Regulation 7–1 - Calculation of the factor pi
Clasification Society 2024 - Version 9.40
Statutory Documents - IMO Publications and Documents - Circulars - Maritime Safety Committee - MSC.1/Circular.1226 – Interim Explanatory Notes to the SOLAS Chapter II-1 Subdivision and Damage Stability Regulations – (15 January 2007) - Annex - Interim Explanatory Notes to the SOLAS Chapter II-1 Subdivision and Damage Stability Regulations - Part B - Guidance on Individual Regulations - Regulation 7–1 - Calculation of the factor pi

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 b11 , b12 , b21 and b22 are coefficients in the bi- linear probability density function on normalized damage length (J). The coefficient b12 is dependent on whether or not Ls = L*, the other coefficients are valid irrespective of Ls .

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 Ls .

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:

  • j: the damage zone number starting with no.1 at the stern;
  • n: the number of adjacent damage zones in question where j is the aft zone;
  • k: the number of a particular longitudinal bulkhead as a barrier for transverse penetration in a damage zone counted from shell towards the centreline. The shell has no.0;
  • K: total number of transverse limits;
  • P j,n,k : the p-factor for a damage in zone j and next (n-1) zones forward of j damaged to the longitudinal bulkhead k.

Pure transverse subdivision

Single damage zone, pure transverse subdivision:

pj,1 = p(x1j,x2j)

Two adjacent zones, pure transverse subdivision:

pj,2 = p(x1j,x2j+1) - p(x1j,x2j) - p(x1j+1,x2j+1)

Three or more adjacent zones, pure transverse subdivision:

pj,n = p(x1j,x2j+n-1) - p(x1j,x2j+n-2) - p(x1j+1,x2j+n-1) + p(x1j+1,x2j+n-2)

Parent topic: Part B - Guidance on Individual Regulations

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 b1/b2 with ½ = b1/b2 = 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 ds + 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 {b1 , b2 , …, bn }
where:
n = number of wing compartments in that group;
b1 , b2 , …, bn = 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 = Kj,n
pj,n = ∑pj,n,k
k = 1
where
= the total number of bk 's for the adjacent zones in question

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:

(bj,1 ; bj+1,1 ; bj+n-1,1 ; bj,2 ; bK )

Because of the expression for r(x1, x2, b) only one bK should be considered. To minimize the number of calculations, b's of the same value may be deleted.

As bj,1 = bj+1,1 the final b's will be (bj,1 ; bj+n-1,1 ; bj,2 ; bK )

The total accumulated p

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 (b1 < b2 = b3 ) generating three damages:

1) to b1 with R11, R21 and R31 damaged

2) to b2 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 H2 leads to a flooding of both wing compartment and hold; for b and H1 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.

Copyright 2022 Clasifications Register Group Limited, International Maritime Organization, International Labour Organization or Maritime and Coastguard Agency. All rights reserved. Clasifications Register Group Limited, its affiliates and subsidiaries and their respective officers, employees or agents are, individually and collectively, referred to in this clause as 'Clasifications Register'. Clasifications Register assumes no responsibility and shall not be liable to any person for any loss, damage or expense caused by reliance on the information or advice in this document or howsoever provided, unless that person has signed a contract with the relevant Clasifications Register entity for the provision of this information or advice and in that case any responsibility or liability is exclusively on the terms and conditions set out in that contract.