

and the line load

The boundary C0flditjos to

and

Q =

q EI/L3 y'(o,)

= Y'''(ó) =

= o: (2) At

_{the flexible}

support (x=l)

the

_{linear relations}

between

the deformations

_y(l)

y'(l) and the

_bending

moment and the

shear force for a given axial

_{force N}

can be determined

_from

Eqs. _(lOOa-e) _in

M.

_Retenyj'3

book.:

_Beams

_on

Elastic

Foundation"

_{Thus, the}

remaining two boundary

_{Conditions have}

the form:

y''(l)

_{+ Siiy'(l)}

+

Stzy(1)

O

(3)

- Xy'(.l)

_-

_5ztY'(l) _{- s22 y(l)} = o

where the

_stiffness

coefficients

S

_EI/L

functions, of N

_and

the foundation

stiffness.

aß are known

For a

_given

_value

of X the _general

solution

to _the

differential

eq.

(1) can be written as:

for X. > Y(x) ₌

_C

_Cosh

_IX

x

+ C2 sinh

¡r

₂ 2X for X < O: (4) y_{Cx,) =}

_{ci co}

fX

+ c2

ir

+ 3 X + q _C

₎

_{express symmetry}

for

(6)

N0.

where the

_{integration constants CC4}

_{can be determined}

from

the four boundary conditions

_{(2) and}

_(3). _.Due _to

che

present, choice of coordinate System the

_{symmetry conditions}

(2) lead to C2 = C3 = O

Therefore

only C1 and C4 neèd

_{to be}

determined from two

mear equation obtained by

_substituting

(4) into Egs. (3.).

:1,

1!

I

N

L.L1i.ij I'! T IVI I LT! UIL! I

I F ! I '

Figure 3. The axial compressive force

_{as function of' the}

axial coordinate

In order to achieve equilibrium between the axial force Ñ0 in

the und.isturbed pipeline and the áxial force

'N

in the free

span of

the

pipeline

section, we have to

cOnsider axial

displacements and the associated frictional

_{fOrces. At}

_the

boundary of

the. tree span the axia,l shortening u(l). of

the

pipeline can be expressed as:

u(l) =

U" {4

[ ]

-

N0E: N}dx

₍₅₎

This axial displacement müst

now be

related to

the axial

feed-in

to the free span from the

supported part

of the

pipeitne.

l'o

relate the axial force N in the

_{free span part of}

_the

pipeline to the axial. force N0 in the undisturbed buried

part

of the pipeline the axial force eqii1ibrium

at the boundary

can be expressed as, cf Fig. 3.:

(7)

-C a.

2v

C o.-= 0. 'J

:

a'-u' Axial displacement u

Figure 4 Axial, friction force per unit length as function of

axial displacement

u.

N = N0 - P(u) - RX2

(6)

Here P(u)

is

the drop in axial force due

_{to the axial feed}

-in u(1)

from the buried part of

_{the pipeline, RX2. is}

_the

Mct lanai axial force due to the concer- trated

_{support force}

R at the polt.t where the pipe leaves the foundation.

In

the present non-linear calculation model

_{we. shall} _take.

into acóunt that a certain axial displacement

_{is necessary}

before full mobilization of

_{the Coúlomb force}

_{Is achieved.}

Thus,

the axial friction force

_{per unit lefigth between the}

outer pipeline surface and

the

Surrounding so1. Over

the

length between

th.e

_{lift-off point and the}

_{region with}

_the

completely restrained pipelIne

_{sée Fig. 4.} _{is taken as:}

q0

Íuc)

_{for u(X) <}

_Uj

for

(g)

U1

(7)

Here q0 is the sum of the

_{pipe weight and the soil}

_presSure

for zero uplift.

_Similarly,

_{the horizontal}

_friction

_force

caused by the concentrated support force R is taken as:

8ilinar Aporozimatson

s'

Experiments

(8)

where the reaction fOrce R is giienas:

R qL

The axial

displacement

ü(x) of the

pipeline

bètween

the

boundary point ánd the axially fully restrained

_{pipeline is}

governed by

where AE denOtes the axial stiffness of the pipe.

In

the elastic range x

<. x1

where u(

solution to eq.(9) is:

u(x) = C1atx

+

C2tX

where

-

ru(i) RX2 R?2 ÄE = 4o?t dx2

From the boundary condition

i im [u ] O

we find that Ç

= O.

=Jc;x7i

a1 -

u1 AE

for

u(l) <u2

for u(l) > u2

(new x; O

x ()

(S)

(9)

In the Coulomb range O

x

X1 where ù(x)

u1 the so1utjo

to eq. (9) is: u(x) 4 b1 x2 + b2x + b3

where

b1 q0X1 AE constants.

At the transition point x1 between these two regions we must

have continuity in u and in du/dx:

C2ealxt ₌ _i6

_b1x

_{+ b2x1 + b3}

₌ _u1

a1 x1

a1C2e =

b1x1 + b2

If we further use that at the edge Of the free

span of

the

pipeline the axial deformation is u(l) then we have the

fol-lowing equations:

where x1. b2 and b3 are unknowns.

The solution i

b3 =

u)

b2 =

a1u1 - b1x1

afld

xl =

and b2,

b3 aré integration

= a1u1 xl o xl i o i O b1 b2 b3 = UI atu u (i)

(10)

or

Still assuming u(i)>ü1 then the

_{strain for x} ₌ _O s:

and

Thus, the axial force P(u) is:

P(ú) = EA =

_{1X1EA(2u(i) - u)}

P(u) b1 0 + b2 = a1 u

/qpX1

-

.AE.U1

-

/qoXj -. AE

q01EA

u(1) 2u(i).i1

i

-

b1x1 g0X1

_{/_AU I}

_ri-;-

-AE

_{q0X1 L.}

2u(l)-for u(l)> (10)

for u(l) u (li)

Let us now first consider the

case Where u(1')

> u1 and u(l)

>

u2.

In this case a cOmbination of Eqs. (10)

and (6) leads to:

qo'X1 EA (2u(1) - u1)

= (Ñ0 -

N - RX2)

fu

-L

- Af2)

+

-

2qX1 EA

2

Equating (12) and (5) léads to:

(N0 - N)2 + R2X

- 2R2 (N0 - N)

+

u1q'X1EA

=

q01EA

¡j

[]

dx. - 2Lq01 (N0 - N)

(11)

u(l)

(N

very simple

N0 - N

[L2

u2 +

/XEAJ

ut

Equating (14) and (5) wé find;

Iq!L.

J0X1EA)

i

--

N)

EA -L I

U r =

il

íi

EA [qL!X2 ₊

_1q0X1EA

u2 _ut -2

-

ut4o.iAJ

_{T.. .}

₍₁₃₎

But even

when q2

_q0'X1 we get a

re1atvely

simple

expression. The integral in (13)

_{can easily be evaluated}

analytically using (4)

Let us also consider the case where

(i) _ú

_atid

_ü(1)

u2.

Then a combination of Es. (11)

_{and (6) leads to}

(14)

(15)

If. we. re.s.0 i

t

now assume

N = N0 that [qo qA2 1EÄ

U'

=

qoj

2

we get

the 2y 21V '-

"2

ldx, br N = N0 EA

-LqX2

_{+ Jq0X1EÀ}

EA -

L

qLX2

+ JqpEA

U2

(12)

The solution procedure

is then the following:

Based on

a

starting value of N the intègra.1

í

d x

is calculated from an analytIcal expression. Then a neW value

of N Is estimated from Eq.

_{(13) or Eq.}

_{(15). This new value,}

of N is used to calculate an improved value N of

_{the axial}

force.

After concergence the static equilibrium form (4).

_{the axial}

compression

N.

and

the moment

distribution

(Eq. (4)

(13)

A PROCEDURE FOR CALCULATION OF VIBRATIONS.

vibrations

fa Straight pipeline

_iS

_{governed by}

_the

linear, homogeneous, non-4imensonaidifferentialéqu'tjon

yr...,.. I

( .)

-X.''(*)..-.y(x)

= O fo.r:O

xr

1»:.::(l6)

where the fréguénc.y _I$EI/(m1

and, where rfl

is the mass per unit length Including the added

mass.

The general solution to Eq. (16) has the form:

.y ( ) =

Cj cash ax

+

C2 sinh ax

+

C3 cos bx

+

C4 sin bx

(17)

where a and b, which may be complex quantities, are given by:

1f

X = a2 - b2:

The boundary

conditions

for x . 1 are the Same as the

boundary conditions (3) for the static problem. At x

O the

boundary conditions express either symmetry;

y'(0)

y'''(Ö) = O

(18)

or anti-symmetry

y(0)

= y''(0).=

0 ₍₁₉₎

In

the case of

_{symmetric vibration modes,}

_Eqs. _(18). _the

integration constants

C2 C4 = O. And In the case of

(14)

antisymmetric vibration modes

= C3 O. Inserting the

remaining two

terms of

the general solution

(17) into the

boundary conditions. (3)

we Obtain tw. linear.: homogeneous

equations In. the unknownC1,

thus,

_{the :atur1}

_{frequencies.. cn}

_either _case _be

determined by equating a Second order determinant

_of _the

coefficient matrix to C1. C3 or C2, C4 tO zero.

Thé. results

can

probably

be

depicted

in nOnT-dj.mensjonál

diagrams of the following form

300

s

200 loo

C3 or C2, Ç4:.:.;

t mode

N

\

_\

-Ï0 _O

₁₀

Figure 5. A proposal for presentation of

natural frequencies as functiofl of effective

(15)

These results can, be used

_{as approximations}

_to

_{the naturäl}

fre4uencies for out- of- plane

vibrations and

fôr higher

order in- plane vibrations. But primarily

_{we Shall}

se _the

mode shapes

generated

by this solution as

coördinate

functions in a

Rayleigh - Ritz procedure for

_{analysis of}

free and forçe4

vibrations

of the sagged pipeline. As

mentioned before the differential

_{equation (16) is only valid}

for small amplitude,

_{free vibrations of perfectly}

_straight

beams.

If we still

_{assume moderately small ampltude vibrations,}

i.e. smaLl rotations, then an approximation

_{to the governing}

differential equation for an

_{initially straight,}

_perfectly

elastic,

pipeline which has

been loaded

into a deElected

shape with a static deflection Y

can be expressed as:

a4y 'e _a2y

EI - + E I

- N(u(1))

ax4 otat4 - ax2

+ c( 82Y .) + (m+mb(x)..) = Q(x,t) 8t at2 a3y âX2Ot (20) where

Y(x,t)

_{Yi(x) + Yd(x,t)}

₍₂₁₎

Q(x,t)

st a t

i

Cx)

X, t)

NN

_static

d

añd

E'e

represents rnate:rial damping, N'e axial.

_{support damping.}

c(x) viscous external damping

_and

_{mb added soil mass at the}

(16)

We. shall again introduce non-dimensional variables:

xX/L

whe,e,ii,

the lowest natural fréquency

ê4uiva

Ientstraightbeam.

Furthermore, we. shall Uçjljz'e. that

fulfils' Eq. (l)..This leadsto

Yd

+ :i

Yd. -

(X, _Xd) "d -. where a1 2 *E y = tL

f,

N,1L2

a2 = -2r.EI

c1L4

a3

-2,rE I -mb(x)LcI)1 EI

4ir2

añd. where' (Xd + X) Is determi4ed by substituting (21) Into

(13) and. and/or (15).

The

boUndary cOñdltlon to

Eq.. (21.) 'are of

the form (18)

(19)

In additión to

(3) possibly with the

addittón of a

of damping

terms.

(,112

w242

Ï r '

the static.. solUtion y

determined for

th y = (22) or sèt + _{a,3. y}

(17)

In order co obtain a solutón to this boundary

_{value problem}

it

Is suggested to use the Rayleigh

_{Ritz method. That Is.}

the solution is expressed as

_{a finite sum.}

1g

(t) y1(x) ₍₂₃₎

whre )(x) are the lowest symmetric and

_{antisymmétric modes}

determined

from (17) and where q1(t) are time

_dependent

functions which have to be determined.

The so far unknown functions q1(t)

_{aré to be determnéd from}

a set of áoupled, órdinary. second order

_differential

equations which can bè obtained by

_{insérting (23) Into (22).}

multiply

_the

_{resulting equation by y(x)}

(j=l n).

integrate the resulting

_{n equations from O to L and finally}

use the

_{crthogonaiity}

_conditions

_which

_are

valid for

eigenfunctions fulfilling

eq. (16). If

_{only one coordinate}

function is used.

i.e. n=l,

_{then only one non-linear}

_second

order diffentia]. equation, in

_q1 _(t)

_{is obtained which will}

have a form similar to

_{eq. (13) in:}

Brus'hi. R.

_{and Vitali. L.:}

_{"Large Amplitude Oscillations}

of

Geometrically

non

Linear

Elastic

Beams

Subjected

to

(18)

CONCLUS ION

The purpose with

the presentéd

_{note has been}

_to _give _an

outline of a reasonably consistent procedure for calculation

of free and forced vibrations of freepanning pipelines

In order

to prOduce non-dimensional diagrams which can be

used

for

design

purpose then the

necessary

detailed

derivations must 'be performed and the few equations presented

here. 'must be checke4.

The main result of the present study is that lt

_{seems to be}

feasible to

_{derive simple. analytical results which}

_{can be}

used for evaluation of the allowable free

_{span lengths for a}

given pipe with given soll çondltiOnS in a given water flow.

Lyngby, February 1988

(19)

P. T. Pederseñ & S. Zhang, Technical Uñiversity of Denmark: Collision Analysis for MS DEXTRA

I Introduction

The present paper describes part of the research work performed in the project Design for Structural Safety under Extreme Loads (DEXTREMEL). The project partners are

Germanischer Lloyd (Project Coordinator), Technical University of Denmark,

Maritime Research Institute Netherlands, National Technical University of Athens,

SIREHNA, University of Newcastle upon Tyne, and Astilleros Españoles. The project is supported by the Commission of European Countries as project BE97-4375.

DEXTREMEL addresses three extreme load scenarios, which must be investigated prior to the evaluation of residual structural integrity of RoRo ferries in adverse

conditions. These scenarios include structurai damage due to 1) collision and

grounding loads, 2) bow door loads, and 3) green water loads.

The present paper deals with the work done during the first 16 months on the

probabilistic prediction

of

the frequency of collisions and the spatial probabilistic

distribution of collision damages.

This part of the research is divided in the following subtasks:

1.0 Definition of relevant design cases

1.1 Developmentofprobabilistic models for collision event

Collision Analysis for MS DEXTRA

By

Paper number (2) SAFER EURORO Spring meeting, NANTES 28 April 1999 1

Preben Terndrup Pedersen and Shengming Zhang Department of Naval Architecture and Offshore Engineering

Technical University of Denmark Build. 101E, DK-2800Lyngby

DeIft University of Technology

Ship Hydromechanics Laboratory

Library

Mekelweg 2 2628 CD Deift

Phone: +31 (0)15 2786873 E-mail: p.w.deheertudelft.nI

Abstract

It is a major challenge to for the maritime community to develop probability-based procedures for design against collision and grounding events. To quantif' the risks

involved in ship traffic in specific geographic areas implies that probabilities as well

as inherent consequences of various collision and grounding events have to be

analysed and assessed.

The present paper outlines such a rational procedure for evaluation of the probabilistic distribution of damages caused by collisions against other ships for a specific ship on

a specific route.

The work described in the paper constitutes a step towards the more long term goal to

develop probability-based codes for design against collision and grounding events,

similar to the present development towards the use of reliability-based procedures for strength designofships subjected to the traditibnal environmental loads.

(20)

P. T. Pedersen & S. Zhang, Technical University of Denmark: Collision Analysis for MS DEXTRA

1.2 Development of models for external ship collision dynamics

1.3 Development of models for internal ship collision dynamics

1.4 Establishment of damage probability distributions So far the effort has been concentrated on Tasks 1.0- 1.3.

During the subtask 1.0 the participants identified a relatively fast (26 knots) RoRo

ferry with a length between perpendiculars equal to 173 m, breadth of 26 m, and 6.5 m draught, see Arias (1998). The project ship, named MS DEXTRA, will during the

project be analysed for two different routes; one of them is between Cadiz in

mainland Spain and the Canary Islands.

srRUci-rj aj

STR UU NG. COLI-I SlOW ANGLE

-

lwr.

OIJTE L1 __: ¿:7

-

E.L,Efl

PRO S,Lrry f C.OW&IOl.J

CHANÑL

/

//

/ /

.uL-rG. YES t..Jo DATA LLCTON

-Fig. 1.1. A schematic illustration of steps in collision analysis

The contents of the remaining four subtasks 1.1- 1.4 are illustrated graphically in Fig.

1.1.

Subtask 1.1 is devoted to a methodology for calculation of the probability of ship-ship collisions for a ship on a given route where the marine traffic is known. Steps in the procedure for this analysis are indicated in the upper part of Fig.1.1.

For each individual collision scenario the subsequent analysis procedure is divided

into two tasks:

(21)

The external dynamics, and The internal mechanics.

Thé external dynamics deal with the energy released for crushing of the involved ship structures and the impact impulse of the collision by analyzing the rigid body motions of the colliding ships taking into account the effect of the surrounding water.

The internal mechanics deals with the structural response and damage caused by the energy released to be dissipated by crushing of the ship structures during the collision.

These two tasks can in most cases be treated independently.

A complete probabilistic collision analysis as the one indicated in Fig. 1.1 involves analysis of several thousands of collision scenarios. Therefore, for practical use in probabilistic models it is necessary that each individual collision analysis can be

performed by relatively fast procedures and thus in most cases simple procedures. Of course, since simplified structural analysis procedures rely on a number of simplifiing assumptions these must be validated by either experiments or detailed Finjte Element Analysis procedures. The following paper by Samuelidis will describe such a detailed

finite element analysis performed for validation of the simplified methods to be

described in the present paper.

2. Collision probability analysis

In recent years there has been a rapid development of new navigational systems. A growing number of VTS systems are established around the world. Extensive trials

have beefl carried out with sole lookout during night on ship bridges. 1MO has

introduced requirements for new ships to fulfil particular manoeuvrability criteria

[1MO Resolution A 751]. And a new generation of large fast ferries has emerged. It is generally agreed that all these activities have considerable influence on the probability

of ship accidents in the form of collisions and grounding. But so far no rational

analysis tools to quantif' the effect of these changes have been available. Instead

nearly all research on ship accidents has been devoted to analysis of consequences of

given accident scenarios. It is with this background that the work in this project,

Hansen and Pedersen (1998), on a rational model for determination of the probability for ship accidents has been carried out.

The main principle behind the most commonly used risk models is to determine the number of possible ship accidents Na, i.e. the number of collisions if no aversive manoeuvres are made. This number Na of possible accidents is then multiplied by a

causation probability P in order to find the actual accident frequency. The causation probability P is the fraction ofthe accident candidates that result in an accident.

For calculatión of the number of possible ship collision candidates Na we shall

consider two crossing waterways Where the ship traffic

is known and has been

grouped into a number of different ship classes according to vessel type, dead weight tonnage or length, loaded or ballasted, with or without bulbous bow etc.

(22)

P. T. Pedersen & S. Thang, Technical University of Denmark: Collision Analysis for MS DEXTRA

Fig. 2.1 shows such two crossing waterways. In Pedersen (1995) is presented a

calculation model for the number Na of possible events where two ships will collide in

the overlapping area Q if no aversive manoeuvres are made. The result from this

reference is that by summing all the class "j" ships of waterway 2 on collision course

with all relevant class "i" ships during the time & the following expression can be

applied: / // Z1

/"

_/ /1 ,/

IIPI

f(z) ,'

¿;.

II

/ lulL /

liii /

"b;

Fig 2.1. Crossing wateiways with rik area of ship-ship collision.

Q(I) Q(2)

Na=J J.

I J

f['(z1)f52(z)VD,jdA&

J _iz)

v' V?

Here QJa) is the traffic flow (i.e. number of ships per unit time) of ship class

_{j in}

waterway no. a, v7 is the associated speed. The lateral distribution of the ship traffic

of class j in waterway a is denoted j7, Dg is the geometrical collision diameter

defined in Fig. 2.2, and finally the relative velocity is denoted V,.

(2.1)

(23)

P. T. Pedersen & S. Than g, Technical University of Denmark: Collision Analysis for MS DEXTRA

Fig 2.2. DefinItion of geometrical collision diameter D.

The expected number of ship-ship collisions is then determined as

Nhzp-hip = Fc Na (2.2)

Here the causation probability P can be estimated on the basis of available accident

data collected at various locations and then transformed to the area of interest.

Another approach is to analyse the cause leading to human inaction or external

failures and set up a fault-tree procedure. In the present work we have applied a new procedure based on a Bayesian Network procedure for calculation of P presented in Hansen and Pedersen (1998).

The methodology for the developed procedure is based on the assumption that the

ship and crew characteristics and the navigational environment mainly determine the

collision probability. That is, technical failures such as engine failure and rudder

failure play a minor role.

The most important ship and crew characteristics are taken to be: ship speed, ship

manoeuvrability, the layout of the navigational bridge, the radar system, the number

and the training of navigators, the presence of a look out etc. The main parameters

affecting the navigational environment are ship traffic density, probability distributions of wind speeds, visibility, rain and snow.

With knowledge of the ship characteristics and a study of the human failure

probability, i.e. a study of the navigator's role in resolving critical situations, a set of causation factors P has been derived.

By application of the Bayesian Network procedure for estimation of the causation probability, it is possible through future research to examine the beneficial effect of

(24)

new bridge procedures, of improved rnanoeuvring capability, of having a pilot on

board, or of introducing a VTS system in certain geographical areas.

Based on the mathematical model for estimation of collision probabilities described above a computer program has been written for calculation of collision probabilities

in specific waterways where the ship traffic distribution is known.

-

ry

_rnrr..a

Fig 2.3 MS DEXJ'RA chosen as design case for DEXTREMEL

The basic idea behind the procedure is that one specific Ro-Ro passenger vessel (MS DEXTRA see Fig. 2.3) is making round trips on a specific rotAte between Cadiz and the Canary Islands.

On this route there is further a given distribution of other types of vessels such that three in principle different types of collisions can occur. One type of collision is a

head-on induced collision due to two-way traffic in the straight waterway segments. Another type of collision oñ the Ro-Ro waterway occurs at bends where two straight

route segments intersects. At such an intersection a ship can become a collision

candidate if the course is not changed at the intersection. This probability of omission Pois taken as 0.01.

Finally, the model calculates the probability for collisions due to an arbitrary number of crossing routes, as indicated in Fig. 2.1. These crossing routes are defined to have one-way traffic only and have a specified distribution of the types of vessels. Thus, if two-way traffic exists in the crossing route, the route must be included twice, one for

each direction of the traffic. This strategy is adopted to allow for different traffic

distribution in the two directions.

For each waterway segment the number of collision candidates related to head-on,

intersections, and crossing situations

is calculated

for each

vessel

type. The

probabilities of collision is then estimated by multiplication with the causation factors verified by the Bayesian Network procedure

P[head on]

= 4.9l0

P[intersection]

= l.310'

P[crossing]

= l.3l 0

(25)

The overall ship traffic in the considered geographical area is divided into different vessel types and into different size categories. For eách of these vessel classes a

certain fraction is assumed to be in ballast condition and the remaining vessels fully loaded. Similarly, the fraction of vessels without bulbOus bows has to be specified

The striking vessels are grouped in the following categories: Bulk carriers Chemical tankers Container vessels Gas tankers Oil tankers Other vessels Passenger vessels Ro-Ro vessels

Each of the categories contains a number of vessels. Each vessel is defined by the

following properties:

Length

Breadth

Depth

And, for both loaded and ballast conditions:

Speed

Draught Displacement Height of deck Height of bulb

Finally the database has data for probability of the vessel: Being in a loaded condition

Being in a ballast condition Having a bulbous bow

Among the routes intended for the RoRo vessel MS DEXTRA is the traffic between

Cadiz in mainland Spain and Las Palmas and Tenerife on the Canary Islands. The

main traffic routes in this area are sho in Fig 2.4.

(26)

P. T. Pedersen & S. Zhang, Technical University of Denmark: Colusión Analysis for MS DEXTRA Mediterranean Sea Canary Is landS Tenerife_/ 280

Figure 2.4. Ship Traffic between Cadiz and the Canaiy Islands.

The geographical position of the six geographical fix-points in Fig. 2.4 is found to be:

Cadiz 36°30' N; 06°20' W Turning 28°28' N L Palmas 28°05'.N; 15°27' W Tenerife 28°28' N; 16°15' W Europe 36°07' N ; 05°21' W C Vinc. 37°00 N ; 09°.00' W

According to Spanish port statistics the total annual traffic to the harbours in Tenerife

and in Las Palmas is 8647 and 8012 ships per year, respectively. The annual ship

traffic out of and into the Gibraltar Straight was according to "COAST 301" in 1981

in total 51138 ship passages. Recently, i.e. 10. Dec. 1998 the total traffic for 1997

became available. Now the traffic has increased to 65597 ships. But since the

composition of the ship traffic in 1997 is not available we have in Hansen and

Pedersen (1998) applied the COST data.

The result of the numerical calculations is a complete distribution of collision

scenarios as indicated in the upper part of Fig 1.1. The integrated result is that the

annual probability of collision involving MS DEXTRA is .042. That is, a collision is expected to take place with a return period of 24 years.

(27)

P. T. Pederen & S. Zhang, Technical University of Denmark: Collision Analysis for MS DEXTRA

3. External Dynamics

3.1 Introduction

Knowing the probabilistic distribution of collision scenarios the next step is to

determine the consequences of these collisions. Here the first step is to determine the

fraction of the available kinetic energy that is released for crushing of the involved

ship structures.

For this, an analytical method for the energy loss and the impact impulse has been developed for arbitrary ship-ship collisions, see Pedersen and Thang (1999). At the

start of the calculation, the ships are supposed to have surge motion and sway motion, and the subsequent sliding and rebounding in the plane of the water surface during the

collision are analyzed. The energy loss for dissipation by structural deformations of the involved structures is expressed in closed-form expressions. The procedure is based on rigid body mechanics, where it is assumed that there is negligible strain

energy for deformation outside the contact region and that the contact region is local and small. This implies that the collision can be considered as instantaneous and each body is assumed to exert an impulsive force on the other at the point of contact. The

model includes friction between the impacting surfaces

so those situations with

glancing blows can be identified.

3.2

Theoretical Analysis Model

To illustrate the assumptions behind the method we shall consider a striking ship (A),

which sails at a forward speed of V and a speed of in the sway direction and

collides with a struck ship (B), sailing at a forward speed of Vb, anda sway speed of

V,2. An XYZ-coordinate system is fixed to the sea bottom. The Z-axis points in a

direction out of the water surface, the X-axis lies in the symmetry plane of the striking ship pointing towards the bow, and the origin of the XYZ-system is placed so that the midship section is in the YZ-p lane at the moment t = O. The origin of a 17-system

is located at the impact point C, the - direction is normal to the impact surface, the

angle between the X-axis and the rj -axis is a, and the angle between the X-axis and the 1-axis is ß, see Fig. 3.1.

(28)

P. T. Pedersen & S. Zhang, Technical University of Denmark Collision Analysis for MS DEXTRA

Fig. 3.1. The coordinate system used for analysis of ship-ship collisions.

The equations of motion of the striking ship (A) due to the impact force components

in the - direction and iñ the T - direction can be expressed as

M0(1+m)v. =F sinaF cosa

(3.1)

M0(l+m,,)v0, =-1cosa+Fsina

(3.2)

MaR02 (1+fa)O)a =

(3.3)

+F[ycosa+(x xa)siflal

Here _Ma

is the mass of the striking ship, (v,v,o0) denote the accelerations

during the collisions of the striking ship in the X- and Y-directions and the rotation

around the center of gravity, respectively. The radius of the ship mass inertia around the center of gravity is Ra the coordinate of the center of gravity of the striking ship

is (Xa ,O), the coordinate of the impact point is (x _,y the added mass coefficient

for the surge motion is m, the added mass coefficient for the

sway motion is m,

and the added mass coefficient of moment for the rotation around the center of the

gravity is Ja

Similarly the motions of the struck ship can be expressed as

Mb(1 + m61)vM =

F sin(ß

-

a) + F cos(fi

_{- a)}

_(3.4)

Mh(l-I-mh2)vb2

= F cos(ßa)+ P sin(ßa)

(3.5)

MbRb2(1 +Jb)wb =

F{(y y)sina (xe xh)cosa]

- F,1[(y y)cosa +(x xh)sina]

(3.6)

(29)

The added mass coefficients m,may,ja and

mbl,mb2,jb, taking into account the interaction effects between the ships and the surrounding water, depend on the hull

form of the ships and the impact duration etc. For simplicity, Minorsky (1959)

proposed to use a constant value of the added mass coefficients of ships for the sway

motion:

m0, =0.4

Motora et al. (1978) conducted a series of model tests and a hydrodynamic analysis on the added mass coefficient for the sway motion. They found that the added mass

coefficient varies during the collision, the value is in the range of =0.4 - 1.3. The longer the duration, the larger the value of the coefficient. However, if the collision

duration is very short, the value of m,= 0.4 assumed by Minorsky may be correct. In Petersen and Pedersen (1981), it is shown that the added mass coefficient for the sway

motion can be estimated from

={m(oo) k[m(0) -m(oo)] }

where m(oo) and m(0) are the threshold values of the addedmass coefficient for the

sway motion when the frequency of the collision approaches infinity or zero,

respectively. The value of the factor k is a function of the duration of the collision

and the ship draught.

The added mass coefficient m related to the forward motion is small compared with the mass of the ship. It is found to be

m =

0.02 to 0.07

The added mass coefficient for the yaw motion of the ship,

j,

is (Pedersen et al.,

1993):

L

=0.21

For simplicity, in

the examples of the present calculations, the added

mass

coefficients are taken to be

m = m,,1 =0.05 (for the surge motion)

m = m,,2 = 0.85 (for the sway motion)

= L =

0.21 (for the yaw motion)

In the examples of the present calculations, the radius of inertia is taken

to be a

quarter of the ship length: R = La /4 and Rb Lb /4.

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By a transformation of the Eqs. (3.1) - (3.6) to the moving _{- Ti coordinate system}

followed by an integration of the relative equations of motion with respect to time, the impact impulse in the - direction and in the 11- direction can be obtained as

'0 =

dz=K,c(OX1+e)D71ri(0)

b .k71 D,1 K

'710 ₌f F71dt

D (0)(I+ e)

Here the coefficients D, K , D71, K are algebraic expressions involving the masses

of the vessels, e is the coefficient of restitution, the added masses and geometrical

parameters describing the location of the impact point, see Pedersen and Thang

(1999).

The ratio ofthe impact impulses, expressed as

D ri(0) - K

(0)(1 + e)

'0

_{K,,(0)(l+e)D71ij(0)}

(3.7)

determines whether the ships will slide against each other or the collision point will be

fixed.

In the case where the ships sticks to each other at the collision point the

energy released in the - direction E can be expressed as

= TF dc = ¡ (3.8)

o

and the energy released in the ? - direction E71 can be expressed as

E71 =71T1d= I i (3.9)

where and are the penetrations in the - direction and in the r - direction at the end of the collision. The total released energy is the sum of the energy released in

the, - direction and in the ij - direction: E,010, = E + E71.

Similar equations have been derived for the case where the ships slide against each

other.

Paper number (2) SAFER EIJRORO Spring meeting, NANTES 28 April 1999 ₁₂

21

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3.3 Verification of Simplified External Dynamics Model

For verification of the procedure let us first consider an example from Brach (1993) using a slender rod (free in air) impacting a surface. The problem is illustrated in Fig. 3.2. The physical parameters are presented in Table 3.1. A comparison of the present results with Brach's results is given in Table 3.2. In the example, the initial velocity of the rod in the normal direction of the surface is (0) = 1.0 mIs. The initial rotational

velocity of the rod is zero and the initial velocities in the direction parallel to the

surface are i(0) = 0.0 m/s, -0.2 mIs, -0.6mIs, and

1.0 mis, respectively. The

coefficients of restitution are e = 0.5 and 0.05, respectively.

The coínparison shows that the present results and Brach's results agree quite well.

Fig 3.2. Diagram of a slender rod striking a massive plane at the poInt C

Table 3.1. Physical parameters of the slender rod

Paper nwnbr (2) SAFER EURORO Spring meeting, NANTES 28 April 1999 13

Mass

M1.Okg

Length (m) L=1.0 m

Moment of inertia

1= ML2

12 Impact angle 45 deg. Initial velocities (mis) &0)= 1.0, u(0) =

ti(0) = 02, 0.6,

0; 0, 1.0

(32)

Table 3.2 Comparison of the present results with Brach 's results.

# Critical value which just causes the glancing to stop.

The second verification example is taken from Petersen (1982) and Hanhirova (1995). The results presented by Petersen were calculated by time simulations and the results

obtained by Hanhirova were calculated by an analytical method. The

case was a

collision between two similar ships. The main dimensions of the ships are given in

Table 3.3.

Firstly, we use the same assumption as used by Petersen. That is an entirely plastic collision where the tWo ships are locked together after the collision. The present calculation results and the existing results are presented in Table 3.4 where d is the

impact location measured from the centre of the struck ship. From Table 3.4 it is seen

that good agreement is achieved except in case No. 4 We cannot explain the

difference in case No. 4, except that the high value given by Petersen for this case

does not seem reasonable.

Table 3.3. Main dimensions of the ships used for validation.

initial velocity Coefficient of restitution (T) Impulse ratio Normal impulse

'

Energy loss 100% Initial energy

Present Brach Presentj Brach

e = -0.0 e = 0.5 0.600# 0.938 0.938 46.9 46.9 e = 0.05 O.600l 0.656 0.656 62.3 62.3 -0.2 e = 0.5 0.507# 0.862 0.862 33.1 33.3 e = 0.05 0.462g 0.581 0.580 47.9 47.9 -0.6 e = 0.5 0.263# 0.712 0.732 17.9 17.9

e=0.05

0.043# 0.431 0.569 29.3 29.3 1.0 _{e = 0.5} _0.905# 1.3 13 1.395 92.2 92.1 e=O.05 0.988# 1.031 1.031 99.9 99.6 Length 116.Úm Breadth 19.Om Draught 6.9m Displacement 10 340t

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Table 3.4. comparison of results for the energy loss, in collisions.

Secondly, we consider the ships sliding against each other. The coefficient of friction

between the two ships is assumed to be p0

= 0.6, and the calculation results are

presented in Table 3.5. The results show that when the ships slide against each other, the energy to be dissipated by the crushing structures is decreased in comparison with the case where the ships are locked together.

Table 3.5. Comparison of results obtained by the present method for the energy loss in cases

_of

ships being locked to.ether or slidin2 against each other.

3.3 External Dynamics Examples involving MS DEXTRA

As a first example it is assumed that an 86 m long high-speed craft strikes MS

DEXTR.A at different collision angles and locations. The main dimensions of the two

vessels are presented in Table 3.6. Before the collisions, the speed of the striking craft is 32 knots (full service speed) and MS DEXTRA is also sailing at its full speed of 27

knots.

Paper number (2) SAFER EURORO Spring meeting, NANTES 28April 1999 ₁₅

Parameters ([V] = m / s)

E(MJ

E(MJ)

Case Va Vb

_{a =}

_ß d Present Petersen Hanhirova Present Petersen Hanhirova

(1982) (1995) -(1982) (1995) 1 4.5 0 90 Ö 70.1 54.4 0.0 0.0 0.0 2 4.5 4.5 90 O 70.1 64.1 54.4 21.4 24.7 41.5 3 4.5 4.5 60 O 35.3 29.8 28.3 0.2 5.2 15.8 4 4.5 4.5 30 0 7.4 71.9 4.0 0.0 49.3 7.2 5 4.5 4.5 120 0 64.9 60.5 41.7 90.4 93.! 115.0 6 4.5 4.5 120 L13 42.9 49.2 74.1 85.4 90.7 102.0 7 4.5 4.5 120 L/6 60.0 64.9 60.6 92.3 91.6 110.0 8 4.5 4.5 120 -L/3 30.8 26.3 74.1 68.0 86.7 102.0 13 4.5 0 120 0 50.1 54.0 40.9 15.0 9.8 14.0 14 4.5 2.25 120 0 57.5 . 60.3 42.8 45.1 40.7 51.5 15 4.5 9.5 120 0 81.4 50.7 28.6 245.3 258.0. 347.0

Parameters ([V] = m /s)' -

E (Mi)

E (Mf)

(T)(m Is)

Case Va \Tb

a=ß

d Plastic p,=0.6 Plastic p0=O.6 Plastic p0=0.6

1 4.5 0 90 0 _70.1 _70.1 _0.0 _0.0 _0.0 _0.00 2 4.5 4.5 90 0 70.1 70.1 21.4 21.4 0.0 0.00 3 4.5 4.5 60 0 35.3 35.3 0.2 0.2 0.0 0.00 4 4.5 4.5 30 0 7.4 7.4 0.0 0.0 0.0 0.00 5 4.5 4.5 120 0 64.9 53.6 90.4 84.1 0.0 -3.44 6 4.5 4.5 120 L/3 42.9 28.9 85.4 54.4 0.0 -5.48 7 4.5 4.5 120 L/6 60.0 45.6 92.3 77.8 0.0 -4.34 8 4.5 4.5 120 -L/3 30.8 24.9 68.0 44.2 0.0 -4.76 13 4.5 0 120 0 50.1 50.1 15.0 15.0 0.0 0.00 14 4.5 2.25 120 0 57.5 53.6 45.1 43.9 0.0 -1.19 15 4.5 9.5 120 0 81.4 53.6 245.3 166.6 0.0 -8.44

(34)

Table 3.6. Main dimensions of the striking- cr

The calculated results for the total kinetic energy loss are shown in Fig. 3.3. It is seen from the results that the effect of the collision angle on the energy loss is significant,

but the collision location has very weak influence on the energy loss. The reason is that the mass of MS DEXTRA is very large compared with the mass of the striking

craft. During the collision, the induced sway motions of MS DEXTRA are very small.

250 200

-,

150 o >1 O) C) w 100 50 o

86m craft (32kn) strikes with MS Dextra (27kn)

Fig. 3.3. Energy loss of an 86 m craft striking with MS DEXTRA vessel at different collision angles and locations.

As a second example involving MS DEXTRA we shall consider a 180 m RoRo ferry

striking MS DEXTRA at different collision angles and collision positions. The

breadth of the RoRo ferry is 31.5 m, the depth is 9.3 m, the draught is 7.3 m, and the

displacement is 27,000 tons. It is assumed that the 180 rn RoRo ferry strikes MS DEXTRA with a speed of 10 knots, while MS DEXTRA is sailing at a speed of 10

knots.

The energy loss in these collisions is shown in Fig. 3.4. The results show that both the collision angles and collision locations have significant influence on the energy loss. It is also noted that the energy loss is large if the collision is in fore part of the struck MS DEXTRA.

Paper number (2) SAFER EURORO Sprinj meeting, NANTES 28April1999 16

- Craft MSDEXTRA Lpp(m) 86.5 173 Breadth (m) 17.4 26.0 Depth(m) * _15.7 Draught Cm) 3.6 6.5 Displacement (ton) 500

1,073

30 deg.

t

o 60 deg.

-9ûdeg.

= 120 deg. -ii-- 150 deg. -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Collision locations (d/L)

(35)

P. T. Pedersen & S. Than g, Technical University of Denmark: Collision Analysis for MS DEXTRA 500 450 400 350 300 2 250 200 Ui 150 100 50 o

180m RoRo vessel (lOkn) strikes with MS Dextra (lOkn) e 30 deg. o--60 deg.

9Odeg.

o!. 120 deg .*-- 150 deg. -0.5 0.4 -0.3 -0.2 -0.1 0 0.1 0.2 Collision locations (d/L)

Fig. 3.4. Energy loss of an 180 m RoRo ferry striking with MS DEXTRA at different collision angles and locations.

4 Simplified methods for analysis of the internal dynamics

4.1 Introduction

Knowing the energy released for crushing in a specific collision scenario the next step

in the present rational collision analysis procedure is to determine the resulting

structural damage.

During this first phase of DEXTREMEL simplified methods for calculating the

collision force and the resulting hole in the ship structures has been established. At the

same time a comprehensive finite element model has been established to verifi the

simplified method, see the following paper by Samuelidis.

The side structure of ships is very complex. The deformed, destroyed and crushed

modes of side structures are also very complex. However, a ship may be viewed as an assembly of plated structures such as shell platiñg, transverse frames, horizontal decks and bulkheads are built in various plates. Observations from full-scale ship accidents and model experiments reveal that the primary energy absorbing mechanisms of the side structure are

Membrane deformation of shell plating and attached stiffeners Folding and crushing of transverse frames and longitudinal stringers Folding, cutting and crushing of horizontal decks

Cutting or crushing of ship bottoms Crushing of bulkheads

(36)

P. T. Pedersen & S. Zhang, Technical Uñiversity of Denmark: Collision Analysis for MS DEXTRA

By analysing damage of each basic structures and adding their contributions together, the total collision resistance and dissipated energy can be determined, Thang (1999).

The simplified method (or the limit analysis) is widely used in engineering analysis

and design. It has been proved that the method is valuable for estimating the collapse load of a structure subject to extreme loads. The collapse load so obtained can be used as a realistic basis for design. However, it should be emphasised that the limit analysis is an approximate method. A basic assumption is that the material is perfectly plastic without strain hardening or softening.

Most of the simplified methods are based on the upper bound plasticity theorem:

If the work rate of a system of applied loads during any kinematically admissible collapse of the structure is equated to the corresponding internal energy dissipation

rate, then the system of loads will cause collapse or be at the point of collapse.

The upper-bound method was used by Wierzbicki and Abramowicz (1983) for axial crushing of plate intersections and plate cutting, by Abramowicz (1994) and Amdahl

(1983) for axial crushing of L, T - and X type elements, by Kierkegaard (1993)

for ship bow crushing, by Paik and Pedersen (1 995a) for plate element crushing, and

by Simonsen (1997) for ship grounding. In the MIT - Industry Joint Program on Tanker Safety the method was applied thoroughly to analysis of the damage of ship grounding. It was shown that the theoretical results are quite close to experimental

results.

4.2 Formulation of the Upper-bound Method

The equilibrium for the external energy rate and the internal energy dissipation rate

can be expressed as

F5=Emt

(4.1)

where F is the external force, 5 is the velocity at the force action point, E1 is the

internal energy rate.

For a general solid body, the internal energy rate can be expressed as

Emt

fCyEudV

(4.2)

where is the rate of the strain tensor, V is the volume of the solid body. By use of von Mises' flow theory, the rate of plastic energy dissipation is given as

Eint =fa0EdV

(4.3)

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P. T. Pedersen & S.Zhang, TechnicalUniversity of Denmark: Collision Analysis for MS DEXTRA

where ee = ei,) and a0 is the flow stress.

For a plane stress condition, the von Mises yield condition gives

a2 +a2 aa +3a2 a2

(4.4)

For a deform ing plate, the rate of internal plastic energy dissipation can be written as the Sum of the bending and the membrane energy dissipation rate:

mt =Eb+Em (4.5)

The bending energy rate can be expressed as

(a,ß=1,2)

(4.6)

A

where A is the plate area, k

_{is the curvature of the plating, and O. and i. are the} rotation and the length of the it/i plastic hinge line, respectively. M _{is the bending} moment tensor, M0 is the fully plastic bending moment M0 =

(2/)(a02t/4), and

t

is the plate thickness. It is seen from the expression that the bending energy contains

the continuous deformation field and the plastic hinge lines. In

some practical applications, simplified velocity fields are assumed so that only the plastic hinge lines

are considered and the continuous deformation of the curvature is neglected. In this

case the bending energy is simplified as

Eb

=M0O1l

_(4.7)

The membrane energy rate of a deforming plate can be calculated from

Em =fNeoßdA,

(a,ß=1,2)

(4.8)

where is the membrane force tensor, eap is the strain rate tensor. By use of von Mises' yield criterion, the membrane energy rate can be expressed as

Em

aotJ

2 2xxy1y2

dxdy (4.9)

In the limit analysis method, a key point is the construction of

a kinematically

admissible velocity and displacement field. This is mainly based on observations from experimental tests, full-scale accidents and existing analysis work.

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4.3 Rupture of Ship Structural Elements

When a structure has been deformed to its limit, it will rupture and be exposed to

failure. It

is an extremely complex problem to predict the rupture of structures

accurately. Different loads may cause different failure modes. Jones (1989) discussed

the rupture criteria of ductile metal beams subjected to large dynamic loads. Three

major failure criteria of the metal beams were discussed. The first is the tensile

teanng fàilure mode, which occurs when the maximum strain equals the critical

rupture strain of the material, and the beam ruptures. Thus

Eniax c

The second failure model is the transverse shear failure mode, which develops ina

beam when large transverse shear deformations occur within a very short region of

the plastic beam. When the total transverse shear displacement W. in a particular

location equals a critical value, the beam ruptures.

The third failure criterion is the energy density failure mode. It is assumed that

rupture occurs in a rigid-plastic structure when the absorption of plastic work per unit volume reaches the critical value O,

e=ec

As Simonsen (1997) mentioned, the simplified methods are based on overall

deformation mechanisms. It is not possible to trace the strain history of material

elements at a very detailed level. Therefore, as _{many authors did, eg. Wang (1995)}

and Paik and Pedersen (l995b), we use the maximum strain failure criteria. That is

when the maximum strain in a structure reaches a critical strain, the structure ruptures.

In practical calculations, we need to know the critical strain of a material to predict

the structural failure. Generally, this depends on axial tensile experiments.

Experiments conducted by Wen and Jones (1993) and Amdahl (1992) showed that the

tensile ductility of mild steel is in the range of 0.20 to 0.35. Amdahl (1995) pointed out that due to scale effects and material imperfections, this value is far too large in

the assessment of fùlL-scale collisions. The critical strain value suggested by Amdahl (1995) for side collisions is between 5% to 10%.

In the minor collision analyses performed by McDermott et al. (1974), the critical

rupture strain for mild steel material in side collisions is evaluated from

E,

E =0.10 ()

0.32 (4.10)

where E is the tensile ductility. It has been indicated by McDermott et al. that this

formula may give reasonable agreement with experimental results in the deformation

of shell plating. So in this work, we either uses McDermott's formula, if the tensile

ductility of the material is known, or assume a value (say

_{e =}

_{5% 10% for mild}

steel) for the critical rupture strain in side collisions.

(39)

When the critical rupture strain is knäwn, the criicaÌ deflection or penetration of the shell plating can be determined. For example, a point load acting in the middle of a plate strip with a span of 2b, the strain in the plate strip due to transverse deflection

can be calculated from

E

=jl()2 1

1(8)2

(4.11)

where 5 is the deflection at the middle point.

When the deflection is large enough, the strain in the plate strip reaches the critical

rupture value. The critical rupture deflection or penetration is then determined from

(4.12)

If the critical strain is e = 10%, then the critical penetration is 5 =0.447b.

4.4 Crushing of Stiffened Decks and Bottoms

Observations from collision accidents show that the damage modes of ship decks are

folding, crushing and tension rupture. Some illustrations of damaged decks from

collision accidents are shown in Fig. 4.1.

iiiHHlHH

1EXTENT OF D4 TO 3rd DECK STBD.11 I I I I

IluIllIllIllIll

11111 I I I I.1

I l/

₁₁₁₁₁ I Striking bow

Fig. 4.1. illustrations of deck damages observed in collision accidents.

(40)

Fig. 4.1 shows the analysis model of a bow crushing into the side of a struck ship. The

stiffened decks or bulkheads are divided into different elements (L-, T-, and

X-elements), that are crushed by the striking bow. With further penetration, more and more elements are crushed and destroyed. Typical axial crushing modes of L , T

-and X - elements are shown in Fig. 4.2.

Upper deck

c;

Deck

.L.

.L

f

'i.-;id

Deck

("i..---Bulkhead Bottom

Fig. 4.2. A striking bow crushes the deck structures and the side of a struck ship. Deck

,

V-.

..L..

.-Deck

Fig. 4.3. Axial crushing modes of L-, T- andX- elements.

Many authors have investigated axial crushing of the basic elements, using theoretical

methods and experimental methods. Amdahl (1983), Wierzbicki and Abramowicz (1983), Kierkegaard (1993), Abramowicz (1994), Paik and Pedersen (1995a) did comprehensive work on the element axial crushing. Paik and Wierzbicki (1997) carried out a benchmark study on the crushing strength by comparing the existing

theoretical formulas and experimental results. Due to the simplicity of the expression

of we have in DEXTREMEL chosen to use Abramowicz's formula to calculate the

deck crushing resistance. The expression is

F = (L + i .2n + 2. 1n )(3.263a0t'67c°33) (4.13)

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For the initial collision phase, it is ieasonable to assume that the deck fails in the folding deformation. The collision location is assumed to be in the middle between

two web frames. The collision situation is shown in Fig. 4.3.

Fig. 4.4. InItial deformation of the deckfolding.

For the initial crushing of the deck the authors have derived the following the

formulas for the mean crushing force:

4.33aOtd'67b°

ô 2H

F=

6.77aOtd167b°3 ô >2H

where

ô is the penetration, H=O.8383(b2td ), t, is the thickness of the deck plate and 2b

is the spacing between heavy transverse stiffeners.

When the striking bow touches the heavy stiffeners of the deck, as shown in Fig. 4.5, crushing of the deck structure (such as the basic elements L-, T- and X-) will happen. Then Abramowicz's formula Eq. (4.13) is employed for analyzing the crushing force. It is assumed that the basic elements are not deformed and crushed until the striking

bow pushes them directly. The deck plate between the touching point and the

non-deformed stiffener suffers tension and bending deformation. Its behaviour is similar to

web folding. Therefore, the collision force for the fùrther deck crushing can be

expressed as .67 0.33

F

=

_{{(L +}

I .2n,. + 2.ln)(3.263a0t1' c, _{) + 4.33a0td'67k°33}} i=J (4.14) (4.15)

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where _L' n are the numbers of the L -, T - and X -elements crushed in each

deck, t, _{is the average thickness of the basic elements in deck i,} _c, _{is the average}

cross-sectional length of an element in the deck j, ta is the thickness of the deck

plate, b. is the distance between the touching póint and the non-deformed stiffeners and Ndeck is the number of crushed decks.

Striking bulb

Fig. 4.5. Further crushing of ship decks.

5 Preliminary Collision Analyses for MS DEXTRA

In this section, we shall demonstrate the outer dynamics procedure described in

Section 3 and the inner mechanics procedure briefly presented in Section 4 to analyse

a few collision scenarios where a 160 m long conventional merchant vessel strikes

MS DEXTRA. The main particulars of the conventional striking ship are presented in Table 5.1. It is assumed that the striking ship collides with the ferry at a forward speed of 4.0 mIs. Two cases are calculated here. One is when the struck MS DEXTRA sails with a forward speed of 4.0 m/s, the other is when the speed of MS DEXTRA is zero

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(V=0). We shall assume that the bow of the striking vessel is considerably stronger

than the side structure of MS DEXTRA.

Table 5.1. Main particulars of the conventional striking ship.

Outer Dynamics: First the collision energy to be dissipated by destroying the struck MS DEXTR.A is calculated. Fig. 5.1 shows the energy loss with various collision

angles where the collision position is located at the centre of the struck ship. Fig. 5.2 presents the collision energy loss with different collision locations where the collision is perpendicular to the struck ship. The results show that both the collision angle and

the collision location

_{influence the energy loss significantly. For the central}

perpendicular collision, the energy loss is 39.4 MJ when the speed of the struck ship is zero, and the energy loss is 50.2 MJ when the speed of the struck ship is 4.0 rn/s.

120

-

100

-80 60 >1 L.

40 n

w 20 o

Collision location at center of struck ship

---

V-strucic=0

.o--V-strucic=4m(s

Fig. 5.1. Energy loss as function of collision angle when the impact is at the centre of the struck ship.

Paper number (2) SAFER EURORO Spring meeting, NANTES 28 April1999 25

Length (m) Breadth (m) Depth (m) Deck height (ni) Draught -(m) Bow angle 20 (degrees) Stem angle ç (degrees) 70 160 24.6 13.3 16.3 5.5 80 15 30 45 60 75 90

_{105 120 135 150 165}

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P. T. Pedersen & S. Thang, Technical University of Denmark: Collision Analysis for MS DEXTRA 60 __ 50

-,

40-w

.230

E2O-G)

w10-o -0.5

Collision angle = 90 degrees

é---. V-struck=0

V-struck=4mfs

Cöllision location (dIL)

Fig.

5.2. Energy loss as function of collision location, when the collision is

perpendicular to the side of the struck vessel.

Inner Mechanics: When the energy loss to be dissipated by destroying the side

structure is known, the subsequent damages to the struck MS DEXTRA can be

calculated using the procedure briefly described in Section 4. The analysis procedure is as follows:

It is here assumed that the collision position is located in the middle between two

transverse frames of the midship, see Fig. 5.3. In the initial phases of the collision, the shell plating of the struck ship is subjected to tension. With increasing penetration, the

striking bow comes into contact with frames, stringers and horizontal decks. The

frames, the stringers and the decks are then subsequently crushed. It is assumed that

frames, stringers and decks are not deformed and crushed until the striking bow touches them directly. By calculation of the resistance of deformed shell plating, frames and decks etc, the collision resistance and the absorbed energy are obtained.

When the calculated absorbed energy is equal to the energy loss, determined from the

outer analysis procedure, the calculation stops. After the maximum penetration has

been determined, the size of a hole in the shell plating created by the striking bow is calculated.

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P. T. Pedersen & S. mang, Technical University of Denmark: Collision Analysis for MS DEXTRA ! _{O O} 50m 22m

¡

720e

lit

¡Ir

-1

t.

fl0000i

Stilking bow

Fig. 5.3. Collision position for a conventional ship strildng the DEXTREMEL vessel.

Fig. 5.4 shows the calculated energy dissipated by the struck ship for various

penetrations when the speed of the struck ship is zero and the collisiön angle is 90

degrees. It is seen from the results that when the penetration of the striking bow into

the side of the struck ship is 5.0 m, the energy dissipated by the struck ship is 39.3

MJ. All the energy loss is dissipated by the struck ship at this penetration (the striking bow is as mentioned above to be rigid). This means that the indentation stops at this penetration. The damage length is 8.38 m. The ratio between the damage length and the vessel length is 4.8%.

OG o

5.7 m

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colislon angle=90 deg. Vstruck=O

Fig. 5.4. Dissipated energy of the struck ship as function ofpenetrations when the speed of the struck ship is zero.

Fig. 5.5 shows the energy absorbed by the struck ship as fi.mction of the penetration when the speed of the struck ship is 4.0 m/s and the collision angle is 90 degrees. The

penetration is measured along the penetration angle ß =90+

a / 2 = 135 degrees.

When the penetration reaches 7.85 m, the energy dissipated by the struck ship is 50.3

MJ. The struck ship at this penetration dissipates all the energy loss. Therefore, the

penetration stops and the max collision penetration is 7.85 m in this case. The

perpendicular indentation is approximately 7.85. siii(l350)

= 5.55 m. The damage

length is 10.5 m. The ratio between the damage length and the ship length is 6.1%.

This result indicates that when a struck ship has forward speed, the collision energy

loss and the resulting damage are larger than when the speed of the struck ship iszero.

60 50 40

E:

10 o

coilsion angIe9O deg. V-struck=4mIs

Fig. 5.5. Dissipated energy of the struck ship as function of penetration when the

struck ship has a forward speed of 4.0 rn/s and the collision angle is 90 degrees.

o i 2 3 4 5 6 Penetration (m)