PIAS Manual  2021 Program for the Integral Approach of Shipdesign
Probdam: probabilistic damage stability
With this module the damage stability can be computed on a probabilistic basis. Summarized briefly, the probabilistic method encompasses that, assuming the vessel is damaged, the probability that this damage is located in a certain area is determined, as well as the probability that damage in that area is survived. The product of these two represents the probability of survival in the event of damage in that area. By calculating these probabilities for many areas and adding them up, the total probability of survival is determined (at multiple drafts). This probability of survival must be greater than the minimum as imposed by the regulations.

# The background of the probabilistic damage stability method

This PIAS manual is not the proper place for an in-depth explanation of the probabilistic damage stability and all its merits, for that purpose the space is lacking and the subject is simply too extensive. Furthermore, a program manual is not a tutorial. For the background we therefore refer to the reference list, which is included at the end of this chapter. Recommended is the nice book of Pawlowski [1], which gives a complete and thorough overview. The backgrounds of specific configurations and calculation methods are discussed in the papers [2], [3], [4] and [5], which themselves also refer to elder literature. And last but not least, the relevant regulations and their explanatory notes will have to be kept at hand.

You will neither find design advices here, nor recommended practices or configurations. This module must be regarded as a toolbox, from which the user can select his preferred tool. However, we do pursue that the properties of all choices and options are clearly explained, and that the calculation process itself is clear.

# Introduction to the module

## General

To calculate the probabilistic damage stability only a few characteristics of the intact vessel, some configuration parameters and the damage cases have to be defined (which can be generated automatically). A damage case consists essentially of a number of selected compartments which are damaged. These compartments have to be defined using module Layout. This module is capable of automatic determination of damage cases and damage boundaries, but the price that must be paid for this comfort is that the vessel has to be defined in compartments totally and uniquely. This means that every point in the vessel must be part of one compartment and may not be part of more than one compartment. Layout contains tools to assist in this process, for example the one discussed in Difference between internal and external geometry.

## External compartments

PIAS subcompartments are available in two flavors, of the type ‘bulkheads’, which are bounded by plane bulkheads or the side shell or bottom shell, and of the type ‘external’, which can have any arbitrary shape and for which we refer to the discussion in Shape definition external subcompartments on the definition method. For probabilistic damage stability subcompartments of both types can be used, however, for external subcompartments the following remarks apply:

• With subcompartments of the type ‘bulkhead’ the damage boundaries will always be determined by one of the four bulkhead locations. With ‘external subcompartments’ such single points can not be assumed a priori, so the whole shape of the external subcompartment has to be taken into account. As such, this will obviously not be a problem, but the effect is that much more points have to be considered, and that the processing time of the automatic determination of damage boundaries might increase considerably.
• A similar effect does occur with the generation of damage cases: with subcompartments based on bulkheads the program can assume that any of those bulkheads might possibly be a damage case boundary. with external subcompartments such assumptions do not apply, so internally the subcompartment is subdivided into may smaller portions, and it is investigated whether each of those portion boundaries is a damage case boundary. On the one hand this leads to increased processing times, while, on the other hand, it will never be certain that the amount of portions is sufficient to find each and every damage case boundary.

• External subcompartiments can be used, but as sparsely as possible.
• When using external subcompartments, be sure to verify the generated damage case thoroughly.

# Main menu of the module

Probabilistic damage stability

This menu contains the main options, which will be discussed just below. Additionally, this menu contains details of the damage cases. Not only the number of cases claims is presented, but also whether there are useful results from previous calculations present, either the probabilities of damage (the prv values) or the probabilities of survival (the s values). These results may be reused later, so that any subsequent calculations could be performed significantly faster.

## Calculation method, configurations and ship parameters

This is the central configuration part of this module, and contains the following sub options:

Calculation method, configurations and ship parameters

### Calculation method, configurations and ship parameters

Attention
In this section many choices and options for probabilistic damage stability are being discussed. The number of options is large, options even exist which are not backed by one of the regulations. Nevertheless these options are kept, they might have a historic origin and, as such, required for elder projects of vessels. At some options question marks can even be put, but each one has its own background and if this field would be pruned not each variant would be available anymore. That would be user-unfriendly.

When this option is selected an setup window will appear. This screen has a variable layout, its content is dependent on the selected calculation method and applied regulations. In this window all kinds of choices and configurations can be made, with the aim to offer the user as many options as possible and (consequently) to pre-program as few as possible. Fortunately, this menu offers the function button <Default>. With this function the configuration is chosen according to the selected regulations, at least, in the opinion of SARC, in January 2012. Of course other institutions or persons may prefer or prescribe other choices, so the is advised to verify that you agree with the programs ‘default’ choices, which are:

 Option SOLAS 1992 & IMO A.265 SOLAS 2009 & 2020 TCG in intact condition Upon no heel Upon no heel Reference point for penetration depth Waterline Waterline Application penetration limitation (b1,b2) Apply rule, except at damage to center line Apply rule, except at damage to center line Type of penetration limitation b1,b2 < 2.min(b1,b2) bmean < 2.min(b1,b2) ‘Mean’ or ‘Minimum’ penetration Mean Mean Penetration rule multicomparttment damages Local Local Damage penetration over CL No Yes, except with calculation method ‘numerical integration’ r outside brackets in product r x (p123-p12-p23+p2) No No Probability of damage never negative No No With intermediate stages of flooding Yes Passenger vessels: yes, cargo vessels: no Generate including horizontal subdivision SOLAS 1992: yes, IMO A.265: no Yes

The exact background and intention of each options will be discussed in the paragraphs below.

#### Ship type

Exclusively for the SOLAS 2009 and 2020 regulations here the choice can be made between ‘cargo vessel’ and ‘passenger vessel’.

#### Applied regulations

Here the user can select one of the following regulations:

SOLAS 2020
The regulations for cargo vessels over 80 m as well as passenger vessels, into force from January 1, 2020.
SOLAS 2009
The so-called harmonized regulations, for cargo vessels over 80 m as well as passenger vessels. Applicable from January 1, 2009 to January 1, 2020.
SPS 2008
The probabilistic ruleas according to the Special Purpose Ship Code from 2008.
SOLAS 1992
For cargo vessels with a length over 80 m. Has been replaced by SOLAS 2009
IMO res. A.265
The equivalent rules for passenger vessels, according to IMCO res. A.265 of 1973. Also replaced by SOLAS 2009
Reconstructed SOLAS 1992
This ‘regulation’ is not relevant in regular ship design, partly because it is only applicable to the calculation method ‘numerical integration’, The background of this pseudo-regulation is discussed in [3] and [5], and its existence originates from the inconsistent processing at combinations of transverse and longitudinal subdivision in SOLAS 1992. As a consequence of this anomaly the results from numerical integration may differ from those as obtained by direct application of the SOLAS rules. Because numerical compatibility has its practical side, we have derived a new probability function by means of reverse engineering, and this is the reconstructed SOLAS 1992. Summarized, ‘SOLAS 1992’ is theoretically in agreement with the regulations, but gives numerically deviant results, while the ‘reconstructed SOLAS 1992’ is theoretically nonsense, but provides answers in line with a conventional calculation. By the way, this whole aspect does not come into play with SOLAS 2009, because the whole foundation of this method is much more solid, thanks to a common treatment of p and r.
Attention
This modules only takes the probabilistic damage stability aspect of these regulations into account. Other matters, such as possible additional deterministisc damage stability requirements, are not considered. Thus, it is necessary to verify whether there are any other than probabilistic demands, and to treat them separately.

#### Calculation method probability of flooding

Here a choice can be made between four methods for the calculation of the probability of damage p.r.v: ‘numerical integration’, ‘1 damage per compartment’, ‘1 damage per subcompartment’ and ‘1 damage per zone’. The backgrounds of these methods are discussed in [3], [4] and [5], summarized briefly it touches a corner stone of the probabilistic method, which is the assignment of a probability of damage to each portion of the vessel. In principle it is indifferent which atomic (i.e. undividable) portion is taken, as long as the sum of all portions covers the whole vessel. In practice a number of choices for these portions has surfaced, enumerated from coarse to fine:

• A zone, whereas a zone is a portion of the vessel between two longitudinal boundaries (e.g. transverse bulkheads). he use of the zonal concept forces the subdivision model into regularity, thus avoiding certain pitfalls of a more refined subdivision. However, the zone-model is artificial; it is an abstraction of the actual subdivision, and as such will produce a less accurate result. It is funny to see that the zonal concept is rather popular, although it is not even mentioned in SOLAS 1992 (however, it is mentioned in the explanatory notes). In SOLAS 2009 the terms ‘zone’ and ‘compartment’ are entangled, however, the ‘zone’ is not defined at all.
• A compartment. This is the most obvious choice, for it corresponds to the actual subdivision.
• A subcompartment. A compartment as an atomic entity may not even be small enough. It might occur that there exists no single damage (with flat aft, forward and inner boundaries) which affects a compartment completely, ut that a finer subdivision is needed to cover the entire compartment. An example is shown in the figure underneath, where the assumption that each compartment is affected by a single damage does not hold for compartment 1. A further division of this compartment, for instance along the dotted line, will make it affected by two damages: B-C and D-E. More complex compartment shapes are by nature in PIAS composed of subcompartments, so they can readily be used as an atomic entity. Obviously, for the determination of the probability of survival the compartment is always flooded in its entirety.
• Tiny pieces of volume (“voxels”), which together fill the whole vessel, and which are independant of the subdivision as such. If the PDF s are not a priori integrated, the whole usage of crisp boundaries disappears. Consequently, there is no need for any atomic portion concept. If, as proposed in [3], the probability functions are integrated numerically in combination with the actual geometry of the compartment (or a group of compartments), then any compartment shape can be processed, including possible niches, irregularities and warped or even curved compartment boundaries. On this numerical integration method some additional remarks can be made:
• The application of numerical integration in this context can be compared with the developments in the area of structural strength; initially analytically determined standard solutions for the deflection of beams were utilized, but for more complex structures the division into very small, Finite Elements proved to be more flexible.
• The option to determine the probability of damage using numerical integration is not mentioned in SOLAS, however, it is a recognized method for the calculation of outflow from oil tankers, see res. MEPC.66(37) of September 14, 1995, and the “Revised interim guidelines for the approval of alternative methods of design and construction of oil tankers under Regulation 13F(5) of Annex I of MARPOL 73/78, MEPC.110(49)” from July 18, 2003.
• [6] and [7] report on the same approach, at the University of Hamburg.
• This method is also known as Monte Carlo method.
• The combination of this method with the IMO res. A.265 regulations has not been implemented in PIAS.

A space where ‘zone’ and ‘compartment’ are too coarse

#### Damage at

Here SB or PS can be given. According to the explanatory notes of (at least) SOLAS 1992, if asymmetry is present in hullform or compartmentation, the calculation must be made to SB as well as to PS, where as attained subdivision index A both the lowest value and the average value of the two may be used. Apart from that, with Config it can be specified for damage stability calculations whether the inclination is to SB, to PS or is determined automatically, but this setting is not applicable for the probabilistic damage stability calculations.

#### Light ship draft or light service draft

Here the light ship draft (for SOLAS 1992 and IMO A.265) or the light service draft (for SOLAS 2009 and SOLAS 2020) must be given. These drafts are the same as can be given Hulldef, as discuseed in Particulars for SOLAS chapter 2, part B1.

#### Subdivision draft

Here the subdivision draft must be given, with the same remark about SOLAS and Hulldef as in the previous paragraph.

#### TCG in intact condition

Here one can choose between Coincides with centerplane and Is determined upon no heel. This switch is only relevant with vessels with an asymmetrical hull shape, With the first choice, TCG at centerplane, an initial list will occur, while with the second option the Transverse Center of Gravity is determined so that no list will occur.

#### Reference point for penetration depth

The penetration depth b is the penetration (in meters) up to the inner damage boundary. However, if this distance is measured from the waterline, and the inner boundary is located beyond the local waterline breadth, a logical inconsistency pops up, see the cross section figure below. With damage to compartment C only, the matter is that Bc > Bwl, so the penetration = Bwl - Bc < 0, and consequently the probability of damage p will be zero. And that is the problem, because we have a real damage case at hand, with a real penetration through real steel, which is not included arithmetically as a damage case in the context of probabilistic damage stability. For this reason this option offers the choice between two alternatives:

• Waterline, where the penetration is determined from the waterline at deepest subdivision draft.
• Upper boundary of damage case, where the penetration is determined at the level of the upper boundary of the damage case under consideration, in our figure height Hc in our figure. In that case a realistic probability will be assigned to the damage case of compartment C. According to the table above, the default for this options is ‘waterline’, however, between the end of 2009 and the first week of January 2011 it was configured as ‘upper boundary of damage case’.
Penetration depth in cross section

#### Application penetration constraint (b1,b2)

According to the explanatory notes of SOLAS 1992 the penetration at side compartments is limited by the rule that the maximum penetration shall not exceed twice the minimum penetration. This constraint can limit the penetration depth very severely, see e.g.the figure below, where only compartment 1 is damaged. The evident penetration is according to the angled line, with the penetration depth indicated by b. However, in this case the minimum penetration, b1, is zero, so the maximum penetration b2 is always greater than twice the minimum penetration, violating the penetration constraint rule. The only solution to comply with this rule is to set b2 also to zero, which results in (an unrealistic small) penetration depth b as depicted in the second figure. With a hollow waterline b would even become negative, which leads to a probability of damage of zero for this rather realistic damage case.

Penetration limitation rule with b1=0

At this option ‘Application penetration constraint (b1,b2)’ it can be specified whether, and how, this rule is applied. The options are:

• Without this penetration constraint rule.
• With this rule, except for damages which extend to CL.
• With this rule, except for damages with an inner boundary // CL (where ‘//’ denotes ‘parallel to’).
• With this rule, also for damages to CL (so the rule is always applied).

#### Type of penetration constraint

As mentioned, the penetration constraint rule of SOLAS 1992 reads that the maximum penetration may not be larger than twice the minimum. For SOLAS 2009 & 2020 the rule is slightly different, it reads that the mean penetration may not be larger than twice the minimum. Therefore the program offers a choice of two alternatives, where the SOLAS 1992 version is formulated as ‘b1,b2 < 2.min(b1,b2)’, and the 2009 & 2020 one ‘bmean < 2.min(b1,b2)’.

#### At exceeding of the (b1,b2) constraint

Here it can be specified (onl;y with the zonal method) what the program should do when the (b1,b2) penetration constraint is exceeded:

• Warning only, which implies that the dimensions of the specified zonal boundaries are still used, and that the program specifies on the output in which damage cases the (b1,b2) rule is violated. It is up to the user to adapte the zonal boundaries for these cases.
• Let the program adapt the penetration automatically to the (b1,b2) constraint rule.

#### Mean' or Minimum' penetration

This choice is related to the determination of penetration breadth b (see e.g. SOLAS 1992 reg. 25-5.2.2). Three choices are possible here:

• According to (at least) the Dutch Authorities it can be interpreted as ‘b = the minimum transverse distance...’. See also the report SLF 34/WP.11, item 5.3.10, and the paper ‘IMO Circular letter 1338: Interpretations by the Netherlands Administration’ ( Appendix 1: mean/minimum penetration, IMO circular letter 1338, in Dutch).
• Literally ‘b = the mean transverse distance...’, an interpretation which is likely to be used by many authorities.
• In 1991 we have propounded the question how to handle in case b1=0 or b2=0 to the Dutch authorities (see Appendix 2: mean/minimum penetration, the question and Appendix 3: mean/minimum penetration, the reply by NSI, in Dutch). The answer was that in such a case always minimum penetration depth may be used. That leads to the (for the ship designer) most favorable situation, because in cases where b1 as well as b2 are greater than zero the mean depth can be used (which is in general the largest), while in case of b1=0 or b2=0 one can switch to minimum, so there is no penalty of a small penetration depth. This choice is incorporated in PIAS, and is called ‘Largest of mean and minimum’ (This ‘Largest of mean and minimum’ rule has not been implemented. The same effect can be obtained with a is not applied on damages with an inner boundary which is parallel to centerline.).

The choice between the three alternatives is left to the user, we can only mention to have observed that the minimum interpretation requires the least computation time.

#### Determine r at multi-compartment damages

This option determines the way how with a multi-compartment damage with longitudinal subdivision the reduction factor r is determined. The first possibility is local dimensionless penetration b/B, where the r of each to-be-subtracted damage case is determined with the individual b/B as measured halfway the waterline of that case. The other choice has the effect that the r of each damage case (so the main damage as well as the damages to be subtracted) is determined on basis of a common dimensionless penetration b/B. The exact implementation depends on the selected calculation method:

• With the (sub-)compartment methods cases the global dimensionless penetration b/B is applied, which is determined halfway the main damage (In paragraph 3.5 of [2] it is explained why this is the only viable option for the general case).
• With the zonal method the minimum dimensionless penetration b/B is applied, where each reduction factor r is determined with the b/B which is the minimum of all involved b/B's (so the b/B from the main damage as well as from the damages to be subtracted). The explanatory notes suggest this approach in fig. a-3 (IMO res. A.684 for SOLAS 1992) respectively paragraph 1.2 (SLF 49/17 for SOLAS 2009). Because with the zonal method the subdivison is regular, this method is in this particular case feasible. SARC has no preference for one method over another. By the way, until November 2001 PIAS had no external mechanism to select this calculation option. Before that date always local b/B was used. From the viewpoint of logic and consistency the use of local penetration can be favored above global or minimum, because with local penetration the probabilities of the damages to be subtracted are always equal to the probability with which they have been added. With global or minimum this is not the case, which might be confusing. Another aspect is illustrated in the figure alongside. With local penetration is p12 = p12.r12 - p1.r1 - p2.r2, with r12 determined on the basis of b12, r1 on the basis of b1 and r2 on the basis of b2. In the example b1<0, so r1=0, b12<0, so r12=0 and b2>0, so r2>0. Consequently p12 becomes negative, regardless the nature of the formula of r as a function of b. With global or minimum penetration r1, r2 and r12 are all determined on a common basis (With global dimensionless penetration on basis of b12, and with minimum dimensionless penetration on basis of the minimum of b1, b2 and b12), so a negative probability cannot occur (at least not due to an anomaly in to the processing of to-be-subtracted damage cases).
Attention
This whole subject with mean, minimum, penetration rule multi-compartment damages and b1/b2 penetration rule does not play a role, cannot even play a role, with the numerical integration method.

#### Damage penetration over CL

In 1992 SOLAS and IMO A.265 the penetration of side damage was limited to centerline. In SOLAS 2009 the penetration is B/2, without further addition. That implies that with SOLAS 2009 in the regions of the narrowing waterline, in foreship and aftship, the damage will extend beyond centerline. With this option one can choose between these two variants. This option is, for now, not available for the ‘numerical integration’ method. It is not because this would be infeasible, on the contrary, but the combination has never been required.

#### r outside brackets in product r x (p123-p12-p23+p2)

Here is can be specified how to process a combined longitudinal and transverse subdivision. If r is taken outside brackets than the equation for a three-compartment damage reads pr = r123.(p123 - p12 - p23 + p2), if r is not taken outside brackets pr = r123.p123 - r12.p12 - r23.p23 + r2.p2 is used.

#### Probability of damage never negative

If ‘no’ is answered to this question then for the probability of damage the true outcome of the calculation is used. With ‘yes’ it will be maximized to zero (in other words, if the probability is less than zero, zero will be taken).

#### Including intermediate stages of flooding

With ‘no’ the damage stability calculations are only performed for the final stage of flooding, with ‘yes’ also the intermediate stages 25%, 50% and 75% are computed.

#### Combine damage case generation with the calculation

Besides for the computation of probabilities of damage, the numerical integration method can rather well be used for the generation of damage, so a distinct generation step can be considered a bit redundant. With ‘yes’ at this option it is specified that the generation of damage cases and the execution of the calculation must be combined, with ‘no’ they are separated.

#### Maximum damage length for damage case generation

So that no damages will be generated larger than this value. At SARC, in general we see little reason to deviate from the default value of hald the ship length.

#### Maximum number of damaged (sub-)compartments per damage case

With Z given here, no damage cases will be generated with more than Z simultaneously damaged compartments (or subcompartments, in case of the subcompartment method). This parameter should be chosen with care; a too small value will result in a limited number of damage cases, and hence a low attained subdivision index. A high number will produce the maximum A, but that may require a high number of damage cases with a corresponding long computation time. A scrisp number cannot be recommended, that depends on the nature of the subdivision and on the number of compartments. At SARC we tend to start at ten, and may adapt this parameter dependant from the computation result.

By the way, the absolute maximum number of damage cases for this module is 5000, that has nothing to do with this parameter.

#### Maximum number of damaged zones per damage case

When using the zonal method the program will generate all combinations of adjacent zones. This could lead to large amounts of damages, from which the majority of multi-compartment damages could preponderantly not contribute to A. In order to limit the number of damage cases, with this option the maximum number of zones per damage can be specified.

#### Accuracy index numerical integration (0-100)

With the numerical integration method the integration step size plays a role; it may be clear that the accuracy increases with a decreasing step size. The real step size is determined by means of an algorithm, on the basis of the number of compartments etc., and it is not considered relevant to confront the user with it. For that reason the accuracy is defined by means of an index, where 0 indicates the least accuracy and 100 the greatest. One might be inclined towards the greatest possible accuracy, but that of course also induces a longer processing time. The exactly required accuracy cannot be indicated in general. Apart from your patience it does also depend on the vessel's layout. In order to give an impression one can imagine two extremes, one extreme is a barge with completely rectangular subdivision. Because the program applies an integration step at each (sub-)compartment boundary, all spaces are already taken into account completely, so the result will (approximately) be independent from the accuracy index. On the other hand, in case of a vessel with angled bulkheads the number of integration steps does play a role, for illustration purposes we have created a barge with only a single longitudinal bulkhead, which is extremely twisted. The required subdivision index A as a function of the accuracy index is plotted in the figure. It will be obvious that in reality the quantitative effect of this index will be somewhere between these extremes.

Accuracy index

#### Generate including horizontal subdivision

With ‘yes’ the damage cases will be generated including horizontal subdivisions (decks) between the compartments. With ‘no’ the horizontal subdivisions will be ignored, so the damages will extend from baseline to the uppermost deck, leading to less damage cases.

#### Damage cases generation including "progressive flooding"

If compartment connections have been defined (see Define compartment connections for more details), then with this option it can be specified that at the generation of damage cases the program will model the flooding of compartments through pipes by means of complex intermediate stages of flooding. This option is only relevant for the (pre 2021) method of complex stages, please refer for that method to Complex stages of flooding (before 2021).

#### Intermediate results in text file incl. all integration steps

For verification purposes the contributions of all numerical integration steps are also included in the text file with intermediate results (When PIAS' multithreading function is active, multiple integration cycles are executed concurrently. A side effect is that these intermediate results are not included in the textfile in a synchronized fashion. If synchronicity is required, multithreading should be switched off, which can be done with the external variable no_multithreading, as discussed in Fairway). With a large accuracy index the number of integration steps can be quite large, and the text file might consequently rise to enormous proportions. To avoid that, it can be specified at this option to exclude the integration steps from the text file.

#### Store intermediate results in spreadsheet file

The text file, as discussed in the previous option, is intended for human interpretation. For an analysis of the computational results it might be handy to have the figures available in a spreadsheet. If that is specified at this option, the results will be written in a .CSV (Comma Separated Values) file, such a file can be read with most spreadsheet programs. However, such a spreadsheet file can only be generated if the configuration options listed below (Re-use unmodified results of former calculations) is set ‘off’. The reason is that all results actually must have been calculated before they can be included in the spreadsheet file.

#### Re-use unmodified results of former calculations

In order to increase the computation speed this module keeps quite some results of former calculations in memory. These can be re-used in subsequent calculations. For example, for each damage case the KNsin(φ) are stored, so that in case of future modifications of VCG' the GZ in damaged condition can be rapidly determined. In general it is recommended to use this facility, but if it is not desired for whatever reason it can be switched off at this option.

• Nb 1. The results of former calculations can also be removed explicitly, see Remove all results of former calculations.
• Nb 2. The re-usage of existing results is a completely module-internal matter. It has nothing to do with the intermediate calculation data (see Show intermediate calculation data which is intended solely for human interpretation.
• Nb 3. When this option is used and PIAS uses the results of former calculations, this will be mentioned in the intermediate calculation data as statement instead of the full data for the corresponding damage case(s). When you require complete intermediate calculation data, this option should not be used or results should be removed explicitly as described above.

#### Orientation damage case plots

With Create plots of damage cases (and possibly Create plot of zonal boundaries) damage cases can be plotted. With the option under consideration it can be specified if these plots are requested in portrait or in landscape format.

#### Wind pressure for calculation of heeling moment (kg/m2), passenger moment, life boat moment & selected wind contour

Here the data for the various components of the heeling moment can be given, which are required for the calculation of passenger vessels.

### Drafts, trims and VCG's

In Calculation method, configurations and ship parameters the subdivision draft and light draft have already been specified, with which the calculation drafts are fixed. In the present menu for each of those calculation drafts the following particulars can be entered:

• The trim.
• The VCG' / MG' combination. Enter the VCG', and the MG' is automatically adapted, and the other way round.
• Whether this is the draft on which the VCG' must automatically be determined, in order for the attained subdivision index to become equal to the required subdivision index: A = R (see Determine the VCG' for which A=R).

### Define hopper stability particulars (incl. pouring in or out)

This module is also capable to calculate the probabilistic damage stability according to the regulations ‘Agreement for the construction and operation of dredgers assigned reduced freeboards’, a.k.a. dr-68. See for all details of hopper stability Probabilistic damage stablity.

### View scheme of standard permeabilities

According to SOLAS 2009 & 2020 the several kinds of space types must be computed with different permeabilities (μ). The μ's which belong to each type of space can be inspected with this menu option. With module Layout the correct space type will have to be assigned to each compartment, where one should verify that all subcompartments of such a compartment have their ‘Autopermeability’ set to ‘yes’, otherwise the static μ will still be used.

Permeabilities for different types of spaces

### Edit scheme of user-defined permeabilities

The permeability list of View scheme of standard permeabilities covers those types of spaces which are explicitly defined in SOLAS 2009 & 2020. Possible additional categories can be defined with this option.

### Define compartment connections

This function co-operates with PIAS' system for complex intermediate stages, the development of which was frozen in 2021 because a more capable system called Consecutive flooding was released by then. When using that, this specific Probdam function is no longer needed, so it will be inactive then. For more information, see Background from tools for ship-internal connections in PIAS.

In this menu for each compartment can be specified which compartments are flooded in the case of damage to the compartment under consideration. In particular one should consider the flooding via damaged pipes. This menu consists of a matrix of connections, between all compartments, where in the first column all compartments are linked to a number that corresponds to the numbers in the header line. With these numbers it is clear which two compartments are connected. The connection between the two compartments can also be found textually in the status bar.

In almost every cell, two connections can be specified for two compartments by using the ‘Connections popup’. All settings in the popup are explained in more detail below, using the compartment connections menu. In addition to the popup, an ‘Open’, ‘Pipe’ or ‘A-bulkhead’ connection can be made between the two compartments using the <O>, <P> and <A> keys respectively.

Connections popup

Double-clicking on a compartment name — in the second column — opens a menu where all connections to and from the selected compartment can be defined. For each connection it must be specified whether it exists by means of ‘Selected’. If such a connection exists, the type of connection must be defined, which can be one of the following types:

• [Open], means a connection with a large cross-section, through which the seawater can pass freely, also in intermediate stages of flooding.
• [Pipe], means a connection with a small cross-section through which the seawater cannot pass freely in intermediate stages. When a connection is of the [Pipe] type, a complex intermediate stage of flooding will be generated (if this option is purchased), in which the compartment connected by means of a pipe will not be flooded.
• [A-bulkhead], is treated exactly the same as a connection of the type [Pipe] in the damage stability calculations.

Additionally, it can be specified whether the connection always exists, or only after the water level at a certain point is exceeded. The latter can e.g. occur with a bulkhead which does not extend over the full height. In this case the column ‘Cr.pnt’ — an abbreviation of Critical Point — should be marked ‘yes’, while at Lcrit, Bcrit and Hcrit the longitudinal, transverse and vertical coordinates of that point must be specified. In addition to the ‘Critical Point’, one can also specify whether the connection should be checked against the criteria for the intermediate or final stage of flooding. Normally this setting is No, but in some situations it may be desirable or mandatory to test against the criteria for the final stage. Please note that, if the regulations have not specified criteria for intermediate stages - eg for cargo vessels under SOLAS 2009 - the criteria for the final stage will be used.

In this menu there is also a ‘Copy’ column with which the entire connection between the compartments to be connected can be copied. Normally the following applies to a connection; if one compartment, say compartment A, is connected with another compartment, say compartment B, it does not imply that B is automatically connected with A. In reality that does also not necessarily has to be the case, e.g. when a pipe is fitted in compartment A which will flood B if damaged. That mechanism does not apply reversely, so at damage of compartment A, B will also be flooded, but at damage of compartment B, A remains unaffected. If there exists a permanent opening between A and B, one should explicitly specify that A is flooded in case of damage to B. This explicit task can be taken care of by setting the copy to ‘Yes’. It is worth noting that in the connection popup, from the matrix menu, copying can be done in a more selective way.

The values specified here are solely used at the generation of the damage cases, in other words, these data are not used independently, but are processed in each individual damage case. So modified data will only be processed after re-generation of damage cases.

Two more remarks can be made on this option:

• The generated data can be inspected at the damage cases ( Select and edit damage cases) after choosing the function <Flooding stages>. With the second menu option a list appears which shows which compartments are connected with a damaged compartment from that particular damage case, and what is the position of a possible critical point. This specification can also be edited, see for more details the section on complex stages of flooding, Complex stages of flooding (before 2021).
• If one defines the following two pipe connections, A to B and B to C, then in the event of direct damage to A, compartment B will be flooded, but as a result of the flooding of B, compartment C will also be flooded. This situation ultimately translates into two complex intermediate stages of flooding, whereby compartment B is 100% filled after the first stage and compartment C is 100% filled after the second stage. This is shown below in a short table, in which the percentages of flooding are printed under compartments A, B and C. The Final stage is printed purely for completeness.  Stage A B C 1 100 0 0 2 100 100 0 Final 100 100 100

### Define zonal boundaries

The compartments defined in PIAS are the real enclosed spaces of the ship. However, in a vessel on a less detailed level often ‘primary spaces’ or ‘zones’ can be identified, where multiple compartments can be situated in such a zone. An example of a zone is the engine room, regardless the exact layout of the many consumable compartments. So a zone boundary is an abstract notion, it does not always have to coincide with a physical bulkhead. Zonal boundaries can be specified for two purposes:

• For a calculation with the zonal method. For this purpose the user will have to define transverse zonal boundaries, and possibly also longitudinal and horizontal boundaries. At the aft and forward extremities (more precise, at aft terminal and forward terminal) obviously zonal boundaries should be placed. If these are not present the program will add them at the aftmost and foremost frame of the hull.
• With the (sub-)compartment method the damage cases can be collected into zones with Collect damage cases into damage zones. Other software for the calculation of probabilistic damage stability may work on a zonal basis, and for comparison purposes it might prove handy if the output of PIAS is also sorted in zones. (see also Print calculation results, subtotalized by zone).

### Notes (free text)

With this option a window appears where free notes about the calculation or the configuration can be written. These notes are saved with the calculation. It can also be specified whether these notes must be printed on the output, or in the text file with intermediate results.

Notes

### Determine the VCG' for which A=R

Except from the subdivision, the attained subdivision index A is also dependant on the vertical center of gravity VCG'. In general the index will rise with decreasing VCG', so there will be a single VCG' for which A is exactly R. With this option this critical VCG' is determined. One aspect is that multiple drafts with belonging VCG's are in play, and that no general rule can be postulated for the distribution of the VCG' variation over the drafts. Therefore, at Drafts, trims and VCG's the user has to assign the single draft for which the critical VCG' is determined.

## Generation of damage cases

### Generate ALL possible NEW damage cases

With this option all existing damage cases will be deleted and all possible (up to a maximum of 5000) damage cases will be generated. This might, notably with a large number of damage cases, take quite some time, which significantly be reduced with the use of multithreading see Speed enhancing mechanisms in PIAS: PIAS/ES

The name by which each generated damage case is identified depends on the setting of ‘Identification of the compartment in the tank sketches’ as discussed at Settings for compartments and tank sounding tables :

• Automatic tank number : The name will be ‘N.M’ where N is the number of simultaneously flooded compartments and M is the sequence number in the series of damage cases with the same number of flooded compartments.
• Name/second name/abbreviation compartment : The name consists of the selected identification of all flooded compartments. Within Calculation method, configurations and ship parameters the maximum damage length and maximum number of damaged compartments can be given, in order to keep the amount of damage cases within reasonable limits. It is recommended to start the calculations with a short damage length and a modest number of simultaneously flooded compartments. Only in cases when the vessel does not comply with the required subdivision index R the damage length and a number of compartments can be increased.

With the (sub-)compartment method the progress will be shown in the Multithreading task monitor. Damage cases are generated from the forward- to the aft side of the vessel and per thread it is shown from which forward boundary damages are generated.

With the (sub-)compartment method the damage cases as generated with this option are only based on the (sub-)compartment geometry. So the initial damage cases are determined by the extreme compartment boundaries. During the computation of the damage stability calculation the true damage boundaries will be determined, taking into account all boundary conditions. For this reason it might occur that a generated damage case after all is not able to exist, within all statutory limitations. Especially the penetration constraint rule b1/b2 plays a role to this effect. If a damage case does not exist, the warning ‘Damage impossible’ is included in the .PD0 file with intermediate results, please also refer to Warnings.

With the previous option all existing damage cases will be removed and new damage cases are generated. The present option also generates damages cases, but these will be added to the existing damage cases. Four selection criteria must be specified here:

• Aft boundary of the area within which the damage cases have to be generated.
• Fore boundary of the area within which the damage cases have to be generated.
• Minimum number of compartments per damage case.
• Maximum number of damaged compartments per damage case. The criteria ‘maximum damage length’ and ‘maximum number of flooded compartments’ are not applicable here.

### Generate high sub damages as complex stages of flooding

With this option complex intermediate stages of flooding — a concept for which reference is made to Complex stages of flooding (before 2021) — are generated, where compartments below a certain, user-specified,height get a zero percentage of flooding. This option can be used to simulate minor damages. In this menu, the [Create] function will automatically create a list of candidates for the relevant heights.

### Collect damage cases into damage zones

With this option, which sorts and collects damage cases, is only relevant for the (sub-)compartment method, it is not related to the zonal method as distinct calculation method for the probability of damage. This sorting is based on the zonal boundaries as specified in Define zonal boundaries. After sorting, those damage cases which fall into the same zone are collected, while subtotals for all zones and combinations of zones are printed. One must recognize the fact that the calculation process of PIAS does not change when using zones, it only affects the presentation of the output. While sorting, the damage cases can be renamed if you have chosen to do so. In that case the name of a damage case is composed in the form of K.L.M.N, where:

• K: A number which indicates the number of zones in which the damage case extends.
• L: The zone number (at a single-zone damage), or the first-last zone number combination (at a multi-zone damage). The first zone number is always zero.
• M: A number to indicate the number of PIAS compartments which are flooded in the damage case.
• N: The sequence number of this damage case.

## Select and edit damage cases

With the (sub-)compartment method the user has to specify all damage cases (where a damage case is a collection of one or more damaged compartments) which can occur within the vessel. That can be done with the present option, which is discussed in general at Input and edit damage cases. In addition, here at the probabilistic damage stability a number of additional columns are present in the text window:

• Aft: Aft boundary of the damage.
• Fore: Forward boundary of the damage.
• Upper: Upper boundary of the damage.
• Ainside: Inside boundary of the damage, at the aft boundary.
• Finside: Inside boundary of the damage, at the forward boundary.
• Fixed: If this column is set to ‘yes’, then the damage dimensions are not calculated automatically but the boundaries defined in the previous columns are taken. Fixing the boundaries is discouraged for normal use. The columns ‘Aft’ up to ‘Finside’ can only be edited by the user if the column ‘Fixed’ is answered with ‘yes’. The values presented in the columns ‘Aft’ up to ‘Finside’ can only be precise for each damage case which has been fully calculated, otherwise approximate values are displayed. If a new calculation is performed, the former exact determined values can sometimes change a bit. The listed values are used as the initial values for the new calculation of damage boundaries, and if the initial values for two calculations differ to some extent, the final boundaries may also slightly differ, plase also refer to Computing the damage boundaries.

Because the probability of survival is composed of the sum of the probabilities for all damage cases, all cases normally have to be included, so all ‘Slct’ values should be set to ‘yes’. In certain (investigation-)cases it could be handy to leave cases temporarily out of the calculation, in which case those cases can be set to ‘no’.

If the text cursor is placed over a damage case and <Enter> is pressed, a window appears where can be specified which compartments (as defined in Layout) are simultaneously flooded in the damage case under consideration. this is also described at Input and edit damage cases. In the context of the probabilistic damage stability an additional column is present, ‘valid for pi, vi and ri’, which is normally to be set at ‘yes’, which means the flooded compartment is taken into account for the determination of the probability of damage, pi, vi en ri. If this column is set to ‘no’, this compartment is not included in the probability of damage, however, it will obviously still be included as a flooded compartment. This might be the case when a compartment is not located in the damaged region but yet flooded anyway, for instance if it is connected to a damaged compartment by means of a pipe.

## Output of input data of damage cases

This option also contains a number of sub-options:

### Print permeabilities and selected damage cases

With this option lists of damage cases and of permeabilities assigned to the distinct types of spaces are being printed.

### Define sections for plots of zonal boundaries and damage cases

As preparation for the plots of damage cases, in this menu theirparticulars can be given, which is discussed into detail at Sketches of tanks, compartments and damage cases.

### Create plots of damage cases

With this option a plot will be created for each damage case, according to the layout as specified with the previous menu option. The sections contain the damaged compartments (in blue) as well as the damage (hatched in red). If, in case of the zonal method, it is specified that a probability triangle should be plotted, than it will contain the location and length of the damage (as red bar on the horizontal axis) and the corresponding probability triangle or trapezoid colored sand-yellow, see the example at the below.

Preview damage cases

### Create plot of zonal boundaries

This menu only appears in case of the zonal method. It will generate a single plot, which contains the compartments and the zonal boundaries, at the sections as specified with Define sections for plots of zonal boundaries and damage cases. See the example below.

Preview zonal boundaries

## Remove (parts of) saved information

### Remove all results of former calculations

As mentioned, in the module Re-use unmodified results of former calculations a mechanism is incorporated with which the results of unmodified formerly calculated damage cases are maintained. With this mechanism the processing time at repeating calculations with slight variations can strongly be reduced, because only those damage cases which have actually been modified need to be re-calculated fully. This module automatically keeps books of compartments, damage cases and intact particulars. A prerequisite for a correct bookkeeping is that the computer clock functions well. With the present option the user can remove all these accumulated results.

### Remove all complex intermediate stages of flooding

With this option all complex intermediate stages of flooding will be removed (see Complex stages of flooding (before 2021) for the ‘complex stages’ mechanism).

### Remove all damage cases with a non-positive probability of damage

Damage cases with a negative probability of damage will decrease the attained subdivision index A. With this option they can be removed. Please realize that as a consequence of this action the whole damage case constellation will change, and, with the (sub-)compartment method, consequently the whole damage case subtraction scheme, which might on its turn cause other damage cases to get negative probabilities.

### Remove all damage cases which do not contribute to "A"

Damage cases with a non-positive ai are removed. It is better not to use this option when in a later stage the VCG' could be decreased. Damage cases with zero probability of survival will be deleted with this option, and it might very well be that with a lower VCG' some of those damage cases would have a positive survival probability.

### Remove all damage cases

With this option all existing damage cases are simply removed.

## Execute and/or print calculations

### Execute and print the calculation

With this option the computation is executed, and printed. This might, notably with a large number of damage cases, take quite some time, which significantly be reduced with the use of multithreading, see Speed enhancing mechanisms in PIAS: PIAS/ES.

The progress will be shown in the Multithreading task monitor. Each thread shows the damage case it is calculating with the start-, and elapsed time and the status of the thread.

An example of the output of a dry cargo vessel is given in Appendix 4: output of an probabilistisc damage stability calculation for a dry cargo vessel, and that of an open hopper vessel (according to regulation dr-68) in Appendix 5: output of an probabilistisc damage stability calculation for a hopper dredger.

In this output pi is the probability of damage, so including the effect of reduction factor r (which accounts for transverse penetration) as well as reduction factor v (which accounts for horizontal watertight subdivision above the waterline). Probability of survival si, as included in the output, is what the name implies; the pure survival probability, based on properties of the residual GZ-curve. This is a slightly different grouping than applied in the regulations — where p×r is baptized “probability of flooding” and s×v “probability of survival” — however, that is a bit of an awkward formulation because it stirrs up the two types of probability. Fortunately, for the final results this grouping issue is irrelevant, because the contribution ai to the Attaineed Subdivision Index A will be p×r×v×s in all cases.

As mentioned earlier, in order to enhance execution speed as much as possible, the program keeps track of results of former calculations. Under the condition that the underlying ship data are not changed, these results can be re-used, entirely or partially, at subsequent computations. This mechanism can be the reason that where an initial calculation can take hours to complete, repetitive and identical calculations can be finished within minutes.

Furthermore, please observe that at the end of the table with damage cases, on the output, the sum of all probabilities of damage p is listed. This sum, Σp, represents the ‘total probability of damage when damaged’ and gives an indication whether the selected damage cases are correct and complete. There are three possibilities:

• Σp is exactly 1. In this case all possible damage cases are well covered by the selected damage cases, because the ‘total probability of damage when damaged’ should theoretically exactly be 1.
• Σp is smaller than 1. In this case the set of selected damage cases is incomplete, it does not represent all possible damage cases. This is not incorrect, but more damage cases could be defined, which could contribute to the attained subdivision index A.
• Σp is larger than 1. In this case overlapping or identical damage cases have been defined, which is incorrect.

Unfortunately, the practice is often stronger than the doctrine, will σp often not become exactly 1, also with a correct set of damage cases. So, in practice we are already satisfied if it is in the neighbourhood, e.g. between 0.9 and 1.13 — or 1.15, or 1.17, this limit is not strict.... This phenomenon applies in particular to the (sub-)compartment method, and is discussed in [2], [3] en [8].

Finally, it must be mentioned that besides probabilistic criteria the regulations could also contain deterministic damage stability criteria,. The computation of deterministic damage stability is not a task of this module, that can be performed with the separate dedicated PIAS modules (Damstab). For instance reg. 25-6.2 of SOLAS 1992 requires essentially that every fore peak damage must be survived with s=1 (that is, GZMAX ≥0.10 m, range ≥20° and θstatical ≤25°). That this rule is not verified is reported by the program by the message “Compliance with SOLAS reg. 25-6.2 has not been verified”.

### "Only execute" and "Print the complete calculation"

These options will speak for themselves. With the first computations are only executed, nothing is printed. With the second the available results of the last computation are printed (again).

### Print calculation results, subtotalized by zone

Complete calculation results can, because of their extent, be rather indigestible. It could be convenient to structure the results within zones. If with Define zonal boundaries zonal boundaries have been defined, then with the present option the output can be arranged and printed in zones, see Appendix 6: collecting the output in zones for an example. This restructuring within zones can be applied with the zonal method and the (sub-)compartment method. In principle the numerical integration method does not use the ‘damage boundary’ concept, and consequently it cannot be determined in which zone a damage case is located.

### Show intermediate calculation data

At the execution of the computations many intermediate results are generated. In order to provide the possibility to analyse or verify a certain computation, these intermediate results are saved. With this the intermediate results are shown. Only the intermediate data of the last calculation will be available via this option (the file will be re-written when a new calculation is performed). If the user desires, these intermediate results can be saved for future reference. Please note that if the option Re-use unmodified results of former calculations is used, this will be mentioned in the intermediate calculation data as statement instead of the full data for the corresponding damage case(s). When you require intermediate calculation data with all information, this option should not be used or be removed explicitly, see Remove all results of former calculations.

## File and backup management

Backups of the Probdam data can be made and restored here. Here is also the option ‘Quit module without saving the data’. See for the details Data storage and backups.

# Computing the damage boundaries

There is a direct relationship between compartment boundaries and damage boundaries, however, when applying the zone method the user will have to define the zonal boundaries manually, so no use will be made of the already available compartment geometry. When the compartment method (as weel as with the subcompartment method, see Calculation method probability of flooding for those methods), the relationship is evaluated, so that the damage boundaries can be automatically determined. For this prupose, a search algorithm is used that ‘plays’ with the damage boundaries so that:

• All compartments intended to be flooded are indeed struck bij that damage.
• All other (non-flooded) compartments are not struck.
• The damage is as large as possible.
• Required constraints are taken into account, such as the b1/b2 ratio as discussed in Application penetration constraint (b1,b2).

Incidentally, multiple damage bondaries may exist, which all lead to the flooding of the intended compartments, and all have more or less the same size. One of those is chosen, by “coincidence”. It can even occur that at one time a slightly different combination of damage boundaries is found than at another time, which might be caused by one of these reasons:

• By a slightly different set of start values for the search algorithm, which will lead to a slightly different result.
• Because PIAS in last instance may search the boundaries by means of a genetisch algoritme, which is a process where change plays a role.

This is not disturbing, both solutions can (and will) both be correct. Moreover, by application of this genetic algorithm it might occasionally occur that damage boundaries are sometimes found and sometimes not. Could that be made more consistent? Yes, but only against a serious increase in calculation times, which is a (too) high price to pay for this occasional effect in damage cases with a marginal effect.

# Warnings

During the calculation Windows might occassionaly report that the program is not responsive anymore. This message is incorrect and can be ignored, for its background please see Frequently asked questions. Sensible warnings may be included in the text file with intermediate results (.PD0 file), and have the following meanings:

• Warning: This damage case is redundant. Means that this damage case is multiply defined. The multiple definition of damage cases always gives incorrect results, due to incorrect subtraction of subdamages. Please be informed that the sequence of definition of damage cases, as well as their being ‘selected’, is irrelevant for the mechanism of subtraction of subdamages, so it is also irrelevant for this warning.
• Warning: This case contains incomplete subtracted subdamages. Means that subdamages are being subtracted which combined do not cover the entire damage. This might be correct, but special attention must be given to this case, because it may indicate that damage cases are missing.
• Warning: Damage impossible, pi, ri and vi are zero. Means that for this damage case no boundaries can be found (also see the remark at the ‘Generation of damage cases’ chapter). This damage case is left out in the calculation.
• Warning: The damage boundaries enclose also negative subcompartments. In combination with the current calculation method (a single damage per subcompartment) such damage boundaries can be expected to be correct, but it is not guaranteed. Please check. This message, which can only be given with the subcompartment method, indicates that negative subcompartments are damaged within the boundaries of the damage cases. Essentially that should not be problematic, but one should realize that negative subcompartments cannot be damaged on their own, their damage is only realistic where they overlap a positive subcompartment. So it is advised to verify the damage boundaries as found by the program.