Probabilistic Stability Overview

Introduction

A probabilistic approach was presented in IMO Resolution A265(VIII) in November 1973 as an alternative to the deterministic method of assessing damage stability in SOLAS 60 for passenger ships. Perhaps because of the fact that use of these regulations was optional, they were largely ignored by U.K. designers. However, the idea of a probabilistic approach remained and has finally proved to be the only way of dealing with dry cargo ships where a strictly deterministic approach would be difficult to apply and would almost certainly lead to the need for substantial extra subdivision.

The scope of A265 has now been incorporated within Calc, in addition to the Cargo Ship Rules SOLAS Chapt. 11-1, Part B-1, Regulation 25-1 - 25-12. The new Harmonized Regulations, which will eventually replace both the Cargo and passenger ship rules, have also been included.

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 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.

The main difference noticeable to designers 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 to horizontal subdivision, in the form of decks and flats above the two drafts considered, in improving the survivability of the ship.

Assessment

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

Determine the degree of subdivision required by calculating the Required Subdivision Index "R".

This index "R" has been derived after much discussion and research over a considerable period by representatives of Government bodies at IMO. The current value has been set as a compromise to try to avoid the need for major changes in subdivision for existing designs. It is very likely that the index will be changed over the years in the light of experience gained in its use. It is even possible that different values may be set for different types of dry cargo ships, but at present it is the same for all. The formula has been deliberately kept simple to allow for such adjustments. Surveys of samples of the world's fleet of dry cargo ships indicate that existing multi-hold bulk carriers will meet the required index comfortably, tween-deckers and container ships will be split approximately 50-50 between compliance and non-compliance as will ro-ro ships greater than 140 meters long. Ro-Ro ships less than this will probably require extra subdivision.

The next step is to calculate the Attained Subdivision Index "A" which is calculated 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 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. The "s" value is multiplied by a reduction factor "v".

v - represents the probability that the spaces above the horizontal subdivision will not be flooded.

Factors "p" and "v" are derived by formulae which take into account the location of the main transverse, longitudinal and horizontal subdivision. Early results indicate that improvement of the attained index is best achieved by 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. To calculate "s", it is necessary to perform damage stability calculations of the traditional kind for all possible combinations of compartments 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 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 use one, two, three or four compartment damage. 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 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 drafts used in the calculations. If this strategy produces an unacceptably low critical KG, then the only 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 equal to the intact stability critical KG. If "A" is less than "R" the system derives a limiting KG such that "A" = "R", if this is possible.