Probabilistic Stability

A probabilistic approach was presented in IMO Resolution A265(VIII) in November 1973 as an alternative to the deterministic method of assessing damage stability as given in SOLAS 60 for passenger ships. The probabilistic method has now been modified and extended to apply to dry cargo ships over 80 meters in length, and for these vessels it is now mandatory to comply with the regulations specified in SOLAS Chapt. 11-1, Part B-1, Regulation 25- - 25-12, Cargo Ship Rules.

Both the Passenger Ship A265(VIII), the SOLAS Chapt. 11-1, Part B-1, Regulation 25- - 25-12, Cargo Ship Rules and the new SOLAS Harmonized Rules are incorporated within Calc. In this approach the survival capability is assessed through comparing the Required Subdivision Index, R, and the Attained Subdivision Index, A. The condition to be satisfied is that A > R. Probabilistic stability assessment addresses only the vessel in the postulated damaged condition. It is to be noted that the probabilistic stability approach does not rely upon information regarding the disposition of deadweight material in the vessel.

For some time now, the IMO hav e been formulating new regulations to replace both the cargo and passenger ship rules. These so-called Harmonized Regulations are now in the final stages of acceptance, and are now incorporated in Calc. The Calc implementation is based on the IMO document SLF 47/17 - Draft Revised SOLAS Chapter II-1 Parts A, B and B-1, plus changes to the 'p' factor (Regulation 7-1) specified in MSC 80/3/5 (21/01/2005).

The main difference noticeable to designers who are familiar with the usual method for damage stability calculations, is that indication of loss of the ship either by capsize or sinkage in any given damage scenario does not automatically necessitate altering the subdivision. The new regulations take a global view of the subdivision of the ship in association with a range of possible damage situations, giving credit to every case in which a particular watertight compartment or group of compartments is damaged and the ship survives. Greater credit is given if the ship survives after suffering damage to a high risk region, such as the fore end. Obviously, the longer the compartment is, relative to the subdivision length, the more probable it is that the compartment will be breached during the ship's lifetime and the less likely it will be that the ship will survive if damaged in that region. The constraints on the designer in placing bulkheads are therefore more subtle than in the deterministic calculations, where a good initial attempt at longitudinal subdivision could be obtained from considering the floodable length curves. Although longitudinal subdivision is still important, almost equal importance is attached, in the probabilistic approach, to horizontal subdivision, in the form of decks and flats above the two designated drafts to be considered, in improving the survivability of the ship.

The method of assessing the probability of survival of a proposed design is outlined below:

1. Determine the degree of subdivision required by calculating the Required Subdivision Index "R", according to the appropriate regulations.

2. Calculate the Attained Subdivision Index "A" which is determined as being the summation of the product of the following factors for each watertight compartment or group of compartments along the length of the ship:

   

   i	represents each compartment or group of compartments under consideration.
   pi	is the probability that only the compartment or group of compartments under consideration may be flooded, disregarding any horizontal subdivision.
   si	is the probability that the ship will survive damage to the compartment or compartments under consideration, including the effects of any horizontal subdivision.

The summation is done for each single compartment or two or more adjacent compartments along the ship's length, at two designated drafts - the deepest subdivision loadline, (subdivision loadline that corresponds to the summer draft to be assigned to the vessel), and a partial loadline, (corresponding to the lightship draft plus 60% of the difference between lightship draft and the deepest subdivision loadline draft).

The "s" value is a function of the shape of the righting lever curve after damage and also depends on the position of any unprotected or protected openings. For cargo ships, the "s" value is multiplied by a reduction factor "v", where.

Factors "p" and "v" are derived by formulae which take into account the location of the main transverse, longitudinal and horizontal subdivisions. Early results indicate that improvement of the attained index is best achieved by providing extra horizontal subdivision.

The "s" value is derived from a consideration of the equilibrium angle of heel after damage, the range of the residual GZ curve, and the maximum residual righting lever, GZ. In order to calculate "s", it is necessary to perform damage stability calculations of the traditional kind for all possible combinations of compartments and assuming an initial KG equal to the limiting KG obtained from the appropriate intact stability criteria (usually IMO Res A749) at the draft being used. If the ship survives a particular damage case, factor "s" will have a positive value less than or equal to unity and that case will contribute to the Attained Subdivision Index "A". If the ship fails to survive, "s" becomes zero and no contribution occurs from that particular case of damage.

A designer can assume one, two, three or more compartment damage conditions. Although the likelihood of the ship surviving such severe damage is low except in the case of a Ro-Ro ship. Alternatively, the designer may find that the attained subdivision index will reach the required value only by using the contributions from single compartment damage and in which case there is no point in performing higher compartment damage calculations, unless the insurance premium depends on the Ai value. Some dry cargo ships can be unsymmetrical about the centreline due to cranes on one side only. Calc can create cases of damage for starboard, port, or both starboard and port and the worst Ai contribution is used. All unprotected and protected openings should be included, since they are used to compute the "s" values.

If it proves impossible to achieve the required subdivision index "R", using every possible damage scenario then two actions can be taken. The first is to re-compute factor "s" at a lower KG. This implies that damage rather than intact stability governs the shape of the limiting KG/draft curve. If this strategy is successful then the new rules state that the damage limiting KG values are to be superimposed onto the existing intact stability curve by constructing a straight line between the damage KG values at the two designated drafts used in the calculations. If this strategy produces an unacceptably low critical KG, then the second possible alternative is to improve the subdivision of the ship.

The program checks a ship design to assess whether the Attained Subdivision Index "A", meets the Required Subdivision Index "R" using an assumed KG, which is usually equal to the intact stability critical KG. If "A" is less than "R" the ship does not comply with the regulations.

In this release of Calc, all set-up and calculation operations concerning Probabilistic Stability, have been consolidated into a new folder node, called Probabilistic Stability.

In order to perform a probabilistic stability analysis, you should proceed as follows:

3. Select and edit the Probabilistic Setup node to select the appropriate probabilistic regulations that must be satisfied, together with the maximum level of compartment combinations and other data including the bulkheads and decks that form zone and sub-zone boundaries.

4. Edit the Probabilistic Scenarios node to create the damage cases that must be run. The damage cases can be generated automatically by the program using the zone boundary information and their is a button on the dialog for doing this.The program assesses the locations of the main bulkheads and decks in relation to the zones to be damaged and from this produces data for input to the Probabilistic Stability module covering all possible combinations of single and multi-zone damage cases.

5. Edit the Probabilistic Stability calculation node and make sure that all damage cases that need to be run are selected.

6. Calculate the Probabilistic Stability node. This will run all damage cases for the designated drafts and calculate the attained index according to the regulations. For each designated draft conditions in turn, the damage cases created above, are analyzed on the basis of each compartment being free to flood to the outside waterline at the stated permeability. The program balances the ship for trim at each specified angle of heel and from the results is able to determine the final angle of equilibrium and the curves of residual GZ, GM, trim and waterline radius. The damage cases are analyzed to determine "s". Similarly from the disposition of the internal subdivision the program is able to calculate "p", "r" and "v" values, and finally the attained index "a" for each zone. The attained index is then calculated by summation of the individual zones and compared with the required index "R".

If "A" is less than "R", the KG used in the damage stability calculations is lowered in an attempt to obtain equality and the program successively iterates until a solution is obtained. If equality is impossible to achieve, the program prints out an appropriate message recommending an increase in subdivision.