Density is the mass of a substance (expressed in kg) per unit of volume. The standardized unit of volume is the cubic metre (m3). The density unit is therefore kg/m3.

Density is a property that is unique to each type of matter. Liquids, solids and gases all have their own density.

A few examples of densities:

Pure water has a density of 1,000 kg/m3. This means that one cubic metre of pure water has a mass of 1,000 kg. Saltwater has a mean density considered to be 1,025 kg/m3. Steel has a density (depending on its composition) of about 7,430 kg/m3. If one metric tonne represents 1,000 kg, then 1 cubic metre of steel has a mass of 7.43 metric tonnes.

Bodies with a density lower than that of water will float, whereas the others will sink.

Density is expressed by the Greek letter ρ (rho).

Specific gravity is an expression that simplifies the use of density. Specific gravity is defined as the ratio of the density of a substance to the density of pure water (1,000 kg/m3).

The specific gravity of a substance is always the ρ of the substance divided by 1,000. Note that no units follow the transformation of density into specific gravity. The value of specific gravity is shown without units.

A few examples of the specific gravities of liquids:

Pure water = 1

Lubricating oil = 0.9 (can vary)

Fuel = 0.72 to 0.75

Average fuel = 0.86 to 0.92

Heavy fuel = 0.92 to 0.95

Fuel oil = 0.95 to 0.99

The mass of a body or a substance is a value that represents the quantity of matter of the body or substance. The mass will not change if the body is placed on the moon, the top of a mountain or at sea level.

The force exerted by this mass when it is subject to the Earth's gravitational attraction is the weight of the body or substance. Since gravitational attraction is variable (it is lower at higher altitudes than at sea level, the Moon's gravitational attraction is weaker than the Earth's, etc.), the weight of a body or substance will vary according to its location, but its mass will always remain the same.

In the International System of Units (SI), the force exerted on a mass that is subject to the Earth's gravitational attraction is expressed in a Newtons. The weight of a body or substance should thus be expressed in Newtons, which is rarely the case in practice. However, in theory, this distinction is very important.

The value of the weight of a body or substance can be found using the formula

F = m × a

The force (in Newtons) will be produced by a mass (in kilograms) that is subject to the acceleration created by the Earth's gravitational attraction (in m/s2), which is an average value of 9.81 m/s2 at sea level.

Displacement: Mass of the volume of water that a ship displaces. This mass is equal to the ship's mass. Displacement is expressed in tonnes. Symbol: Δ

Displacement volume or underwater volume: Volume of the underwater part of a ship. It is expressed in m3. Symbol: V

Draft: Depth of the underwater part of a ship. There is forward draft, aft draft and mean draft. It is expressed in metres or centimetres. Symbol: d

Deadweight: The mass that a ship can carry. This mass represents the cargo, fuel, water and everything required for proper operation of the ship. Specifically, cargo deadweight represents the mass of the cargo that can be loaded.

Deadweight is the total mass of goods that a ship can carry at its maximum permissible draft (including fuel, fresh water, gear, provisions, etc.)

Lightship displacement: Mass of a ship in light condition.

Loaded displacement: Mass of a fully loaded ship ready for sea. Loaded displacement equals lightship displacement plus deadweight.

Loaded displacement: Lightship + deadweight = loaded displacement

Waterplane area: Area at the intersection of the surface of the water and the waterline of a ship. It can vary according to the ship's draft. Symbol: Aw.

Amidships: Amidships is the midship section of a ship taken at its widest breadth. This is the reference for transverse stability calculations. It also allows you to visualize the transverse structural members of the hull.

Lightship weight: Real weight of an empty ship

Archimedes' principle: Any body immersed in a liquid is subject to an upward vertical force equal to the weight of the displaced mass of water.

With respect to stability, Archimedes' principle can be expressed as: the mass of a body immersed in a fluid is equal to the mass of the volume of fluid displaced.

When a ship is floating freely at rest, the mass of the ship (displacement, Δ) is equal to the mass of the volume of water displaced by the ship.

Coefficient of fineness of the waterplane area (Cw): This coefficient (variable according to the ship's draft) can be expressed as the ratio of the waterplane area to the area of a rectangle having the same length and breadth.

Block coefficient (Cb): Coefficient (variable according to the ship's draft) that represents the ratio of the underwater volume of a ship to a rectangular block having the same length, breadth and depth.

Tonnes per centimetre (TPC): This is the mass required to increase or decrease a ship's mean draft by 1 cm. This value varies only according to the waterplane area (Aw), and the waterplane area can vary according to the ship's draft. Therefore, the TPC can vary according to the ship's draft.

TPC = Tonnes per centimetre immersion TPI = Tonnes per inch immersion

If the specific gravity of the water in which the ship is floating changes without any changes to the ship's displacement, the ship's draft will change. The draft will change because the ship must displace the same mass of water, which no longer has the same density. Δ = × ρ

If the displacement (Δ) remains constant and the density (ρ) changes, the underwater volume () must change; therefore, the ship's draft will change automatically.

If a ship goes from fresh water to salt water, buoyancy will increase and the draft will decrease. Inversely, a ship that goes from salt water to fresh water will see its draft increase.

A ship that loads to its marks (summer load lines S) in salt water, will see its draft increase as it goes into fresh water and will exceed its marks. This situation is accepted because the absence of heavy weather on bodies of fresh water compensates for the ship's increased draft. This situation has even been made official by adding an additional mark (F) to the load lines.

Fresh Water Allowance (FWA): Inversely, a ship that loads in fresh water can load up to its "F" line, so that when it is in salt water it will float at its regular marks.

This allowed increase in draft is called the "Fresh Water Allowance". The FWA is therefore the change in draft when a ship goes from salt water to fresh water.

Transverse statical stability is the ship's stability at small angles of inclination. The rules of statical stability are considered to apply to angles of inclination less than 15°.

To be able to calculate stability, first the fixed and mobile points on a ship's cross section must be identified.

Reference point K is applied to the ship's lowest point, which is the keel. This is a fixed point.

Transverse centre of gravity G: Before applying this concept to a ship, a few short definitions of a general nature are necessary.

The centre of gravity of a body can be defined as:

- The point at which the force of gravity is exerted vertically downwards;

- The point where a pivot can be placed that will keep the body balanced;

- The geometric centre of a uniform body.

For a ship, the centre of gravity is the point at which the force generated by the ship's mass is exerted vertically downwards. The position of the centre of gravity changes according to the ship's loading conditions.

Effects of changing load: The height of the centre of gravity depends on the vertical distribution of the ship's mobile masses such as cargo, fuel and ballast). The height of the centre of gravity is measured from reference point K. The height of the centre of gravity is identified as a ship's KG.

For a ship to float without an angle of list, point G must be on the same vertical axis as K. As soon as G leaves this vertical axis, an angle of list is produced and the ship will no longer float upright.

The centre of gravity can thus be shifted vertically and horizontally by transferring, adding or removing mobile masses.

If mass is added to a ship, the centre of gravity shifts towards the position of the added mass. For example, fuel added to a ship's double bottom tanks will lower the centre of gravity. Deck cargo generally raises a ship's G. A concentration of cargo on a ship's port side will shift G to port, which will cause the ship to list to port.

The inverse is true when mobile masses are removed. For example, fuel consumption reduces the mass in the storage tanks, and if they are double bottom tanks, the loss of mass shifts G upwards. If deck cargo is unloaded, G is shifted downwards. Unloading cargo on the starboard side will shift G to port, which will result in a list to port.

If a mass already on board is moved, the position of the ship's centre of gravity will shift in the same direction as the mass. For example, shifting port ballast to starboard will shift G to starboard, which will tend to cause a list to starboard.

Transverse centre of buoyancy B: A ship's centre of buoyancy can be defined as the point through which the force of buoyancy acts vertically upwards. You can also say that the centre of buoyancy is the geometric centre of the ship's underwater volume. The height of the centre of buoyancy is measured from reference point K and is thus KB.

Ship's inclination: If a ship is floating upright, its symmetrical construction will place point B on the same axis as K and G. The only way to shift the transverse centre of buoyancy is to change the ship's inclination.

The underwater volume will be a different shape and B will shift to reach the new geometric centre of this underwater volume. Since the position of B depends solely on the geometry of the cross section, if the hull shape is known, it is easy to identify B based on the load and list conditions.

If a ship is floating upright, points K, G and B will all be on the same vertical axis. If the ship is inclined by external forces (wind, waves, tight mooring at the wharf), G should not change position (no mass has shifted), but B will shift to the geometric centre of the new underwater volume.

Points B and G will no longer be on the same vertical axis, consequently the ship's weight will act vertically downwards through G and the force of buoyancy will push upwards from B1.

Righting moment: When a ship is inclined, these two forces are no longer on the same vertical axis and a righting moment is created. The righting moment tends to bring the ship back to an upright position. This moment is equal to a force multiplied by a distance. The value of the force is the same for the upwards and downwards vectors, and is equal to the ship's displacement.

Righting lever (GZ): The distance between the two vectors is called GZ and represents the righting lever. The larger the righting lever, the higher the righting moment. The size of the righting lever increases with the ship's inclination. In other words, up to a certain angle of inclination (usually between 40° and 60°), the more the ship lists, the greater its tendency to return to an upright position. If the maximum righting angle is exceeded, the righting lever decreases and the ship's ability to right itself also decreases until it reaches an angle where the righting lever is zero and the ship is in serious danger of capsizing.

Inversely, if G is located high on the centreline, the righting lever will be smaller so the righting moment will be weaker. The ship will right itself more slowly.

The value of the righting moment (also called the moment of statical stability, MSS) is calculated by the formula MSS = Δ ×GZ

To find the value of GZ at small angles of inclination, the following trigonometric equation is used: GZ = GMsinΘ. Θ being the ship's angle of inclination.

Metacentre (M): Looking at the inclination diagram, you can see that a point M has appeared. Point M is located at the intersection of the buoyancy vector and the centreline and is called the metacentre.

For small angles of inclination (less than 15°), M is considered to be fixed. The presence of M allows us to introduce a new concept that actually controls stability at small angles of inclination.

Metacentric height (GM): This is the distance between G and M, which is identified as distance GM, also called the metacentric height.

The position of G in relation to M is crucial in a ship's ability to right itself. Under normal conditions, G should always be below M. The GM is then said to be positive. The greater the distance between these two points, the higher the positive GM. As stated in the previous paragraph, the larger the GM, the larger the righting lever. If G approaches M, the righting lever decreases and the righting moment is weak.

If GM is zero, meaning that G coincides with M, the righting lever is non-existent. If an external force then makes the ship heel to a small angle, the ship will remain heeled at this angle because there is no righting moment.

If GM is negative, meaning that G is above M, not only is the righting lever non-existent, but it also becomes a capsizing moment. If the ship is then subjected to a light external force, it will incline sharply and, depending on the shape of the hull, may even capsize completely. In any case, a negative GM is a situation that must absolutely be avoided.

Abrupt shifting of G: Two situations have a radical effect on the position of G. In both situations, an abrupt rise in G occurs, which in some extreme cases can lead to a situation where GM becomes negative. Both situations are a result of the free surface effect and the effect of suspended weight.

Suspended weight: When cargo is handled using cranes or cargo booms mounted on a ship, the centre of gravity of the mass being handled is considered to be at the point of suspension, which is the end of the crane arm or cargo boom. For example, if a crane lifts a mass of 5 tonnes from the bottom of a hold, as soon as the mass leaves the surface it was resting on, the centre of gravity of these 5 tonnes is instantly transferred from the bottom of the hold to the head of the crane arm. This causes an instant and sometimes significant rise in the ship's G. If the GM was already small, this change in position can result in a negative GM.

Free surface effect: The other situation is the occurrence of the free surface effect. If a ship's tank is partially filled and the ship rolls, the mass of liquid in the tank moves uncontrollably. The centre of gravity of the liquid mass shifts from side to side, and the change in "shape" of the liquid can also cause the G of the moving mass to rise radically. In addition, the inertia of the liquid mass moving around affects the ship's transverse stability and the position of its G. The effect of the inertia of the moving liquid is applied by making a virtual change to the position of G. This change in the height of G of the liquid mass can have a radical effect on the ship's height of G, which can result in a negative GM. To reduce the free surface effect, anti-rolling devices are placed in the tanks.

A combination of these two situations can occur when a ship is loading or unloading. Cargo handling is often combined with ballast handling. While in port, fuel or storage transfers can be done. Free surfaces can appear in ballast as well as fuel tanks. When this situation occurs during cargo handling with cranes or cargo booms, a negative GM can easily be created.

As an engineer, in some situations during a layover in port, you must check with the officer in charge of the ship's stability before transferring liquid masses.

Longitudinal stability: The principles of transverse stability apply partly to create longitudinal stability.

Points K, G, B and M from transverse stability are used with a ship's longitudinal section. They become points GL, BLand ML.

The longitudinal metacentre (ML) is found the same way as for transverse stability. It will be located at the intersection of the vertical lines passing through points BL and BL1 when the trim is adjusted.

In longitudinal stability, trim can be considered as the equivalent of list in transverse stability. Trim represents the longitudinal inclination of the ship and instead of being expressed in degrees, it is given as a difference between the forward and aft drafts.

When aft draft is greater than forward draft (usual situation), the trim is positive. When forward draft is greater, the trim is negative.

For example: a ship with a forward draft of 5 m and an aft draft of 5.75 m will have a positive trim of 75 cm.

Shifting, loading and unloading of a mass will affect a ship's trim. The change in trim when handling masses is measured using the concept of MCTC and MCTI.

MCTC is the Moment to Change Trim 1 centimetre.

MCTI is the Moment to Change Trim 1 inch.