Briefly exploring Damage Stability of Ships

As we already had a brief insight from one of our previous articles, 'Stability' is defined as the phenomenon of a ship to resist external or internal loads on it and to acquire its original upright state on removal of the external or internal loads. Stability is a crucial phenomenon governing ship design and seakeeping performance of a vessel.

Moreover they can be classified into two types: 
·   Intact stability

·  Damage stability

But how much are we sure of the fact that a vessel becomes unstable only when there is undamaged condition? In fact, the reverse is in often more likely, that is the ship has suffered a breach or damage and is structurally affected. This in turn has triggered flooding of water leading to loss in stability. Loss of stability can also be caused by other factors without involving damage, various causes being explained in the previous article. 

Figure 1: (Copyright: Wordpress)

DAMAGE STABILITY

This type of stability concerns with stability of the ship when it is damaged, usually hull is breached. It includes of flooding of ship compartments when hull is damaged leading to sinkage of the ship below margin line or even total sinkage or capsize of the ship. In this article we would focus mainly on damage stability.

     A ship gets damage or suffers a breach mainly because of one of the following causes or sometimes their combination:

•  Collision: This is a very common reason often leading to adverse effects. Collision may be with another vessel (remember Titanic?) or some landmass like harbor, port, reef or island. Most of them are caused by compounded operator error, carelessness, technical flaws, machinery and equipment failure, problems in maneuverability, accidents or sometimes unavoidable circumstances leading to damage.
• Grounding: Grounding is often caused by improper draft considerations in water bodies, excessive trim or in shallow draft conditions. 
• Structural problems: Sometimes there is lack of structural soundness due to manufacturing defects, improper behaviour, lack of maintenance, fatigue or unprecedented loading.
• Environmental Vagaries like rogue waves, cyclones, sea-storms, heavy rainfall or sometimes cold weather conditions leading to ice accretions. It may be worthwhile to mention that icebergs which are very much prevalent in northern seas are very big problems for navigation which can often lead to precarious collisions just like in case of Titanic.  

CLASSIFYING DAMAGE STABILITY

         So far we have introspected upon the causes that hampers the stability of the vessel after wreaking damage. But broadly, damage stability may be classified into two groups: 

•     Deterministic Damage Stability
•     Probabilistic Damage Stability

        Deterministic Damage Stability

   

      This is the traditional old-school technique for assessing stability of the ship when it is  flooded. In this process the ship is divided into several subdivisions along its length with the help of transverse watertight bulkheads. Now the stability of the ship is calculated when one or more compartments gets flooded due to breach of hull.

WAYS IN WHICH A SHIP SINKS

A ship can sink in usually 3 ways when its hull is damaged giving way for flooding. 

FOUNDERING

It is the case when a ship runs over some reef or rock and it damages its bottom keel and consequent flooding occurs. Due to flooding the ship’s draft increases to compensate the lost buoyancy and sinks when Weight > Buoyancy. One important thing is that foundering does not necessary lead to heeling of ship is the weight distribution is still properly maintained. However, due to flooding of the damaged compartments may lead to direct sinkage and sometimes trim. 

   

                                                                                  

                                                                            

Figure 2: Damage in hull due to Foundering (Courtesy: NEEC)

CAPSIZING 

This is the most common problem in regard to stability. However, it is treated differently in case of intact and damage stability. 
In intact stability, the preliminary condition is that the ship remains 'undamaged'. Now what could be the cause for a vessel to capsize without suffering physical damage? The answer is simple. As the loss of stability is solely caused by the loss of equilibrium of forces, the most occurrent cause is due to shifting of cargo or injudicious distribution of weights (cargo, ballast, machinery etc.) which triggers of the ship to heel to one side in a local phenomenon termed as Listing. However, in dealing with damage stability problems, we isolate the causes caused due to internal effects and merely concentrate on the damaged aftermath stability conditions. 
In damage stability criterion, capsizing is caused due to the breaching of hull after suffering damage sideways which can cause water flooding in that region only (tanks, cargo spaces etc.). This sudden flooding of water causes the ship to heel to one side accounting to its loss in equilibrium. 
As illustrated in the figure, unwanted flooding of a compartment or space leads to drastically altered buoyancy forces along with their lines of action. This difference in the line of action of the overall buoyancy of the damaged ship with respect to the weight still acting through its center of gravity creates a large heeling moment causing it to topple and finally capsize. As a result, it heels over large angle such that its righting moment is insufficient and it topples over. This happened in the disaster of ship COSTA CONCORDIA due to the collision.   
      

Figure 3: The capsizing of ship due to heeling moment

                                     

 PLUNGING

This depicts damage on a longitudinal basis. This aspect of damage generally deals with flooding in the fore and act regions. The causes can be in plenty from head-on collisions (like in a case of Titanic) to leakage in the hull skin. The unwanted seepage of water in the fore and aft is not as severe as capsizing. Moreover, with the development of watertight bulkheads dividing the hull into numerous watertight compartments from fore to aft, the risk is less posed (as even if the fore or the aft peak bulkhead got flooded, the remaining watertight compartments would remain safe due to their watertight integrity). Plunging only has a negative impact of trim (either by bow or stern). Trim is much more acceptable as compared to sideways heel when it comes to stability criterion but still can have detrimental consequences if deck immersion takes place, that is waterline reaches up to the weather deck and then flooding it. In the worst case scenarios, this creates a high trimming moment beyond revival making the ship succumb to its damage, sinking it by bow or stern. 

The following figures illustrate. 

                                                           

     

Figure 4: Gradual Capsizing due to trim 

EFFECT OF FLOODING ON STABILITY

We have so far expressed concerns about flooding which can affect stability. But have we thought about the simple physics that governs them?

When a ship floods its buoyancy is lost but displacement V remains same. So the draft increases to regain the lost buoyancy. As the draft increases the waterplane area (A_W) decreases. Thus 

              View  =  V_{old , the volume remaining the same}

                                    _{Aw(new)<Aw(old), accounting to the hull form}
                                        
                                           So,

                                            I_T(new) < I_T(old) , Area Moment of inertia of waterplane

                                        We know, from stability calculation relations BM= I_T /V .

                                        So BM_new < BM_old .

                                        GM = KB + BM – KG .

Figure 5 : Courtesy: NEEC

Now due to flooding KB will increase (due to increased draught) and BM will decrease (see figure). As a combined result GM decreases. Now as GM decreases the righting lever GZ  decreases and the stability of the ship as a whole,decreases (refer figure 5). 

Figure 6: Conditions of Transverse Statical Stability 

SUBDIVISION AND FLOODABLE LENGTH

In deterministic approach of damage stability, the ship is divided into several subdivisions. This is done to restrict flooding of the ship on hull damage.

Figure 7: Division into watertight bulkheads (Image Courtesy: NEEC)

Essentially in all ships, the hull is divided into a number of watertight compartments by the means of watertight bulkheads. The physics behind this development is intrinsically related to damage stability. Say, on a particular occasion, the ship hull gets breached. It will get flooded, of course. But with the subdivision into compartments, the risk of sinking is very much lowered. Even if flooding occurs, it is limited to one or a few bulkheads. On the contrary, if there were no bulkheads, the entire hull (Cargo holds, machinery spaces, engine room, accommodation etc) would have flooded beyond limits, making the ship the to sinkage.However, if damage takes place across numerous watertight compartments beyond the maximum limits of flooding (change in draft, of course), the ship may be in the precarious conditions of sinkage.Remember Titanic? It had grazed past the iceberg in an attempt to maneuver the past, inflicting damage to many of its compartments comprising for its watertightness, resulting in its deadly aftermath. Instead, if it would have directly rammed into the iceberg, the Fore peak tank and at most a couple of subsequent compartments would have flooded saving the rest. This could have prevented the catastrophe. 

Thus, invariably all modern designs maximize the number of watertight bulkheads to account for 'Damaged Stability.'

Here all subdivisions are done according to its ability to resist flooding in damaged conditions to the safest limits. Floodable Length is an important parameter taken into account here. It is defined as the maximum length of the compartment that can be flooded such that the draft of the ship remains below the margin line. Thus, maximum division of bulkheads is the best solution. But, other factors such as minumum required size of hold, improper cargo stowage, more number of required outfittings or increased steel weight hinders the possibility to some extent. Thus, optimizing the safe limits of floodable length to the minimum required length of watertight compartment are done in most cases. 

Figure 8: Watertight Subdivision of a ship (Courtesy: Wikipedia)

FLOODABLE LENGTH CURVE

Suppose the ship is divided into a certain number of  transverse bulkheads. Now each compartment has its floodable length. The lengths of the floodable lengths are plotted vertically from the midpoints of the horizontal floodable lengths and these points are joined together. 

Figure 9: Floodable Lengths (Image Courtesy: NEEC)

The end and starting points of the curve is joined with the aft and forward keel of the ship (profile view). This gives us the floodable length curve.

Figure 10: Floodable Length curve (Image Courtesy:NEEC)

At first for a vessel the factor of subdivision (FOS) is calculated. The FOS depends on the length of the ship and other numerical factors. The inverse of FOS gives the compartment standard of the ship. One compartment standard means the ship will survive if one compartment is flooded. The product of FOS and floodable length gives the permissible length. The length of a compartment cannot be greater than permissible length. The area under floodable length curve is the maximum extent a ship can be allowed flooding to prevent the ship to sink. For a single compartment, flooding triangles are formed from the edges of the bulkheads with the height equal to permissible length. This is the actual floodable length curve for single compartment flooding. This curve also gives the stability of the ship if one or multiple compartments are damaged.

Figure 12: Image Courtesy:NEEC

If for any single or group of compartments flooding the area of the actual floodable length curve exceeds the allowable floodable length curve then the ship will sink. 

Thus if the third bulkhead from aft of the ship damages the ship will sink since it’s floodable length curve exceeds the allowable floodable length curve.

Figure 11: Floodable Length estimates for a damaged hull profile (Image Courtesy: NEEC)

PROBABILISTIC DAMAGE STABILITY

In deterministic approach we had subdivision to restrict flooding and the sinkage of the ship if one or more compartments are flooded can be retrieved from floodable length curve. But in reality, we don’t know whether a ship will be damaged in a voyage or not. If it’s damaged how many compartments will be flooded? Will it be damaged by aft or forward? Will it plunge or capsize or founder? There is a lot of uncertainty involved. 

The theory of probability solves this problem. With the help of probability, we find,

• How ships are damaged?
• How often is a part damaged?
• What is the chance of survival if that part of ship is damaged?

We find the probability of each case and multiply the probabilities of each constraint. Finally, we add the total probability to get the chance of survival of the ship according to the damage occurred.

Figure 12: Sample Probabilistic  Tables

ATTAINED SUBDIVISION INDEX

Figure 13

Attained subdivision index is the sum of all probability of surviving for 1 compartment flooding, 2 compartment flooding, etc.

A = P(one compartment flooding) + P(2 compartment flooding) + etc.

A > R.

     

Where R = the required subdivision index.

Thus the damaged stability of vessels is calculated which decides the overall stability of the ship when it is damaged and tell whether the ship will survive sustaining the respective damage.LSD

Article By: Rijay Majee

ADDED MASSES IN SHIPS

A very typical feature associated with ships or any other floating bodies is that unlike any land borne object, it has to overcome the effect of the fluid it is floating in. Let us take a simple example. Suppose you are wading through knee-deep water on a waterlogged road after a spell of heavy rainshowers. Does it feel the same to walk the same stretch as compared any other normal dry day?

The answer is a big no. Not only you take more time to tread the same distance, but you also need somewhat extra effort to make your way. The same physics applies to vessels. Water has a finite value of density. Furthermore, when a vessel surges through the water, it creates a disturbance to the surrounding fluid. The fluid which already has some ‘flow’, thanks to its velocity potential gets an added acceleration triggered off by the ship motion.

Figure 1: Graphical simulation of the added mass effect condition of a vessel (Courtesy: http://www.scmdt.mmu.ac.uk/cmmfa/images/ship.jpg)

It is seen that when a vessel surges through water, it creates a boundary layer in the surrounding fluid which always is viscous. What is Boundary layer? A very critical aspect in fluid mechanics which ascribes the phenomenon of flow separation due to the motion of a body in a real fluid. A boundary layer is formed which demarcates the normal surrounding flow with the ‘flow affected by the body motion’. Thus the fluid content in this boundary layer gets influenced by the ship motion.

Figure 2: (Copyright: Learn Ship Design)

This accelerated fluid particles create forces on the surface of contact of the vessel!  As the ship motion has already triggered off a disruption to the potential flow of the fluid, the fluid particles possess some amount of kinetic energy. Thus apart from accelerating itself, it also has to expend some amount of extra ‘kinetic energy’ upon the surrounding fluid. This is realized in terms of propelling with some extra amount of mass which gives rise to the concept of added mass or virtual mass. Added masses arise simple from hydrodynamic considerations triggered mainly by waves and other external disturbances and have no correlation with the structure or propulsive parameters of the vessel.

Figure 3:  Representation of added mass due to surrounding fluid (Courtesy: Googleimages)

Though added mass is essentially a wave phenomenon, it depends on several factors. Now what does added mass of a body depend upon? 

• Displacement of the object: 

As well predicted, the added mass is a function of the mass displacement of the body under consideration.  The larger is the displacement, more is the added mass measured. Though it may be common to confuse it with buoyancy forces, it may be well noted that buoyant forces are static properties of a floating body which solely depend upon the geometry of the body and the fluid density. In other words, it is a hydrostatic effect. This has no relation with the added mass which is a hydrodynamic phenomenon limited to finite sized floating bodies surging in water with some acceleration. For symmetric bodies such as cylinder, cube etc., the added mass is mathematically the displaced volume times the density of the fluid. But a ship being a complex geometric object, the displaced water plus some extra amount of fluid in its wake is taken into account for determination of added mass.

• Motion of the body: 

·        A ship or any other body has some definite value of velocity and acceleration as well. This, in turn is reciprocated by the pressure field of the displaced velocity in terms of kinetic work. To put it simply, the more the velocity of the body, more is the reaction forces generated by the fluid.

•  Density of the Fluid:

·         A ship or any other body has some definite value of velocity and acceleration as well. This, in turn is reciprocated by the pressure field of the displaced velocity in terms of kinetic work. To put it simply, the more the velocity of the body, more is the reaction forces generated by the fluid.

• Hull Form:

·         As the ship has a complex geometry, a detailed analysis of the hull form exacts to the accurate determination of the added mass. The sections at each station from fore to aft and their interaction with the displaced fluid is studied in detail which gives the exact idea of the added mass. It has been observed that finer hull forms have a reduction in added mass. 

Figure 4: Finer and fuller hull form 

• Boundary Conditions: 

·         There is an interesting physics behind the behaviour of added mass properties in effect to the boundary conditions persistent. In shallow waters, the added mass of any floating body is seen to increase considerably. This effect is more pronounced when the vessel is floating through a restricted water body like channel or canal. The reason is quite simple. As the restrictions increase, the waves which are formed from the moving body is reflected back continually until damped. This increasingly high rate of wave incidence increases the added mass effect. Thus, it is worthwhile to say that a ship finds more difficulty in passing through a canal or channel as compared to the open sea.

Figure 5: Ship moving through channel (Courtesy: Wikipedia)

DETERMINING THE ADDED MASS

For symmetrical perfect objects like a solid sphere, cuboid, ellipsoid the added mass in normal conditions is the mathematical equivalent of its displacement. However, for complex shaped objects like a ship, the added mass determination may be a difficult task to achieve. Here the added mass comes out to be more than the mass displacement. Also the values are variant from ship to ship and from time to time. 

Figure  6: (Copyright : ADDED MASSES OF SHIP STRUCTURES)

Analogous to the variation in buoyancy when a ship encounters waves, the added mass varies from situation to situation. Thus it is valid to reason that there’s no fixed algorithm to determine the added mass. As the added mass is directly related to the kinetic energy of the surrounding fluid, it would involve terms related to energy components. 

In Newtonian terms, the drag, velocity and energy vartiation may be expressed as:

This ρ*I term is the measure of the mass equivalent related to the hydrodynamics of the disturbed fluid. 

The evaluation of this mass term is carried out using these classical techniques: 

• Analytical/Empirical Approach. (also including strip theory)

• Numerical Approach 

The methodology of these two approaches is a matter of scope beyond our discussion. But now in recent the development of softwares such as FEM packages, ABAQUS, NASTRAN and so on, the determination process has become easier. The above two methods include intricate mathematical deductions along with basic physical principles of boundary element method and infinite potential flow. 

Figure 7: 3D Modelling Patch(sample) for added mass calculation using FEM (Courtesy: Analytical and Numerical Computation of Added Mass in Ship Vibration Analysis)

·        IMPLICATIONS OF ADDED MASS

Now all of you must be wondering: why is the determination of added mass so important?

The answer lies in its applicability. Added mass is mainly aggravated by waves and other fluid inter actions in open water. And as the ship has to face them inevitably, added mass calculations must be taken in account. As added mass effects are virtually realized as ‘entrained’ mass, design considerations must be taken in to account in the following ways:

• ·      Resistance of the hull. Thus the propulsive characteristics are modified accordingly to meet the surplus power required for added mass counterbalance.
• ·       Structural design modifications. As obvious, due to the hydrodynamic forces generated in the form of added mass incident upon the hull, the structural reliability factor must be enhanced manifold in strength and load-bearing capacity.
• ·       Design of hull form. Based on the type of ship, the hull form is decided where the minimization of added mass is also given adherence.
• ·       Maneuvering Characteristics: Added mass also causes difficulty in manoeuvring; extra rudder forces and more time required for a change of heading. Thus during fabrication of basic control surfaces like rudders or stabilizers, the added mass effect is taken for a better estimate of the manoeuvring characteristics and thus modifying its design.
• ·      Cost estimation and economy: As we know in ships, the resultant profit is the final aim for all parties. Added mass consumes more fuel, expends more engine power and also increases the time of voyage in the long run. But in shipping industry, economy is hard money. Thus the estimation of excess fuel consumption, cargo-carrying safety limits, voyage charter timings and increased engine power required due to added mass is mandatory as to give a better idea of the net expenditure.LSD

   Article by : Subhodeep Ghosh