4 The tables of probability for side and bottom damage

  4.1 To ease application of the probability density functions, the probability density distributions for damage location, extent, and penetration have been converted into a set of tables and simple equations. These tables indicate the probability that the damage is bounded on one side by a given longitudinal, transverse or horizontal plane.

  4.2 For example, the function p_b(d) is the probability that damage is restricted to less than d, the normalized damage location, given g(y), the probability density distribution of extent of damage, h(x), the probability density distribution of location, and c, the maximum extent of damage. Similarly, p_a(d) is the probability that damage is restricted to more than d.

	(4.2-1)
	(4.2-2)

  4.3 These equations are repeated for all of the damage probability calculations. For the cases involving penetration they simplify to single integral equations. For the cases involving both extent and location, special consideration must be given to the ends of the density. The functions define the damage location as the centre of damage. Damage zones towards the ends or sides of the ship can span beyond the vessel. This explains why all the probability tables do not extend to 1.00.

Integration Region for Integrated Damage Probability P_j of j-th Tank

  4.4 To obtain the probability that a region bounded by d₁ below and d₂ above is damaged, one finds p = 1 - p_b(d₁) - p_a(d₂) . Note that this probability includes all damages which include the region, not just those that damage that region alone. To determine the probability of damage for a region in three-dimensional space the appropriate probabilities in each dimension are multiplied together reflecting the independence between the pdfs. To simplify the calculation process each three dimensional region is modelled as an equivalent rectilinear block described by six boundaries.

  4.5 The tables and equations for side damage provide the following parameters:

• P_Sa = the probability the damage will lie entirely aft of location X_a/L;

  P_Sf = the probability the damage will lie entirely forward of location X_f/L;

  P_Sl = the probability the damage will lie entirely below the tank;

  P_Su = the probability the damage will lie entirely above the tank; and

  P_Sy = the probability the damage will lie entirely outboard of the tank.

  4.6 The tables and equations for bottom damage provide the following parameters:

• P_Ba = the probability the damage will lie entirely aft of location X_a/L;

  P_Bf = the probability the damage will lie entirely forward of location X_f/L;

  P_Bp = the probability the damage will lie entirely to port of the tank;

  P_Bs = the probability the damage will lie entirely to starboard of the tank; and

  P_Bz = the probability the damage will lie entirely below the tank.