International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 6

Number 1

March 2012

1 INTRODUCTION

Ship-ship collisions are rare events that potentially

might have disastrous impact on the environment,

human life and economics. To find effective risk

mitigating measures the risk must be reliably as-

sessed. Proper assessment of the ship-ship collision

risk requires understanding on the complicated chain

of events. Simplifying assumptions on certain pa-

rameters are necessary as the research in this field is

not comprehensive. Especially, the important link

between the encounter of the colliding vessels and

the actual moment of impact contain obvious uncer-

tainties.

In this paper a case study is conducted to compare

models found in literature for dynamic parameters in

collision scenario. The case study concerns colli-

sions in which the struck vessel is an oil tanker. The

traffic is simulated by means of a Monte Carlo simu-

lation based on AIS data to obtain realistic encounter

scenarios for the analyzed area. The assumptions are

then applied to encounter scenario to obtain the

complete impact scenario. The deformation energy

released in the collision is calculated by analytic

method (Zhang 1999) and the damage extents are es-

timated with simple formula to normalize the results

of deformation energy calculations. The effects of

assumptions for dynamic parameters to collision risk

are discussed.

2 COLLISION RISK EVALUATION

2.1 Concept of risk

Risk is a product of probability p and consequences

c and is expressed with (Kujala et al, 2010)

∑

⋅=

(1)

where i denotes certain chain of events or scenario.

2.2 Tanker Collisions

In case of ship-ship collisions scenario is a function

of vast number of static and dynamic parameters.

The parameters used in this study are listed in Ta-

ble 1.

Uncertainty in Analytical Collision Dynamics

Model Due to Assumptions in Dynamic

Parameters

K. Ståhlberg, F. Goerlandt, J. Montewka* & P. Kujala

Aalto University, School of Engineering, Department of Applied Mechanics, Marine

Technology, Espoo, Finland

* Aalto University, School of Engineering, Marine Technology, Espoo, Finland

Maritime University of Szczecin, Institute of Marine Traffic Engineering, Poland

ABSTRACT: The collision dynamics model is a vital part in maritime risk analysis. Different models have

been introduced since Minorsky first presented collision dynamics model. Lately, increased computing capac-

ity has led to development of more sophisticated models. Although the dynamics of ship collisions have been

studied and understanding on the affecting factors is increased, there are many assumptions required to com-

plete the analysis. The uncertainty in the dynamic parameters due to assumptions is not often considered.

In this paper a case study is conducted to show how input models for dynamic parameters affect the results of

collision energy calculations and thus probability of an oil spill. The released deformation energy in collision

is estimated by the means of the analytical collision dynamics model Zhang presented in his PhD thesis. The

case study concerns the sea area between Helsinki and Tallinn where a crossing of two densely trafficked wa-

terways is located. Actual traffic data is utilized to obtain realistic encounter scenarios by means of Monte

Carlo simulation. Applicability of the compared assumptions is discussed based on the findings of the case

study.

Table 1. Collision parameters used in this study.

________________________________________________

Description Unit Type

________________________________________________

M Mass [kg] Static

L Length [m] Static

B Width [m] Static

Added mass coefficient, [-] Static

surge motion

Added mass coefficient, [-] Static

sway motion

j Added mass coefficient, [-] Static

rotation around centre of gravity

R Radius of ship mass inertia [m] Static

around centre of gravity

Surge speed [m/s] Dynamic

Sway speed [m/s] Dynamic

x x-position of centre of gravity [m] Static

y y-position of centre of gravity [m] Static

x-position of impact point, [m] Dynamic

in coordinate system ship A

y-position of impact point, [m] Dynamic

in coordinate system ship A

α collision angle [rad] Dynamic

coefficient of friction [-] Static

e coefficient of restitution [-] Static

________________________________________________

The static parameters can be derived from AIS

data, statistics and theory of ship design. Modeling

of the dynamic parameters is often based on statis-

tics of the collisions.

Ship-ship collision risk evaluation schematic is

outlined in Figure 1 for the case of an oil tanker be-

ing struck vessel.

Figure 1. Tanker collision risk evaluation schematic

The first step of the risk analysis is modeling the

traffic in the analyzed area. Modeling may be done

via simulation of individual vessel movements as

proposed by Merrick et al. (2003), van Dorp et al.

(2009), Ulusçu et al. (2009) and Goerlandt & Kujala

(2010) or alternatively by simulating the traffic

flows as proposed by Pedersen (1995, 2010) or

Montewka et al (2010). The encounter scenarios are

obtained as a result of the traffic simulation. The

impact scenarios may be then obtained with the

models discussed in detail in Section 3.3.

Second part of the risk analysis is the evaluation

of the consequences which begins with the estimat-

ing the released deformation energy that is absorbed

by the vessel structures. Collision dynamics models

to calculate the deformation energy can be divided

into two groups, time domain and analytical (Wang

et al 2000), based on applied calculation method.

Analytical closed form methods have been proposed

by Minorsky (1959), Vaughan (1977), Hutchison

(1986), Hanhirova (1995), and Zhang (1999). Mod-

els based on time domain calculations are proposed

by Chen (2000) and Tabri et al. (2009). In analytical

models the external dynamics and internal mechan-

ics are uncoupled while in time domain methods

these are coupled.

3 COMPARISON METHODS

3.1 Traffic simulation and encounter scenarios

The traffic simulation is described here shortly as

the simulation itself is not crucial regarding the

comparison of impact models. The simulation is de-

scribed in detail in (Ståhlberg, 2010)

The traffic in the analyzed area is obtained from

AIS data. The data contains traffic information from

the month of July 2006 in the sea area between Hel-

sinki and Tallinn where densely trafficked water-

ways cross. In Figure 2 the analyzed area and the da-

ta points are presented. The four main waterways in

the crossing area are named after compass quarters

in form of “from-to” as shown in Figure 2. The con-

sidered waterway combinations and resulting en-

counter types are listed with reference numbers in

Table 2.

Figure 2. Plot of AIS data points in analyzed area

The AIS data is filtered to distinguish the traffic

between waterways and ship types. The numbers of

passages through the analyzed area per ship type are

listed in Table 3. The Monte Carlo simulation

flowchart starting from the filtered AIS data is

shown in Figure 3. The result of the simulation is the

encounter situations based on the traffic data.

Table 2. The considered waterway combinations and resulting

encounter types with respective reference numbers.

__________________________________________________

Ref number Route Encounter type

__________________________________________________

1 N-S, E-W Crossing

2 N-S, W-E Crossing

3 S-N, E-W Crossing

4 S-N, W-E Crossing

5 W-E, E-W Head-on

6 E-W, W-E Head-on

7 E-W, E-W Overtaking

8 W-E, W-E Overtaking

_________________________________________________

Table 3. Number of passages per ship type and route.

__________________________________________

Ship Route

_________________________________

Type N-S S-N E-W W-E

__________________________________________

HSC 741 740 0 0

PAX 253 254 26 14

Cargo 5 4 768 742

Tanker 0 0 218 215

Other 3 3 36 35

__________________________________________

* HSC = High Speed Craft, PAX = Passenger vessel,

Cargo = Cargo vessel

The Monte Carlo simulation to create encounter

scenarios is run 10000 times for those combinations

of main waterways in which the tanker may be

struck vessel. In the utilized data set tankers were

recorded sailing only on “E-W” and “W-E” water-

ways. In this study the probability of a vessel in-

volved in collision is weighted with the number of

voyages in the area.

Figure 3. Flowchart of Monte Carlo simulation

3.2 Impact scenario simulation

With the encountering vessels’ characteristics

known the impact scenarios are simulated here by

applying the compared models for the dynamic pa-

rameters. The models may be considered to be the

“evasive maneuvering” model shown in Figure 1.

The compared assumptions are presented in Fig-

ures 4-7 and the distribution parameters are com-

piled into Table 4.

In “Blind Navigator” –model there are no maneu-

vering actions taken to avoid the collision and thus

the speeds and courses are unchanged from the en-

counter scenario. The collision location is assumed

to be uniformly distributed along the struck vessel’s

length. This model is used by Van Dorp & Merrick

(2009) and COWI(2008). Based on the analysis of

collisions in (Cahill, 2002) and (Buzek & Holdert,

1990) it seems extremely rare that neither vessel

takes any action.

Figure 4. Input distributions for collision angle, Lützen: initial

angle 90°, Brown (2002) quasi-equivalent to NRC (2001)

Figure 5. Input distributions for striking ship speed, Lützen

with initial speed of 15 kn

Figure 6. Input distributions for struck ship speed, Lützen with

initial speed of 10 kn, Brown (2002) quasi-equivalent to NRC

(2001)

Figure 7. Input distributions for location of impact along struck

ship’s length, 0 = aft, 1 = fore

Table 4. Overview of impact scenario models.

__________________________________________________

Impact Collision V

Impact

Model Angle, β Point, d

[deg] [kn] [kn] [x/L]

__________________________________________________

Blind β=β

U(0,1)

Navi

Rawson U(0,180) bi-normal idem to U(0,1)

(1998) N(5,1) V

N(10,1)

Truncated {2, 14}

NRC N(90,29) W(6.5,2.2) E(0.584) B(1.25,1.45)

(2001) {0, 1}

Lützen T(0,β

,180) U(0,0.75V

) T(0,V

) Empirical

(2001) T(0.75V

) See FIG 7

Brown N(90,29) W(4.7,2.5) E(0.584) Empirical

(2002) See FIG 7

Tuovinen Empirical Empirical Empirical Empirical

(2005) See FIG 4 See FIG 5 See FIG 6 See FIG 7

_________________________________________________

* Distributions are marked as follows, U=Uniform(min, max)

N=Normal(μ, σ), T=Triangular(min, triangle tip, max),

E=Exponential(λ), B=Beta(α, β, min, max), W=Weibull(k, λ)

Lützen’s (2001) set of assumptions implies that

the struck vessel is more prone to speed reduction

than the striking vessel while the impact angle is tri-

angularly distributed between 0° and 180° with the

tip of the distribution at the encounter angle. The

longitudinal impact location is given by empirical

distribution. Although there is no explanation how

the distributions for collision angle and velocities are

derived these are included into the comparison be-

cause of the existing relation between encounter and

impact scenarios.

Rawson et al (1998) model is based on statistics

of the grounding accidents with assumption that the

collision speed being similarly distributed as

grounding speed. Velocities of the colliding vessel

are distributed according to a double normal distri-

bution in which the averages are described to repre-

sent the service speed, i.e. no speed reduction, and

half of service speed. The same speed distribution is

used for both striking and struck vessel. Collision

angle and collision location are uniformly distribut-

ed between 0°…180° and along the struck vessel’s

length respectively.

Tuovinen (2006) compiled statistics from over

500 collisions. Statistics have been used here as pre-

sented originally, in form of empirical distributions.

Brown (2002) and NRC (2001) give quite similar

distributions. Brown gives lower velocity for the

striking vessel. These models both assume that strik-

ing vessel has higher velocity than struck at the mo-

ment of impact. It is noteworthy that these two mod-

els suggest much lower collision speeds than other

models. Collision angle is normally distributed

around right angle. In NRC model the collision loca-

tion is beta distributed so that midship section is

rammed at higher probability than the fore and aft of

the vessel while Brown suggests empirical distribu-

tion.

Overall, the distributions Lützen suggested are

the only ones taking the encounter into account in

any way and other models give same distributions

for dynamic parameters irrespective of encounter

scenario. None of these models indicate how to de-

termine which vessel is striking and which is struck.

It is assumed here that the probabilities of vessel be-

ing striking or struck are equal for all models as no

other probabilities were suggested in these models.

The compared models do not have the possibility of

initial sway nor yaw speeds, which in case of ma-

neuvering is unlikely.

It can be seen in the Figures 4-7 that models, with

exception of Brown and NRC, give distinctively dif-

ferent distributions for the dynamic parameters.

Considering the struck vessel speed being lower in

all the models expect Rawson it appears likely that

the collision statistics from which the distributions

are derived include collisions in which the struck

vessel is in anchorage or in berth. Tuovinen’s (2005)

statistics include approximately 6% of such cases

and 41% of open seas collisions. Brown (2002)

states that the share is significant as in about 60% of

collisions struck vessel speed is zero.

3.3 Deformation energy calculations

Zhang presented in his PhD thesis (Zhang, 1999) a

simplified calculation method for released

deformation energy in ship-ship collision. Zhang’s

method is based on rigid body mechanics and

conservation of momentum. The method is derived

based on the dynamics of two rods colliding on a

frictionless surface and has three degrees of

freedom. The hydrodynamic effects are considered

as constant added masses. Both vessels may have

initially forward and sway speeds. During the

collision the rotational movements are considered as

small and are neglected. After the collision both

vessels are allowed to have rotational velocity.

Figure 8 illustrates the impact scenario and defines

the used co-ordinate systems. The formulation is not

presented here due to its lengthiness.

Figure 8. Impact scenario and the co-ordinate systems

3.4 Damage calculation

The method of damage calculation used here is pre-

sented in Goerlandt et.al. (2011). The focus in case

of a tanker being a struck vessel is on the possibility

that cargo oil is spilled. That requires penetration of

one or more oil cargo tanks. Thus the penetration

depth must exceed the double side width added with

the dislocation of the inner shell when a rupture oc-

curs. Additionally, the collision location along stuck

tanker hull must be within the boundaries of the car-

go tanks. Smailys & Česnauskis (2006) suggested

following limits for cargo area for tankers operating

in the Baltic Sea.

LdL 94.02.0 ≤≤

(2)

where L is vessel length and d is distance of impact

point from amidships along the centerline.

For the purposes of this study the simple criterion

for oil cargo tank penetration is used and is ex-

pressed as critical energy, E

, with

( )



































≤≤

−−

40000,5

2000

400005000,

1000

15.12

5000,5.12

DWT

(3)

where W

is double side width given in meters in

ABS (2010) classification rules by:











≤≤+

30000,2

3000010000,

20000

5.0

10000,1

DWT

(4)

This criterion is obtained from a simple linear re-

gression in the example cases discussed in (Zhang

1999, Lützen 2001, HSE 2000). It is further assumed

that the effect of striking vessel bow geometry is

negligible and that the energy absorbed by the strik-

ing vessel is taken into account in E

. Even though

the evaluation of the critical energy is based on a

very simplified model and better alternatives are

available in the literature (Brown 2002, Ehlers

2008), this criterion is withheld due to its simplicity.

Application of the simple criterion of (Eq. 3) affects

all impact scenario models in a similar way, such

that the conclusions are still valid. The actual value

of E

is in this respect not essential as it is only used

as a reference to better present the differences in im-

pact models. In this study the collision consequences

analysis is limited to evaluating if the deformation

energy in direction normal to the struck vessel hull,

, exceeds E

that is required to breach a cargo

tank, while neither the actual structural damages nor

the amount of oil spilled are not considered.

4 SIMULATION RESULTS AND DISCUSSION

In this section, the results of the Monte Carlo simu-

lations for the relative velocity, collision energy and

hull breach probability are given for the impact sce-

nario models.

4.1 Relative velocity

The relative velocity V

rela

is considered as the

velocity that the bow of the striking vessel is

approaching the collision point at the struck vessel

side. In vector form it is given with:

rela



−=

(5)

The released deformation energy is highly

depending on the V

rela

at the moment of impact.

Relative velocities obtained from simulation for

“Blind navigator” and Lützen model in selected

encounter situations are presented in Figure 9. The

other four models give similar results for V

rela

irrespective of the encounter situation and thus

results are presented only for waterway combination

The “Blind Navigator” model is giving much

higher values of V

rela

, apart from head-on encounter,

than other models as expected. There are two peaks

in the result distributions of “Blind navigator” for

crossing encounter situations. The lower peak repre-

sents passenger vessel cases and higher peak High

Speed Crafts as striking vessel.

Figure 9. Simulated relative velocity distributions according to

impact models in which encounter is considered

Figure 10. Simulated relative velocity distributions according

to impact models in which encounter is not considered

Pdf [-]

rela

[m/s]

Lützen (2001)

N-

S,…

Pdf [-]

rela

[m/s]

Rawson

(1998)

The angle between N-S and W-E traffic flows is

approximately 120° while between N-S and E-W

traffic the angle is 60°. The effect of angle on rela-

tive velocity can be seen by comparing “Blind Navi-

gator” results in Figure 9, the larger angle results in

higher V

rela

. The Lützen model appears to be rela-

tively insensitive to variation of the encounter angle

as only slight difference can be observed. This is due

to the reduction of the struck vessel speed. The Lüt-

zen model gives the impact speed of the struck ves-

sel to be on average ⅓ of the initial velocity.

The models that are derived from statistics by

Rawson, NRC, Brown and Tuovinen give much

more diverse results for V

rela

than may be anticipated

as the available accident data is limited and one

would expect that the statistics would be practically

based on the same data. It should be noted that these

four model result in similar distributions for all en-

counter scenarios. Thus while the V

rela

is lower in

case of crossing encounter it is higher in case of

overtaking compared to “Blind navigator” and Lüt-

zen models.

4.2 Deformation energy

In here only the transversal deformation energy E

considered because it represents the deformation en-

ergy in direction of penetration depth. The simula-

tion results for E

in each simulated encounter are

normalized by dividing it with respective critical en-

ergy E

. In Figures 11-13 the cumulative distribu-

tions for normalized deformation energy E

ξN

for

each impact scenario model are presented for select-

ed waterway combinations.

Figure 11. Simulation results of normalized deformation ener-

gy for “Blind navigator” and Lützen (2001) impact models.

Figure 12. Simulation results of normalized deformation ener-

gy for “Blind navigator” and Lützen (2001) impact models.

Figure 13. Simulated relative velocity distributions according

to impact models in which encounter is considered

In “Blind Navigator” and Lützen models V

rela

and

impact angle are dominating factors in normalized

as seen in figure 11 when comparing results of

crossing encounters with head-on and overtaking

encounters. For head-on encounters normalized E

little higher than for overtaking but much lower than

in crossing encountering. This is because even if

rela

is high the deformation energy is mostly re-

leased in η-direction along the struck vessel side.

The vessels sailing on W-E and E-W waterways

are often on round trip to Gulf of Finland and thus

the vessels are recorded in most cases on both wa-

terways. Furthermore the loading condition was as-

sumed to be fully laden for all vessels. For these rea-

sons the vessel mass distributions are equivalent.

The same applies for N-S and S-N waterways ex-

cept that the vessels are sailing between Helsinki

and Tallinn. Additionally, the vessel masses on latter

waterway pair are much lower than that of the prior.

The differences in the vessel masses are resulting in

differences between waterway combinations in the

Figures 12, 13 as the distributions of V

rela

are equiv-

alent for all encounter scenarios in these models.

In figures 14, 15 the normalized cumulative dis-

tributions are compiled into same graph for crossing

encounter and head-on encounter respectively with

marked with vertical line.

From Figures 14, 15 similar observations as from

Figures 11-13 can be made. The four models derived

from statistics each result in higher E

ξN

in head-on

encounter than crossing while the opposite occurs

for the “Blind navigator” and Lützen models. The

same is valid for overtaking as was shown in Figures

12, 13.

4.3 Probability of oil cargo tank penetration

The oil cargo tank is penetrated when E

ξN

is greater

than 1 and the impact location is within tank limits

given by Equation 2. The number of simulated im-

pact scenarios in which the impact location is out-

side tank limits are listed in Table 5.

Cdf [-]

[-]

Blind

Navigator

N-

S…

Cdf [-]

[-]

Lützen (2001)

N-

S…

Cdf [-]

[-]

NRC (2001)

N-

S…

Cdf [-]

[-]

Brown (2002)

N-

S…

Cdf [-]

[-]

Rawson (1998)

N-

S,…

Cdf [-]

[-]

Tuovinen

(2005)

N-

S…

Figure 14. Simulated transversal deformation energy relative to

critical energy in crossing encounter.

Figure 15. Simulated transversal deformation energy relative to

critical energy in head-on encounter.

Table 5. Number of simulated collision scenarios of total

10000 simulations in which collision location is outside cargo

tank limits given by Eq 2.

__________________________________________________

Blind Rawson NCR Lützen Brown Tuovinen

Navi (1998) (2001) (2001) (2002) (2005)

__________________________________________________

2626 2602 1700 1772 1223 2449

__________________________________________________

In Table 6 the numbers of simulated collisions re-

sulting in an oil spill per impact model are present-

ed. The same is visualized in Figure 13.

Table 6. Number of simulated collision scenarios in which

oil cargo tank penetration occurs of total 10000 simulations.

________________________________________________

Ref Blind Rawson NCR Lützen Brown Tuovinen

No* Navi (1998) (2001) (2001) (2002) (2005)

________________________________________________

1 7283 1581 1889 4695 926 3153

2 7379 1612 1955 4671 930 3002

3 7335 1648 1930 4743 982 3169

4 7321 1629 1934 4557 977 3033

5 604 3230 4146 3232 2901 4705

6 563 3098 4089 3054 2794 4550

7 105 3121 4107 2842 2738 4551

8 43 3192 4142 2940 2841 4591

________________________________________________

* See Table 2 for explanation of Reference numbers

Figure 13. Number of simulated collision scenarios in which

oil cargo tank penetration occurs of total 10000 simulations

Taking the collision location into account does

change the results but very little. The differences be-

tween the models remain obvious. The collision fol-

lowing crossing encounter results in an oil spill in

three out of four cases according to “Blind naviga-

tor” model. Brown’s model suggest that oil spill

would occur only once in ten collisions.

5 CONCLUSIONS

In this paper a number of proposed models for im-

pact scenarios from literature have been applied to

the output of a maritime traffic simulation model to

create impact scenarios. The released deformation

energy is calculated with an analytical collision dy-

namics model for each impact scenario. Based on

the obtained deformation energy the cargo tank pen-

etration probability is estimated. The simulation re-

sults for relative velocity, transversal deformation

energy and oil cargo tank penetration are compared

between different impact scenario models.

The results of this case study indicate that the

models give a widely varying average hull breach

probability. In particular, the uncertainty on cargo

tank breach probabilities dependence of initial en-

countering is significant, which is an important fac-

tor in the analysis of oil spill risk in specific location

i.e. crossing or merging of waterways.

The distributions of collision energy for models

based on statistics depend almost solely on the strik-

ing vessel mass instead of the actual encounter sce-

nario. In the statistics that the models are based on

there are no collisions where a high speed craft is

involved. Further it is reasonable to assume that the-

se statistics include collisions, in which the struck

vessel is in anchorage, leading to underestimation in

struck vessel speed at the moment of impact in colli-

sions occurring at open seas.

None of the statistics is broken up for cases in

different sea areas nor is the encountering related

with the collision. These lacking in data are partly

N-S, E-W; Crossing

S-N, E-W; Crossing

due to the limited number of collision cases availa-

ble, lack of transparency and unsatisfactory report-

ing standards.

It is very likely that the statistical models are

grossly underestimating the effect of encounter

speed for both vessels in the area concerned in this

case study. This leads to the conclusion that the un-

derstanding of the conditions of ship collision in a

risk modeling framework is very limited at present.

The proposed models for impact scenarios are

moreover burdened with some inherent conceptual

limitations. The most significant limitation is the un-

satisfactory modeling of evasive maneuvering,

which links the initial encounter situation to the im-

pact scenario. The results clearly indicates that espe-

cially the parameters which navigators have a possi-

bility to affect in evasive maneuvering, i.e. vessel

speed and collision angle, play a determining role in

the evaluation of the consequences. Further research

on this matter is needed.

ACKNOWLEDGMENT

The authors appreciate the financial contributions of

the following entities: the EU, Baltic Sea Region

(this study was partly founded by EfficienSea pro-

ject), Merenkulun säätiö from Helsinki, the city of

Kotka and the Finnish Ministry of Employment and

the Economy.

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