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Understanding Survivability

The recent and nearly catastrophic accident of the

Collision and grounding incidents are not artifacts of the past. They happen more often than common perception might suggest. In fact, recent studies show that, statistically, we can expect to witness one catastrophic accident (those claiming more than 1,000 lives) every 20 years. There is human error behind many accidents that have occurred, but even the most advanced operational procedures will not prevent these kinds of incidents. There are countless possibilities of system malfunctions, human negligence, or carelessness leading to disasters, and the risk of a ship sinking or capsizing cannot be eradicated. Instead, it should be understood, measured, and accounted for. The risk is manageable; therefore, it should be managed.

However, treating safety as a business instrument means that there has to be a rationale behind risk management. Safety must serve industry and not prevent its growth. For obvious reasons, this cannot be achieved by lowering standards—these are already debatable—and therefore the only way forward is better understanding of the physics of the problem. The issues of capsizing and sinking cannot be solved, at least in a rational manner, by increasing the number of bulkheads or imposing minimal height of residual freeboard. Nature is far more inventive than scientists and engineers. On the contrary, the legislative instruments regulating safety must be robust enough to ensure that the rightful safety levels that might have prevented past accidents have been set, which also will minimize the risk of future disasters.

The currently in-force probabilistic framework for damage stability, known as SOLAS 2009, is a good example of such thinking. It is based on knowledge derived from the past, but by abandoning deterministic artifacts, it aims at allowing innovative designs. However, shortly after being introduced, the new regulations provoked serious concerns for passenger ships with respect to accuracy in predicting the probability of surviving damage. Specifically, it was postulated that the new SOLAS overestimated the probability of survival in the case of RoPax vessels, while survivability of large cruise vessels was said to be underestimated. In order to address these issues the European Commission founded in 2009 a research project entitled GOALDS (

A major difficulty in modeling damage survivability stems from the complexity of the underlying physical phenomena. More specifically, the problem lies in predicting ship behavior in waves while accounting for floodwater dynamics and the flooding process in complex geometries typical of passenger ships. For individual cases, the prediction can be based on computational fluid dynamics CFD solutions. However, due to high demand in computational power and long computation time, such an approach cannot be followed on an industrial scale where large number of cases and long runs are envisioned (physical experiments share similar limitations). Therefore, studies on diversified samples, necessary for statistical generalization of the results, require simplified mathematical models, while physical tests and CFD experiments provide data for benchmarking of numerical results.

Usually, numerical simulations are carried out by computer codes based on linear or linearized theories of ship motions. This enables employing the superposition principle, which simplifies the process of solving the equations of ship motions to a great extent. A similar effect occurs with the assumptions made with respect to flow-through damage (external) and internal openings; for example, that the flow through an opening commences without delay whenever there is a pressure difference on either sides of the opening, or that water spreads instantaneously in a damaged compartment. Furthermore, while accounting for ship-floodwater interaction, it is often assumed that free surface of the floodwater remains horizontal regardless of the orientation of the ship (hydrostatic modeling) or that the whole floodwater is reduced to its centroid and such defined lump-mass moves within the compartment over predefined surfaces (approximate hydrodynamic modeling).

Inevitably, the computational efficiency of the numerical techniques comes at the expense of accuracy of the results. Moreover, performing formal quantification of intrinsic uncertainties is difficult and consequently the models are usually described as being of “satisfactory engineering accuracy.” In practice, it is virtually impossible to provide accurate assessment for epistemic (systematic) uncertainties for such compound models. Furthermore, quantification of aleatory (statistical) errors is very difficult. This derives mainly from the fact that, unlike in the case of an intact ship, the ergodicity assumption does not hold—in principle—for dynamical processes associated with the motions of a damaged ship. As a result, the quantification of these processes in the time domain as an ensemble domain assessment (i.e., analysis of large collection of short runs) becomes inapplicable. The time domain quantification requires very long runs, making CFD and EFD experiments impractical.

*Figure 1: Numerical simulations of the sequence of flooding stages leading to capsize of large passenger vessels.*

One of the easiest to obtain, yet important characteristics of a damaged ship is the so-called capsize band: a range of sea states within which a transition from unlikely to certain capsize is observed. The width of the band varies with damage characteristics and ship loading conditions; in fact, the band is broader and covers higher sea states in case of ships characterized by larger residual stability. For this reason, the capsize band is often plotted as a family of curves (sigmoids) obtained by varying the vertical position of the center of gravity (KG) with the ordinate corresponding to the rate of observed capsizes.

**Figure 2: CFD computations showing visualization of sloshing in a flooded compartment.**

An important notion originating from the concept of capsize band is critical significant wave height (HS

In the process of formulating amendments, an explicit reference to the HS

*Figure 3: Response amplitude operator (left) and non-dimensional roll damping coefficient (right) in calm-water forced-roll oscillations. Physical experiments carried out at the Ship Stability Research Centre. The dry hull condition corresponds to an intact ship. In damaged condition, part of the solid ballast was removed from the flooded compartment to maintain the same mass and center of gravity of the system (mass + floodwater) as in the case of an intact vessel. The second peak on the RAO of the flooded ship—at approximate frequency 1.125—is practically invisible due to ten-fold increase in roll damping.*

Here is where another important notion emerges from the concept of capsize band–time to capsize. The time to capsize is a random quantity; it cannot be calculated precisely but it can be derived by means of a probability distribution. Without going into unnecessary details, one can conclude, based on the concept of critical significant wave height, that because the HS

It is noteworthy that the rate at which survival time diminishes at higher sea states depends not only on residual stability, but also on the complexity of the internal architecture of the ship. It is much higher in the case of RoPax ships than in the case of cruise liners. The difference derives from the distinct watertight architecture of the vessels and specifically from the presence of large undivided cargo decks/holds typical for RoPax ships. For this reason, RoPax vessels are more vulnerable than cruise ships to rapid capsizes with catastrophic consequences. However, suffering collision or grounding damage in sea states higher than HS

It is a well known fact, nearly a truism, that capsize is caused by accumulation of floodwater in the open-to-sea internal spaces of the ship. Hence, the HS

In order to validate the assumption it was necessary to investigate the process of accumulation of floodwater in damaged spaces. The task, however, is not trivial for a number of reasons. Most importantly, the analysis should be performed for surviving cases because only in the absence of progressive flooding can the process can be stationary; that is, its statistical characteristics do not change with time or position. This makes the analysis simpler and enables reasoning based on collection of short-time realizations instead of individual long runs.

*Figure 4: The graph on the left shows capsize bands recorded during model tests on RoPax vessel. Graph on the right depicts contraction of the capsize band with increased observation time.*

On the other hand, it is usually very difficult to detect capsize on the basis of time histories. With RoPax, in particular, there is seldom a hint that the ship is about to capsize until the process has already begun. Therefore, the first question to be asked is: How to decide whether the particular run survives or not? Indirectly, the answer is already known as, during a surviving run, an average amount of floodwater must remain constant. In GOALDS, however, a slightly different approach was followed. Namely, instead of looking at the average amount of floodwater, the maximum (95th percentile) values were recorded at times corresponding to multiples of wave peak periods, Tp. The characteristic obtained for surviving cases at a sea state corresponding to critical significant wave height was taken along with its upper confidence limit as a reference, with all other characteristics plotted against it.

The results revealed interesting features of the process of floodwater accumulation. First, if the reference curve did not have a horizontal asymptote (limit), this would indicate that some of the surviving runs will result in capsize with longer simulation times. Second, the characteristics obtained for surviving runs in sea states exceeding HS

Interestingly, particularly in the case of RoPax ships, it was observed that capsize would not occur immediately after the limit had been exceeded; in many cases, it would take considerable time before the ship lost stability. Furthermore, even if the ship returned temporarily to the safe band after exceeding the floodwater limit, capsize was inevitable. These conclusions are important for two reasons. One, they provide additional evidence supporting the revisited concept of critical significant wave height, HS

One of the principal goals of the GOALDS Project was to derive new, more robust, and accurate formulation for estimating the probability of surviving collision and grounding damages in waves—the s-factor. Similar to the earlier approach pursued in the Project HARDER, it was decided to base the new formulation on the concept of HS

Numerical simulations, however, are based on simplified mathematical models and, as discussed earlier, the prediction can be significantly affected by uncertainties. Therefore, regression models should be based on unbiased and readily verifiable parameters; for example, instead of referring to the amount of accumulated floodwater, it is more appropriate to account for damage extent, and so on. However, going further in this direction, it soon became apparent that the only reliable parameter that could be derived from numerical simulations is the HS

As indicated previously, the SOLAS formulation for calculating the s-factor employs only two parameters: maximum righting lever, GZ

The first two parameters were taken as a ratio, GZ

However, when applied to experimental data (model tests) it proved to be generally conservative. Bearing in mind that, as a rule of thumb, the numerical prediction underestimates survivability, it was decided to base further refinements on results of physical tests instead of numerical data. The final expression was obtained by very simple manipulation, which resulted in replacing the ratio GZ

The modified formula allowed very high correlation with experimental data and high accuracy of prediction. Furthermore, validation with a DoE technique demonstrated that the choice of parameters is completely sufficient to represent the sample of measured critical significant wave heights.

Predicting survivability of damaged ships in a seaway is a complex task, involving analysis of complex physical phenomena. This, however, can be done only if the available experimental data is of high quality and derived from large and diverse samples. These two requirements are conflicting as, at present, it is practically impossible to perform sufficiently large number of CFD or physical experiments within reasonable time, even if the work can be carried out by a number of research institutions. Numerical simulations, on the other hand, are much more time-efficient, but the efficiency comes at the cost of lower accuracy. However, this can be overcome if the numerical prediction is used primarily in order to derive dominant parameters, while tasks requiring higher precision and validation are performed with the use of more accurate, but less effective, techniques.