Regulation 7-1 – Calculation of the factor pi
General
The definitions below are intended to be used for the application of part B-1 only.
In regulation 7-1, the words “compartment” and “group of compartments” should be understood to mean “zone” and “adjacent zones”.
Zone – a longitudinal interval of the ship within the subdivision length.
Room – a part of the ship, limited by bulkheads and decks, having a specific permeability.
Space – a combination of rooms.
Compartment – an onboard space within watertight boundaries.
Damage – the three dimensional extent of the breach in the ship.
For the calculation of p, v, r and b only the damage should be considered, for the calculation of the s-value the flooded space should be considered. The figures below illustrate the difference.
Damage shown as the bold square:
Flooded space shown below:
Paragraph 1.1
The coefficients b_{11}, b_{12}, b_{21} and b_{22} are coefficients in the bi- linear probability density function on normalized damage length (J). The coefficient b_{12} is dependent on whether or not L_{s} = L^{*}, the other coefficients are valid irrespective of L_{s}.
Longitudinal subdivision
In order to prepare for the calculation of index A, the ship’s subdivision length Ls is divided into a fixed discrete number of damage zones. These damage zones will determine the damage stability investigation in the way of specific damages to be calculated.
There are no rules for the subdividing, except that the length Ls defines the extremes for the actual hull. However, it is important to consider a strategy carefully to obtain a good result (that is a large attained index A). All zones and combination of adjacent zones may contribute to the index A.
The figure above shows different longitudinal divisions of the length L_{s}.
The first example is a very rough division into three zones of approximately the same size with limits where transverse subdivision is established. The probability that the ship will survive a damage in one of the three zones is expected to be low (s-factor = 0) and, therefore, the total attained index A will be lost.
In the second example the zones have been placed in accordance with the watertight arrangement, including minor subdivision (as in double bottom, etc.). The chances of getting good s- factors in this case should be good.
Where transverse corrugated bulkheads are fitted, they may be treated as equivalent plane
bulkheads, provided the corrugation is of the same order as the stiffening structure.
The triangle in the figure below illustrates the possible single and multiple zone damages in a ship with a watertight arrangement suitable for a seven-zone division. The triangles at the bottom line indicate single zone damages and the parallelograms indicate adjacent zones damages.
Figure illustrates the possible single and multiple zone damages in a ship.
As an example, the triangle illustrates a damage opening the rooms in zone 2 to the sea and the parallelogram illustrates a damage where rooms in the zones 4, 5 and 6 are flooded
simultaneously.
The shaded area illustrates the effect of the maximum absolute damage length. The p-factor for a combination of three or more adjacent zones equals zero if the length of the combined adjacent damage zones minus the length of the foremost and the aft most damage zones in the combined damage zone is greater than the maximum damage length. Having this in mind when subdividing Ls could limit the number of zones defined to optimize the attained index A.
As the p- factor is related to the watertight arrangement by the longitudinal limits of damage
zones and the transverse distance from the ship side to any longitudinal barrier in the zone, the following indices are introduced:
Paragraph 1.2
Transverse subdivision in a damage zone
Damage to the hull in a specific damage zone may just penetrate the ship ’s watertight hull or penetrate further towards the centreline. To describe the probability of penetrating only a wing compartment, a probability factor r is used, based mainly on the penetration depth b. The value of r is equal to 1, if the penetration depth is B/2 where B is the maximum breadth of the ship at the deepest subdivision draught ds, and r = 0 if b = 0.
The penetration depth b is measured at level deepest subdivision draught ds as a transverse
distance from the ship side right-angled to the centreline to a longitudinal barrier.
Where the actual watertight bulkhead is not a plane parallel to the shell, b should be determined by means of an assumed line, dividing the zone to the shell in a relationship b_{1}/b2 with ½ = b_{1}/b_{2} = 2.
Examples of such assumed division lines are illustrated in the figure below. Each sketch
represents a single damage zone at a water line plane level ds and the longitudinal bulkhead represents the outermost bulkhead position below d_{s} + 12.5 m.
In calculating r-values for a group of two or more adjacent compartments, the b-value is common for all compartments in that group, and equal to the smallest b- value in that group:
b = min {b_{1}, b_{2}, …, b_{n}}
where: | n = | number of wing compartments in that group; |
| b_{1}, b_{2}, …, b_{n} = | mean values of b for individual wing compartments contained in the group. |
Accumulating p
The accumulated value of p for one zone or a group of adjacent zones is determined by:
k=K_{j},_{n}p_{j,n} = p_{j,n,k}
k=1
j+n-1
where K_{j,n }= K_{j} the total number of b_{k}’s for the adjacent zones in question.
j
The figure above illustrates b’s for adjacent zones. The zone j has two penetration limits and one to the centre, the zone j+1 has one b and the zone j+n-1 has one value for b. The multiple zones will have (2+1+1) four values of b, and sorted in increasing order they are:
(b_{j},1 ; b_{j}+1,1 ; b_{j+n-1,1} ; b_{j,2} ; b_{K})
The total accumulated p
j=T
p = p_{j,n}j=1
where T is the total number of damages.
Examples of multiple zones having a different b
Examples of combined damage zones and damage definitions are given in the figures below. Rooms are identified by R10, R12, etc.
Figure: | Combined damage of zones 1 + 2 + 3 includes a limited penetration to b3, taken into account generating two damages: |
1) to b3 with R10, R20 and R31 damaged
2) to B/2 with R10, R20, R31 and R32 damaged
Figure: | Combined damage of zones 1 + 2 + 3 includes 3 different limited damage penetrations generating four damages: |
1) to b3 with R11, R21 and R31 damaged
2) to b2 with R11, R21, R31 and R32 damaged
3) to b1 with R11, R21, R31, R32, and R22 damaged
4) to B/2 with R11, R21, R31, R32, R22 and R12 damaged
Figure: | Combined damage of zone 1 + 2 + 3 including 2 different limited damage penetrations (b_{1} < b_{2} = b_{3}) generating three damages:: |
1) to b_{1} with R11, R21 and R31 damaged
2) to b_{2} with R11, R21, R31 and R12, damaged
3) to B/_{2} with R11, R21, R31, R12, and R22, R32 damage
A damage having a horizontal extension b and a vertical extension H_{2} leads to a flooding of both wing compartment and hold; for b and H_{1} only the wing compartment. The figure illustrates a partial subdivision draught dp damage.
The same is valid if b- values are calculated for arrangements with sloped walls.
The same is valid if b- values are calculated for arrangements with sloped walls.