Regulation 7-1 – Calculation of the Factor p_i

General

1 The definitions below are intended to be used for the application of part B-1 only.

2 In regulation 7-1, the words "compartment" and "group of compartments" should be understood to mean "zone" and "adjacent zones".

3 Zone – a longitudinal interval of the ship within the subdivision length.

4 Room – a part of the ship, limited by bulkheads and decks, having a specific permeability.

5 Space – a combination of rooms.

6 Compartment – a space within watertight boundaries.

7 Damage – the three dimensional extent of the breach in the ship.

8 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.

Regulation 7-1.1.1

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 L_s is greater or less than L* (i.e. 260 m); the other coefficients are valid irrespective of L_s.

Longitudinal subdivision

2 In order to prepare for the calculation of index A, the ship's subdivision length L_s 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.

3 There are no specific rules for longitudinally subdividing the ship, except that the length L_s defines the extremities of the zones. Zone boundaries need not coincide with physical watertight boundaries. 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. In general it is expected that the more zone boundaries the ship is divided into the higher will be the attained index, but this benefit should be balanced against extra computing time. The figure below shows different longitudinal zone divisions of the length L_s.

4 The first example is a very rough division into three zones of approximately the same size with limits where longitudinal subdivision is established. The probability that the ship will survive a damage in one of the three zones is expected to be low (i.e. the s-factor is low or zero) and, therefore, the total attained index A will be correspondingly low.

5 In the second example the zones have been placed in accordance with the watertight arrangement, including minor subdivision (as in double bottom, etc.). In this case there is a much better chance of obtaining higher s-factors.

6 Where transverse corrugated bulkheads are fitted, they may be treated as equivalent plane bulkheads, provided the corrugation depth is of the same order as the stiffening structure.

7 Pipes and valves directly adjacent or situated as close as practicable to a transverse bulkhead can be considered to be part of the bulkhead, provided the separation distance on either side of the bulkhead is of the same order as the bulkhead stiffening structure. The same applies for small recesses, drain wells, etc.

8 For cases where the pipes and valves cannot be considered as being part of the transverse bulkhead, when they present a risk of progressive flooding to other watertight compartments that will have influence on the overall attained index A, they should be handled either by introducing a new damage zone and accounting for the progressive flooding to associated compartments or by introducing a gap.

9 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.

10 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.

11 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 L_s could limit the number of zones defined to maximize the attained index A.

12 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:

Pure longitudinal subdivision

Regulation 7-1.1.2

Transverse subdivision in a damage zone

1 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 d_s, and r = 0 if b = 0.

2 The penetration depth b is measured at level deepest subdivision draught d_s as a transverse distance from the ship side right-angled to the centreline to a longitudinal barrier.

3 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 1/2 ≤ b₁ /b₂ ≤2.

4 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 d_s and the longitudinal bulkhead represents the outermost bulkhead position below d_s + 12.5 m.

4.1 If a transverse subdivision intercepts the deepest subdivision draught waterline within the extent of the zone, b is equal to zero in that zone for that transverse subdivision, [see figure 1]. A non-zero b can be obtained by including an additional zone, see figure 2.

4.2 If the deepest subdivision draught waterline on the side of a single hull ship includes a part where multiple transverse (y) coordinates occur for a longitudinal (x) location, a straightened reference waterline can be used for the calculation of b. If this approach is chosen, the original waterline is replaced by an envelope curve including straight parts perpendicular to the centreline where multiple transverse coordinates occur, [see figures 1 to 4]. The maximum transverse damage extent B/2 should then be calculated from waterline or the reference waterline, if applicable, at the deepest subdivision draught.

5 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,...,b_n}

• where: n = number of wing compartments in that group;

  • b₁,b₂,...,b_n = mean values of b for individual wing compartments contained in the group.

Accumulating p

6 The accumulated value of p for one zone or a group of adjacent zones is determined by:

where the total number of b_k's for the adjacent zones in question.

7 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)

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

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

Examples of multiple zones having a different b

9 Examples of combined damage zones and damage definitions are given in the figures below. Compartments 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, R22 and R32 damaged.

10 A damage having a transverse extent b and a vertical extent H₂ leads to the flooding of both wing compartment and hold; for b and H₁ only the wing compartment is flooded. The figure below illustrates a partial subdivision draught d_p damage.

11 The same is valid if b-values are calculated for arrangements with sloped walls.

12 Pipes and valves directly adjacent or situated as close as practicable to a longitudinal bulkhead can be considered to be part of the bulkhead, provided the separation distance on either side of the bulkhead is of the same order as the bulkhead stiffening structure. The same applies for small recesses, drain wells, etc.