If you look into any of the recent or past accident cases where a ship's hull developed a crack or entirely split into two, the striking factor will be the region of occurrence of this phenomenon. In all the cases, the cracks or split-offs have originated from the midships (i.e. 25% of the Length Overall from the midships). Rather than discussing much on why the cracks and split-offs developed, this article will discuss more on why do they develop in the midships region only? We will directly delve into the theory that governs the analysis of our question, and then you can read some rare and interesting articles and videos on case studies of such accidents, that I have recommended at the end of this article.
So, why midships only? Why don't hulls generally crack at the aft and for'd regions? Why didn't any ship ever split off from a region closer to the aft and for'd ends? Courtesy to Euler's Simple Beam Bending Theory. In this theory, the boundary conditions are generally fixed-fixed, simply supported, hinged-hinged or fixed at one end. Now compare your ship to one of those beams with any of the above boundary conditions. Which boundary condition do you think will fit this case? The answer is none. So how do Naval Architects use the Euler's Theory to analyse the structure of ships?
In every engineering problem related to structures, the key deciding factor of the analysis is the boundary condition that is to be determined before the analysis started. If the designer is not efficient enough to choose the most suitable boundary condition for a structural analysis, they design is often prone to be more of an engineering disaster. Naval Architects have been very careful in deciding the boundary conditions for a floating ship and over the years, designers have considered the hull to be a beam supported by an elastic foundation. In other words, the water provides support to the hull but in a continuously varying amount. In order to understand the variation of this support or reaction force on the hull (which is basically the buoyancy force) along the length of the ship, lets start with a sample hull form in Figure 1.
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Fig. 1: Hull of a RO-PAX ship.
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The upward reaction (buoyancy) exerted by the elastic foundation (water) longitudinally varies in magnitude. The nature of variation will vary according to the longitudinal distribution of the submerged volume of the hull. The buoyancy force will be more at the midships as the submerged volume in this region is larger and it gradually decreases at the aft and for'd ends, as the submerged volume reduces. This variation of submerged volume can be well visualized from Figure 2. As a result, the longitudinal distribution of the magnitude of reaction force (buoyancy) is somewhat as shown in Figure 3.
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| Fig. 2: The submerged portion of the hull (looking from under the keel) |
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| Fig. 3: Buoyancy per unit length - Buoyancy Curve |
The ship's hull is also subjected to the weight (acting vertically downwards) of components like main engine and machinery, propulsion system, superstructure, ballast, fluids (Lube oil, Fuel Oil, Fresh water, Bilge, etc), mooring and anchoring equipment, piping, cargo (distribution of cargo weight depends on what kind of a ship it is) and the hull's own steel weight. Some of these components are almost point weights (example: anchor weight, windlass weight) and most are distributed weights. All these weights per unit the individual lengths of their distribution are plotted to scale, with the magnitude on the vertical axis and the longitudinal position on the horizontal axis, and what is obtained, generally looks like Figure 4.
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| Fig. 4: Longitudinal distribution of weight per unit length - Weight Curve |
If we superimpose one graph on the other, and subtract the buoyancy from the weight at every single point, we will obtain the net load (often called only load) distribution along the entire hull. Since this is a continuous variation of weight and buoyancy, we use the fundamentals of calculus and express the load in the following way:
Total Load on Hull Girder (beam) = ∫w.dx - ∫w.dx
where,
- w = value of weight per unit length (from weight curve)
- b = value of buoyancy per unit length (from buoyancy curve)
- dx = length of infinitesimally small element of hull girder
What is interesting in case of ships, unlike most beams used in civil structures is that, the net load on the beam (here, the hull) is not always acting downwards. In the areas where the buoyancy exceeds the value of the weight, the net load on the hull is vertically upwards and vice versa. The load curve that is generally obtained is as shown in Figure 5.
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| Fig. 5: Load curve obtained from weight and buoyancy curve |
What you are going to know now, is a concept that a Naval Architect must never afford to forget in his entire career. The load curve of any ship is random and changes with almost every voyage. So we obviously cannot represent it by a particular function to obtain the Shear Force and Bending Moment Diagrams (recall your concepts from Strength of Materials). What we do, apply simple calculus knowing the significance of the following expressions:
Shear Force = ∫(w-b).dx
Bending Moment = ∫∫(w-b).dx.dx
In easier terms,
- Shear Force at a point is the area under the load curve up to that point from the aft end.
- Bending Moment at a point is the area under the Shear Force curve/diagram up to that point.
If the process is followed according to the two point above, the so called SF and BM diagrams of the ship is obtained for the given loading condition as shown in Figure 6 (remember, this is subject to change during its next voyage depending on whether it returns only on ballast or partially loaded).
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Fig. 6: SF and BM Diagrams of a ship (Remember, these graphs will be different for different loading conditions, but their natures always remain same. Always.)
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Now that we have obtained the bending moment diagram for the ship, its time to note a few of the most important aspects of structural design of a ship. As you read each point below, make sure you never let them out of your brains!
- The shear force in the hull girder is always zero at the aft, ford and midships.
- The bending moment in the hull girder is always maximum at the midships.
- Due to the the maximum bending moment occurring at midships, if we can design the hull with a longitudinal strength sufficient enough to sustain the bending moment at the midships, our design is safe! (even then a check is always conducted for every frame of the ship. Prevention as you see, is always better than cure!)
- The three above points will remain the same irrespective of the kind of loading on the ship.
All what we saw till now, was a mere application of Euler's beam bending theory in the way used by hull designers. Now, a ship's hull can bend in two ways depending upon the distribution of loading on the hull:
When the concentration of weight is more at the for'd and aft ends or when the crest of a passing wave is at the midships with the troughs at the for'd and aft ends (therefore providing more buoyancy at midships than at the ends), the ship is said to Hog, as shown in Figure 7. Similarly when more weight is concentrated at the midships or when the trough of a wave is at the midships and the crests at the ends of the ship (therefore more buoyancy is now being exerted on the for'd and aft ends), the ship is said to Sag.
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Fig. 8: Container ship "Fowairet" in a hogged condition due to grounding. Can you guess the position of the grounding impact on the ship, going by the fact that it has hogged?
(Courtesy: Google Images)
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Fig. 9: Oil Tanker "Prestige" that split off due to excessive sagging.
(Courtesy: Google Images) |
What actually happens within the hull girder due to the developed bending moment is that, a bending stress is developed at every transverse section of the hull. The universal expression for bending stress is :
Bending Stress = (Bending Moment)/(Section Modulus)
This brings us to the two most important observations that are kept in mind during the structural design of a ship's hull:
- Maximum Bending Moment at the midship means that the bending stress at the midships will be the maximum, and hence the deciding factor for the design.
- The bending stress at the midship is kept within safe limits by designing the midship section with a sufficient section modulus. (Again assuming you are thorough with the basics of Strength of Materials!)
So, if we look at a midship section, and study the bending stress on it due to the loading on the ship, we will be closer to the answer of the question we are looking for an answer to.
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Fig. 10: Bending Stress distribution at the midship section of a tanker in Sagging condition.
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Observe the above figure and tally them with the points which will follow:
- The neutral axis of the section is generally closer to the keel (due to more construction material at the bottom of the ship)
- Since the ship is sagging, the bottom plate is subjected to tension and the main deck plating is under compression (Visualize!). The opposite happens during hogging.
- The deck plate being further away from the neutral axis, compared to the bottom plate, experiences more magnitude of bending stress than the bottom plate. (Tally with the formula of bending stress!)
If the tension in the bottom plate in this condition exceeds the maximum tensile strength of the hull girder material (Mild Steel: Yield Strength = 490 MPa), the plate fails or in other word, the bottom plating cracks. Whereas, if the ship was exposed to a condition of hogging such that the tensile stress at the deck exceeded the ultimate strength of the deck plating and shear strake material, they fail or develop cracks. When the crack propagates through the side shell, the ship is prone to split off into two.
What you read, was just the entire theory behind why ships have generally been seen to crack or in some cases, split off due improper loading or inefficient design. Go back to the name of this article, and you will now be able to answer the question yourself! LSD
Article By: Soumya Chakraborty
Recommended Readings and Visuals of Accident Cases:
