Flooding Survivability Analysis - Total Risk (Safety Level)

IMO (SLF 47/48) Passenger Ship Safety

2.1.4 Total Risk (Safety Level)

2.1.4.1 Flooding Survivability Analysis

Using the harmonised probabilistic rules for damage stability as basis, substan- tive elements of the risk model (Eq. (2.3)) have been developed, (Jasionowski and Vassalos 2006), as indicated next:

prN(N|hz1) =

∑

3 i

n_{f lood}

∑

wi·pj·ⁿ

∑

^Hs

ek·ci,j,k(N) (2.4)

ci,j,k(N) =

⎛

⎜⎝−ln εi,j,k

· εi,j,k

t_{f ail|}_j(N) 30

⎞

⎟⎠·∂t_{f ail|}_j(N)

30 (2.5)

Where, the terms wi and pj are the probability mass functions of the 3 specific loading conditions and damage extents and nf lood the number of flooding extents, respectively, calculated according to the harmonised probabilistic rules for ship sub- division, (SLF47/17 2004). The term ek is the probability mass function derived from the statistics of sea states recorded at the instant of collision and nHs is the number of sea states considered. The term ci,j,k(N)is the probability mass function of the event of capsizing in a time within which exactly N number of passengers

pdf(Hs)

Hs critical The concept of a capsize band

pcap = cdf(Hs)

1 3 4 5 6

0.1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.2

0 1.5

1 2 2.5 3 3.5 4 4.5 5

0.5 0

Fig. 2.17 Principal the concept of “capsize band”, (Jasionowski et al. 2004), for critical sea states of 0.5 and 4.0 m

fail to evacuate, conditional on events i,j and k occurring, and can be tentatively estimated from Eq. (2.5) based on the harmonised probabilistic rules; the formulation shown accounts for ship geometry, loading and each individual sea state in any given flooding event. The termεi,j,k(withσr) represents the so-called capsize band shown in Fig. 2.17, that is the spread of sea states where the vessel might capsize.

These can be estimated as follows:

εi,j,k=1−Φ

Hsk−Hscrit(si j) σr(Hscrit(si j))

(2.6) σr(Hscrit) =0.039·Hscrit+0.049 (2.7) WhereΦ(.)is the cumulative standard normal distribution and Hscrit(s)is given by Eq. (2.8) below, (Jasionowski and Vassalos, 2006):

Hscrit(s) =Hscollision(s) =0.16−ln(−ln(s))

1.2 (2.8)

The si jis the probability of survival, calculated according to (SLF47/17 2004).

∂t_{f ail}(N) =t_{f ail}(N)−t_{f ail}(N−1) (2.9)

tf ail(N) =N⁻¹_{f ail}(t) (2.10)

Nf ail(t) =Nmax−Nevac(t) (2.11)

Finally, the term Nevac(t)is the number of passengers evacuated within time t, as shown below.

As illustrated in Fig. 2.18, two parameters are of paramount importance in eval- uating risk meaningfully: the first is the time required for orderly evacuation of passengers and crew in any given event(Nevac(t)), derived from numerical simu-

Nmax

tcap t

( )

N_evac

( )

N_fail

Fig. 2.18 Interplay between time to capsize and evacuation time

lations using advanced evacuation simulation software, (Vassalos et al. 2002); the second is the time to capsize/sink(tc), which is evaluated by using two methods, as explained next.

The time to capsize(tc), given a ship hull breach, is a random variable, (Vassa- los et al. 1998) hence only known as a distribution determined through probability methods Moreover, it is dependent upon a number of parameters (e.g. flooding con- dition, sea state, damage extent) all of which are also random in nature. In this respect, accounting only for the damage scenarios implicit in the new harmonised rules for damage stability (normally over 1,000 for a typical passenger ship) and considering the 3 loading conditions, also implicit in the rules, and some 10 sea states per damage case, it becomes readily obvious that brute-force time-domain simulations is not the “route to salvation”. In view of this, two lines of action are currently being pursued. The first relates to the development of a simple (inference) model for estimating the time to capsize, for any given collision damage scenario;

the second entails automation of the process using Monte Carlo simulation, as out- lined next.

Method 1 – Univariate Geometric Distribution – Collision

Considerable effort has been expended over recent years to develop an analytical expression, which could replace the need for expensive numerical simulations, and still provide an overall description of the character of the stochastic process of ship capsize when subjected to collision damage in a seaway. Although not yet considered to be as generic and comprehensive as the time-domain solution, a first pro- totype of such an engineering tool has been proposed in Jasionowski and Vassalos (2006) and Jasionowski et al. (2004, 2006), which is the model underlying Eq. (2.5).

The inference model here is based on a geometric probability density distribution for time to capsize for each flooding scenario. Deriving from the above and shown here in Eqs. (2.12) and (2.13) is the probability density distribution for the time to capsize, using the same terms used in the probabilistic rules for damage stability, as

defined earlier:

p(tcap) =

∑

3 i

n_{f lood}

∑

wi·pj·

n_Hs

∑

e_k·c_i,_j,k(tcap) (2.12) Where

ci,j,k(tcap) =−ln(εi,j,k)·(εi,j,k)^tcap³⁰ /30 (2.13) Convergence between this expression and results from time-domain simulations provides the validation needed for typical damage cases, as shown in Fig. 2.19 for a typical cruise ship, and the confidence that this simple inference model, is capable of predicting the likelihood of a vessel to capsize in any given flooding scenario within given time in fractions of a second. This is a significant development.

Considering all pertinent flooding scenarios for a typical ship, the outcome is the marginal cumulative probability distribution for time to capsize, shown in Fig. 2.20.

A close examination of Figs. 2.19 and 2.20 reveals the following note worthy points:

As a random variable, time to capsize can only be predicted in probabilistic terms. In other words, the deterministic number (3 h) postulated at IMO is in- appropriate. The correct term should be “time to capsize within 3 h with prob ability of x%”.

If a vessel did not capsize within the first hour post-accident, capsize is unlikely, on average.

The marginal probability distributions for time to capsize tend to asymptotic val- ues defined by (1-A).

Ability to estimate the probability of the time to capsize (in fractions of a second) could prove a very important tool to aid decision making in emergencies, particu-

Scenario={displ, KG, damage, Hs}

probability that vessel capsizes within 1 hourif collision takes place

Fig. 2.19 Cumulative probability function for time to capsize (scenario level) – comparison between analytical model and numerical simulation results

40,000 scenarios

probability that vessel capsizes within 3 hoursif collision takes place probability that vessel capsizes within 3 hoursif collision takes place

Probability that vessel survives for 3 hoursif collision takes place.

Fig. 2.20 Cumulative marginal probability distribution for time to capsize within a given time for a ship

larly when knowledge specific to the accident in question is accounted for and resolution of the model is enhanced by the time domain simulation results, shown next.

Method 2 – Monte Carlo Simulation – Collision and Grounding

To overcome problems associated with “averaging” (e.g., average probability of survival) and other approximations and potential weaknesses that might be embedded in the formulation of the probabilistic rules, the random variables comprising loading conditions, sea states and damage characteristics (collision: location, length, height, penetration according to the damage statistics adopted in the probabilistic rules; grounding: location, length, height, width, as determined in-house (The Ship Stability Research Centre – SSRC) using statistics from the EU Project HARDER, (2003) are sampled using Monte Carlo sampling and each damage scenario is sim- ulated using explicit dynamic flooding simulation by PROTEUS3, (Jasionowski 1997–2005). The resolution could be as high as necessary (every second of each scenario) accounting for transient- cross- and progressive-flooding, impact of multi- free surfaces, watertight and semi-watertight doors (relevant to passenger ships) and of course any damage scenario (collision, grounding, raking, etc.). Applications of this method indicate that 500 scenarios would be sufficient (for typical cruise ship / RoPax vessels the absolute sampling error for the cumulative probability of time to capsize derived was of the order of 4%–5%). Examples of a Monte Carlo simulations setup are shown in Fig. 2.21 (generic) and Fig. 2.22, Fig. 2.23 for collision.

Collision Collision

Water Water ingress?

ingress?

yes

Damage case Damage case

Case i=1 Case i=2 Case i=k Case i=342

Outcome Outcome

t(i) t(2) t(k) t(342) Model

tests Model

tests

Minor incident

Vessel unable to survive for 3h

Vessel survives for at least 3h (t ∝) Implication

Implication

Numerical simulations

Performance Performance--based based evaluation and verification

evaluation and verification t = time to capsize Fig. 2.21 Monte Carlo simulation – collision and grounding

Typical results are shown in Figs. 2.20 and 2.24 as cumulative distribution functions of time to capsize within a given time. From the latter it will be seen that differences between the two methods of nearly an order of magnitude have been encountered and this led to renewed scrutiny of the probabilistic rules, as reported in (Vassalos and Jasionowski 2007).

500 scenarios

Fig. 2.22 Monte Carlo simulation set up – collision

500 scenarios example

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 10 20 30 40 50 60

length [m]

CDF for length [–]

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

PDF for length [–]

cdf, data cdf, MC pdf, data pdf, MC

Fig. 2.23 Monte Carlo sampling – Length

4.5% of possible grounding damages (leading to water ingress) will lead to capsize within 20 minutes)

1.2% of possible collision damages (leading to water ingress) would lead to capsize within 1hour.

Analytical (SOLAS 2009)

simulations

Fig. 2.24 Probability distributions of time to capsize

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