Kristiansen Svein. Maritime Transportation: Safety Management and Risk Analysis
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This estimate of the frequency for loss of control compares well with the figure based on
the Japanese studies.
The Norwegian study (Kristiansen, 1980) previously referred to also estimated the
impact accident rate per distance unit as shown in Table 6.6. The coast was split into ten
main fairway segments. The estimated accident frequency given in the right-hand column
varied from 0.3 10
5
to 1.5 10
5
accidents per nautical mile, i.e. by a factor of 5. This
variation may be explained by different dominating ship types, fairway characteristics and
environmental factors. It should also be kept in mind that the characteristic distance (D)
does not necessarily reflect the typical traffic pattern, although the majority of vessels are
assumed to sail along the coast.
The mean value of 0.69 10
5
accidents/nm combined with an assumed geometrical
probability of 25% indicates the following loss of control frequency:
c
¼ P
c
¼
P
a
P
i
¼
a
P
i
¼
0:69 10
5
0:25
¼ 2:8 10
5
ðfailures=nmÞ
Even this e stimate compares well with the previous values.
In certain risk studies it will be of interest to estimate the number of impact accidents
within a specified period. We have the following expression for the accident probability
per passage (vessel movem ent):
P
a
¼
N
a
N
m
or rewritten:
N
a
¼ N
m
P
a
Table 6.6. Collision, grounding and impact accidents on the Norwegian coast for the period 1970^78
Fairway segment Distance Traffic Accidents P
a
¼N
a
/N
m
10
4
a
¼P
a
/D 10
5
(nm) N
m
(1/year) N
a
(1/year) (1/passage) (1/nm)
Oslo fjord 92 28,600 16.9 5.9 0.64
Langesund – Tananger 183 20,900 11.8 5.6 0.31
Tananger – Bergen 101 65,600 30.7 4.7 0.46
Bergen – Stadt 125 50,300 28.2 5.7 0.46
Stadt – Kristiansund 114 28,600 23.3 8.1 0.71
Kristiansund – Rørvik 155 21,600 24.8 11.5 0.74
Rørvik – Støtt 147 8,600 15.8 18.4 1.25
Støtt – Harstad 150 19,500 32.3 16.6 1.10
Harstad – Sørøysund 175 12,500 26.7 26.7 1.22
Sørøysund – Kirkenes 212 7,700 24.1 31.3 1.48
Weighted mean value 129 26,400 — 8.9 0.69
Norwegian coast 1454 263,900 253.2 —
Source: Kristiansen (1980).
14 6 CHAPTER 6 TRAFFIC-BASED MODELS
From the basic definition of the impact accident model we have:
P
a
¼ P
c
P
i
¼
c
D P
i
This gives the following computational expression for ground ing:
N
a
¼ N
m
c
D
B þ d
W
and stranding:
N
a
¼ N
p
D 1
2
W
D
6.6 COLLISION
In contrast to grounding and stranding, collision represents an impact between two
moving objects and not only one relative to a stationary hazard. A collision may also vary
in terms of how the vessels are approaching each other: head-on, crossing or overtaking.
These situations will be modelled somewhat differently, although the basic approach is the
same as for groundings in the sense that the critical impact cross-section is taken into
consideration.
6.6.1 Head-on Collisions
A ship is exposed to meeting traffic as outlined in Figure 6.7. The subject ship is exposed
to head-on approaching ships within a section of a fairway with distance D and average
width W. The modelling approach assumes the own (subject) vessel, denoted by index 1, is
approaching a traffic flow of vessels denoted by index 2. We also introduce the relative
sailing distance D
0
expressing the fact that both groups of vessels are moving.
B
1
¼ Mean beam of meeting ships (m)
v
1
¼ Mean speed of meeting ships (knots)
Figur e 6.7. Modell ing of head-on collision accidents.
6.6 C OLLIS ION 14 7
B
2
¼ Beam of subject ship (m)
v
2
¼ Speed of subject ship (knots)
N
m1
¼ Arrival frequency of meeting ships (ships/unit of time)
D
0
¼ Relative sailing distance (nm)
The mean number of meeting ships within a square nautical mile of the fairway is
referred to as the density of the oncoming ships. This density is calculated as the number of
ships entering the fairway within a time period relative to an area characterized by the
width of the fairway and the sailed distance of the first meeting ship:
s
¼
N
m1
T
ðv
1
TÞW
¼
N
m1
v
1
W
where:
s
¼ Traffic density of meeting ships (ships/nm
2
)
T ¼ An arbitrary period of time (hours)
The subject ship (index 2) spends T
2
time units to sail the specified section of the
fairway and has a relative speed v to the meeting traffic:
T
2
¼
D
2
; v ¼ v
1
þ v
2
The subject ship sail s the distance D
0
relative to the oncoming ships:
D
0
¼ v T
2
¼ðv
1
þ v
2
Þ
D
v
2
The impact diameter of a collision is equal to the sum of the exposed ship’s beam and the
beam of the meeting ships:
B ¼ B
1
þ B
2
Hence the area, A, where the subject ship is exposed to danger of collisions within the
fairway is equal to:
A ¼ B D
0
¼ðB
1
þ B
2
Þðv
1
þ v
2
Þ
D
v
2
The expected number of collisions per passage of the fairway, given that control has been
lost, is given by the product of the exposed area and the traffic density:
N
i
¼ A
s
¼ðB
1
þ B
2
Þðv
1
þ v
2
Þ
D
v
2
D
s
14 8 CHAPTER 6 TRAFFIC-BASED MODELS
or:
N
i
¼
ðB
1
þ B
2
Þ
W
ðv
1
þ v
2
Þ
v
1
v
2
D N
m1
If the main parameters of the subject ship are equal to the parameters of the meeting ships,
the expression is greatly simplified:
N
i
¼ 4 B D
s
A Head-on Collision Case
Problem
The number of reported head-on collisions for a specific fairway has in recent years been
about 14 annually, with little variation. Last year, however, there were 20 reported
collisions. Should we assume that this last higher number of collisions is an exception or
not? Find the expected number of head-on collisions per year on the basis of the general
collision model.
Assum pt i ons
. The lateral position of the ships within the fairway is uniformly distributed.
. The probability of losing control P
c
¼ 2 10
4
per passage of the fairway.
Analysis
The traffic density of meeting ships is:
¼
N
m
V W
¼
25
15 1852 3000
¼ 3 10
7
ðships=m
2
Þ
6.6 C OLLIS ION 14 9
The impact probability for oncoming traffic is:
N
i
¼ 4 B D
s
¼ 4 15 25,000 3 10
7
¼ 0:45 ðaccidents=incidentÞ
The probability of head-on collision for a single vessel in the fairway is:
P
a
¼ N
i
P
c
¼ 0:45 2 10
4
¼ 9 10
5
ðaccidents=passageÞ
The expected number of head-on collisions considering that we have the same traffic flow
in each direction is:
N
a
¼P
a
N
m
¼ 9 10
5
25 ð24 365Þ¼19:7 ðships=yearÞ
This estimate indica tes that the last year registration of 20 head-on collisions is not
exceptional.
6.6.2 Overtaking Collision
In overtaking encounters the vessels involved are sailing in the same direction but at
different speeds. The estimation of overtaking collisions is basically identical to head-on
collisions apart from the expression for relative speed. Assuming that the subject vessel
(subscript 2) is exposed to a uniform traffic flow in the same direction (subscript 1) we
have that the number of potential accidents is:
N
i
¼
ðB
1
þ B
2
Þ
W
ðv
1
v
2
Þ
v
1
v
2
D N
m1
Alternatively, we may compute the number of overtaking accidents within a unidirectional
traffic flow with a distribution of speed:
N
1
¼
ðB
1
þ B
2
Þ
W
D N
m
X
f
x
f
y
1
v
x
v
y
where f
x
and f
y
denote fractions of the total traffic flow N
m
with speed v
x
and v
y
respectively. The summation is taken over all combinations of different speeds within
the traffic flow.
An Overtaking Situation
Problem
A straight fairway of distance 25,000 m and width 3000 m has a co-directional traffic of
25 vessels per hour. Mean breadth of vessels is 15 metres. How many overtaking collisions
can be expected on an annual basis?
150 CHAPTER 6 TRAFFIC-BASED MODELS
Traffic observations have shown that the speed distribution is roughly as follows:
Fraction of traffic (%) 30 50 20
Speed (knots) 12 15 18
Solution
The expected number of encounters is given by:
N
i
¼
2B D N
m
W
X
f
x
f
y
1
v
x
v
y
¼ k
X
The constant term is:
k ¼ 2 15 25,000 25=3,000 ¼ 6:25 10
3
The summation part in the expression is:
X
... ¼
0:30 0:50
1
12
1
15
þ 0:30 0:20
1
12
þ
1
18
þ0:50 0:20
1
15
þ
1
18
2
6
6
6
4
3
7
7
7
5
1
1852
¼ 2:9 10
6
This gives:
N
i
¼ 6:25 10
3
2:9 10
6
¼ 0:018 (encounters/passage)
Assuming a probability of loss of navigational control P
c
¼2 10
4
, we get the following
estimate for the overtaking collision frequency:
N
a
¼ 0:018 2 10
4
25 8760 ¼ 0:79 ðaccidents=yearÞ
Comment
Let us make a comparison with the previous head-on collision case. We found then that
the two opposing traffic flows generated 19.7 accidents per year. Two similar flows would
have generated 0.79 2 ¼1.6 overtaking accidents per year. This means that the ratio
between head-on and overtaking is: 19.7/1.6 ¼ 12, which is a substantial difference.
6.6 C OLLIS ION 151
6.6.3 Crossing Collision
A collision between crossing vessels is sli ghtly more complex to analyse. This stems from
the fact that the scenario represents two encounter situations. As shown in Figure 6.8, we
have that the subject ship (subscript 2) is exposed to crossing ships within a section of a
fairway of lengt h D and width W . Making a distinction between the striking and the struck
vessel, it is immediately clear that both sets of vessels may have either role.
The description of the traffic situation is based on following nomenclature:
B
1
¼ Bea m of crossing ships (m)
L
1
¼ Leng th of crossing ships (m)
v
1
¼ Speed of crossing ships (knots)
B
2
¼ Me an beam of subject ship (m)
L
2
¼ Me an length of subject ship (m)
v
1
¼ Me an speed of subject ship (knots)
N
m1
¼ Arr ival frequency of meeting ships (ship/unit of time)
The mean number of crossing ships within a square nautical mile of the fairway is
referred to as the density of the crossing ship. Analogous to the meeting situation
described in the previous section, the density of the crossing traffic is given by the
following equation:
m
¼
N
m1
T
ðv
1
TÞW
¼
N
m1
v
1
W
where T is an arbitrary selected time period. The subject ship (2) takes T
2
hours to pass the
section of the fairway where it is exposed to the crossing traffic:
T
2
¼
D
v
2
The relative speed between the vessels is given by vector summation (Figure 6.9).
Figure 6.8. Crossing vessels situation.
152 CHAPTER 6 TRAFFIC-BASED MODELS
The crossing ships sail a distance D
1
in the course of the period that the subject ship is
exposed in the critical fairway section:
D
1
¼ v
1
T
2
¼ v
1
D
v
2
The following development of the model is split into two owing to the fact that we have
as already mentioned two collision situations:
1. A crossing vessel hit the subject vessel.
2. The subject vessel hit a crossing vessel.
Let us consider the first situation. The impact diameter of a collision is equal to the
sum of the exposed ship’s length and the mean beam of the crossing ships:
Q
1
¼ðB
1
þL
2
Þ
Hence the area where the subject ship is exposed to the collision hazard (Figure 6.10) is:
A
1
¼ Q
1
D
1
¼ðB
1
þ L
2
ÞD
v
1
v
2
Figure 6.9. Relative speed of crossing vessels.
Figure 6.10. Exposed area to collision for situation1.
6.6 C OLLIS ION 153
The expected number of collisions per passage of the fairway is given by the product of
the exposed area and the traffic density:
P
i1
¼ A
1
m
¼ B
1
þ L
2
ðÞD
v
1
v
2
N
m1
v
1
D
¼ðB
1
þ L
2
Þ
N
m1
v
2
The same line of argument can be followed for the second collision situation except
that the roles of striking and struck ship are changed:
D
2
¼ D
Q
2
¼ L
1
þ B
2
A
2
¼ Q
2
D
2
¼ðL
1
þ B
2
ÞD
This results in the following expected number of collisions:
P
i2
¼ A
2
m
¼ðL
1
þ B
2
Þ
N
m1
v
1
The total expected number of side collisions is the sum of the two calculated figurers P
i1
and P
i2
:
P
i
¼ P
i1
þ P
i2
¼ðB
1
þ L
2
Þ
N
m1
v
2
þðL
1
þ B
2
Þ
N
m1
v
1
¼
N
m1
v
1
v
2
ðB
1
þ L
2
Þv
1
þðL
1
þ B
2
Þv
2
½
Assuming that the subject ship and the crossing ships all have identical characteristics, the
expression is further simplified and visualizes the basic model, which says that the
potential number of encounters is equal to the traffic density times the exposed area:
P
i
¼
N
m1
v
2 ðB þ LÞ¼
m
2 ðB þ LÞD ð6:9Þ
A Cr ossi ng Encounte r Situati on
Problem
Two traffic lanes cross each other at 90
. Seven collisions have been recorded on an annual
basis in recent years. The local community is concerned by the considerable amount of oil
154 CHAPTER 6 TRAFFIC-BASED MODELS
spilled as a result of these accidents. They demand implementation of new safety measures.
The proposal has been challenged by the coast administration, which claims that the safety
level is acceptable. Give an assessment of the situation. For simplicity, identical traffic
volumes and same ship characteristics can be assumed.
Solution
Assess the safety standard of the vessels navigating the seaway by estimating the
probability of loss of control.
Data
Width of fairways: W ¼D ¼3km
Ship length: L ¼75 m
Speed: v ¼15 knots
Traffic: N ¼5 ships/hour
Ship beam: B ¼15 m
Analysis
Traffic density:
¼ N=ð DÞ¼5=ð15 1852 3000Þ¼6:0 10
8
ðships=m
2
Þ
Impact probability:
P
i
¼ 2 D ðL þ BÞ ¼ 2 3000 ð75 þ 15Þ6:0 10
8
¼ 0:032 ð1=incidentÞ
On the basis of extensive traffic studies it has been shown that the probability of losing
navigational control is P
c
¼5 10
4
(per passage).
The probability of impact accident is:
P
a
¼ P
c
P
i
¼ 5 10
4
0:032 ¼ 1:6 10
5
The expected number of collisions per year is therefore:
N
a
¼ N
m
P
a
¼ 25 ð24 365Þ1:6 10
5
¼ 3:5
Assessment
The fact that the fairway in question has twice as many collisions as might be expected,
indicates that there is a problem related to the crossing of traffic. The local community
therefore has a good case in their demand for further investigation of the conditions.
6.7 CO LLISION 155