Section 12.1 Principles of Transverse Stability (GM, GZ Curves)

Transverse stability refers to a vessel’s ability to resist heeling (listing or rolling) forces and to return to an upright position when such forces are removed. It is perhaps the most fundamental aspect of ship safety, as a loss of transverse stability can lead directly to capsize. The key parameters used to assess and quantify transverse stability are the Metacentric Height (GM) and the Righting Lever Arm (GZ) curve.

1. Basic Concepts and Definitions:

To understand transverse stability, several key points of reference within the ship’s geometry and weight distribution must be defined:

A. Centre of Buoyancy (B):

This is the geometric centre of the ship’s underwater volume (the volume of displaced water).

It is the point through which the total buoyant force is considered to act vertically upwards.

As the vessel heels, the shape of the underwater volume changes, and therefore the position of B shifts. For a stable vessel, B shifts towards the heeled (immersed) side.

The vertical position of B above the Keel (KB) and its longitudinal position (LCB) are found in the ship’s hydrostatic tables for any given draft.

B. Centre of Gravity (G):

This is the geometric centre of all the weights that make up the ship, including the hull, machinery, equipment, cargo, fuel, water, stores, and crew.

It is the point through which the total weight of the ship (which equals its displacement) is considered to act vertically downwards.

The position of G is determined by the distribution of weights onboard. Adding weight low down lowers G; adding weight high up raises G. Shifting weight vertically changes the Vertical Centre of Gravity (VCG or KG). Shifting weight transversely changes the Transverse Centre of Gravity (TCG), causing a list.

The KG value is a critical input for all stability calculations. It is calculated by taking moments of all weights about the keel.

C. Metacentre (M):

Transverse Metacentre (MT or simply M): When a vessel is heeled to a very small angle (typically up to about 7-10 degrees), the lines of action of the buoyant force (acting upwards through B) for successive small angles of heel intersect at a point called the transverse metacentre.

For small angles of heel, M can be considered a fixed point for a given displacement and waterplane shape.

The height of the Metacentre above the Keel (KM) is a geometric property of the hull and can be found from the ship’s hydrostatic tables for any given draft. KM is composed of KB (Keel to Centre of Buoyancy) + BMT (Centre of Buoyancy to Transverse Metacentre).

BMT (Transverse Metacentric Radius): This is a measure of the initial stability provided by the shape of the vessel’s waterplane. It is calculated as IT / V, where IT is the transverse moment of inertia of the waterplane area about its centerline, and V is the underwater volume of displacement. A wider waterplane area generally results in a larger BMT and thus a higher M.

2. Metacentric Height (GM) – The Initial Stability Indicator:

The Metacentric Height (GM) is the vertical distance between the Centre of Gravity (G) and the Transverse Metacentre (M).

GM = KM – KG

Significance of GM:

Positive GM (M above G): The vessel is initially stable and will tend to return to the upright position if heeled by a small external force. This is the normal, desired condition.

Negative GM (G above M): The vessel is initially unstable and will not return to the upright. It will either capsize or loll to one side to an angle where B shifts sufficiently far outboard to bring G vertically below B again (angle of loll). This is an extremely dangerous condition.

Zero GM (G coincides with M): The vessel is in neutral equilibrium and will remain at any small angle of heel to which it is disturbed.

GM as an Indicator of “Stiffness” or “Tenderness”:

Large Positive GM (“Stiff” Ship): The vessel has a strong tendency to return to upright. It will roll with a short, quick, and often uncomfortable motion. Excessive stiffness can put stress on the hull and cargo lashings.

Small Positive GM (“Tender” Ship): The vessel has a weaker tendency to return to upright. It will roll with a long, slow, and often more comfortable motion. However, very small GM values mean the vessel has less reserve stability to resist heeling forces.

Fluid Free Surface Effect (FSE):

If there are slack tanks onboard (tanks that are not completely full or empty, allowing the liquid within to move freely as the ship heels), the center of gravity of the liquid in the tank shifts towards the low side. This has the effect of reducing the vessel’s effective GM.

This reduction is called the Free Surface Correction (FSC), and the effective GM is the GMFluid (or GMCorrected) = GMSolid – Total FSC.

FSC for a single rectangular tank is calculated as: FSC = (i x ρ_liquid) / (Δ x ρ_sea) where i is the transverse moment of inertia of the free surface of the liquid in the tank, ρ_liquid is the density of the liquid in the tank, Δ is the ship’s displacement, and ρ_sea is the density of the seawater. For practical purposes, FSC (metres) = (l x b³ x ρ_liquid) / (12 x V x ρ_sea) where l and b are length and breadth of the free surface, and V is volume of displacement.

Loading instruments automatically calculate and apply FSE for all slack tanks. Minimizing the number of slack tanks, especially those with large surface areas (like wide double bottom tanks), is crucial for maintaining good stability.

IMO Minimum GM: The IMO Intact Stability Code specifies a minimum GM of 0.15 meters for most cargo ships in most conditions, after correction for free surface effect.

3. Righting Lever Arm (GZ) and the Curve of Statical Stability:

While GM is a good indicator of initial stability (at very small angles of heel), it does not describe the vessel’s stability at larger angles. For this, we use the Righting Lever Arm (GZ).

Righting Lever Arm (GZ):

When a vessel with positive initial stability is heeled to an angle (θ), the Centre of Buoyancy (B) shifts to a new position (B₁). The buoyant force acts upwards through B₁, and the weight of the ship acts downwards through G.

These two forces (buoyancy and weight, which are equal in magnitude) form a couple. The perpendicular distance between their lines of action is the Righting Lever Arm (GZ).

GZ = GM sin(θ) is a valid approximation for small angles of heel only (typically up to about 7-10°).

For larger angles, GZ must be calculated using more complex formulae (e.g., Wall Sided Formula, or more accurately from KN values obtained from the ship’s hydrostatic data – see below).

Righting Moment (RM): The moment that tends to return the ship to upright is RM = Displacement (Δ) x GZ.

Curve of Statical Stability (GZ Curve):

This is a graph plotting the GZ values (in metres or feet) against corresponding angles of heel (θ, in degrees) for a specific condition of loading (displacement and KG).

Key Features of a GZ Curve:

Initial Slope: The slope of the curve at 0° heel is determined by GM (specifically, tan(slope) ≈ GM in radians).

Angle of Maximum GZ (θmax): The angle at which the righting lever reaches its maximum value.

Maximum GZ Value (GZmax): The largest righting lever the vessel can generate.

Range of Stability (θvanish or θv): The angle at which GZ becomes zero again and the vessel loses all ability to return to upright (i.e., the angle of vanishing stability). Beyond this angle, the vessel will capsize.

Area Under the GZ Curve: Represents the vessel’s dynamic stability – its ability to absorb energy from external heeling forces (like wind or waves) without capsizing. The area up to a certain angle (e.g., 30°, 40°, or the angle of flooding) is often assessed against IMO criteria.

Calculating GZ using KN Values (Cross Curves of Stability):

Ship’s stability data usually provides KN values. KN is the righting lever calculated assuming the ship’s Centre of Gravity (G) is at the Keel (K).

To find the actual GZ for a given KG: GZ = KN – KG sin(θ)

This formula is used by loading instruments to generate the GZ curve for any given loading condition.

4. IMO Intact Stability Criteria (A.749(18) as amended, or 2008 IS Code MSC.267(85)):

The IMO sets minimum criteria for intact stability that most cargo ships, including bulk carriers, must satisfy. These criteria are assessed using the GZ curve for the specific loading condition. Key criteria typically include:

Area under the GZ curve:

Shall not be less than 0.055 metre-radians up to θ = 30° angle of heel.

Shall not be less than 0.09 metre-radians up to θ = 40° or the angle of downflooding (θf) if this angle is less than 40°.

Shall not be less than 0.03 metre-radians between the angles of heel of 30° and 40° or between 30° and θf if this angle is less than 40°.

The righting lever GZ:

Shall be at least 0.20 metres at an angle of heel equal to or greater than 30°.

The maximum righting lever GZmax:

Should occur at an angle of heel preferably exceeding 30° but not less than 25°.

The initial metacentric height GM:

Shall not be less than 0.15 metres (after free surface correction).

Specific Criteria for Certain Ship Types or Conditions:

For ships carrying timber deck cargoes, there are additional, more stringent criteria.

For ships engaged in grain carriage, the International Grain Code specifies its own stability criteria (as discussed in Chapter 8.3).

There are also criteria for severe wind and rolling (weather criterion).

Loading instruments will automatically check the calculated GZ curve against all applicable IMO criteria and indicate whether they are met.

5. Factors Affecting Transverse Stability:

Many factors can influence a vessel’s GM and GZ curve:

Vertical Distribution of Weight (KG): This is the most significant factor controlled by the ship’s staff.

Loading heavy cargo low in the ship (e.g., dense ores in lower holds or double bottoms) lowers KG, increasing GM and generally improving stability (making the ship “stiffer”).

Loading heavy cargo high (e.g., deck cargo, light cargo filling upper parts of holds) raises KG, decreasing GM and reducing stability (making the ship “tender”).

Consumption of fuel/water from low tanks raises KG; consumption from high tanks lowers KG.

Displacement (Δ): Changes in displacement affect KM (and KB, BMT). Generally, for a given KG, KM increases with draft up to a certain point.

Free Surface Effect: Slack tanks always reduce effective GM. This effect is more pronounced for wider tanks.

Beam of the Vessel: Wider vessels generally have a larger BMT and thus a larger initial GM for a given KG and KB (more form stability).

Freeboard: Higher freeboard generally leads to a larger range of stability and greater reserve buoyancy, delaying deck edge immersion and downflooding.

Superstructure and Deck Erections: Intact, weathertight superstructures can contribute to righting moments at very large angles of heel (after the main deck edge is immersed), extending the range of stability.

Suspended Weights: Lifting heavy weights with ship’s cranes suspends the weight’s center of gravity at the head of the derrick/crane jib, effectively raising the ship’s overall KG and reducing GM. This must be accounted for during lifting operations.

Icing: Accumulation of ice on decks and superstructures in cold weather adds weight high up, raising KG and reducing GM. It also increases windage area.

Water Trapped on Deck: Large quantities of water trapped on deck (e.g., due to blocked freeing ports) add weight high up and create a significant free surface effect, both detrimental to stability.

Cargo Shifting: A transverse shift of cargo directly causes a list and reduces effective stability.

Grounding / Bilging: These are damage stability scenarios (covered later), but they drastically alter stability.

6. Practical Stability Management by Ship’s Officers:

Accurate Calculation of KG: This is the foundation. Requires careful accounting of all weights and their VCGs.

Proficient Use of Loading Instrument: Essential for calculating GM, generating GZ curves, and checking compliance with IMO criteria for all loading conditions (departure, arrival, and any critical intermediate stages).

Minimizing Free Surface Effects: Keep the number of slack tanks to a minimum. If tanks must be slack, prioritize narrower tanks or press them up/draw them down completely as soon as operationally feasible.

Awareness of “Stiff” vs. “Tender” Conditions:

Aim for a GM that provides adequate stability without being excessively stiff (which can be uncomfortable and strain the ship/cargo) or too tender (which reduces safety margins). Experience and company guidance play a role here beyond just meeting minimum IMO GM.

The “feel” of the ship (its rolling period) can give an experienced mariner an indication of its stability condition. A very short, snappy roll indicates stiffness; a long, lazy roll indicates tenderness. (Approximate natural rolling period T ≈ (0.44 x Beam) / √GM for bilge keel ships, or other similar formulae depending on constants used).

Understanding the Full GZ Curve: Don’t rely solely on GM. The GZ curve provides a complete picture of stability across all angles of heel. A good range of stability and a healthy GZmax at a reasonable angle are just as important as initial GM.

Verification and Cross-Checks: Regularly verify the accuracy of the loading instrument against manual calculations for a sample condition or by comparing with approved test conditions.

Analysis for the Master (Transverse Stability): The Master has ultimate responsibility for the stability of the vessel at all times.

Ensuring Competence: Confirm that deck officers (especially the Chief Officer) are fully competent in stability theory, calculations, use of the loading instrument, and understanding of IMO criteria.

Review and Approval: Personally review and approve all stability calculations for departure and arrival conditions, and ensure critical intermediate stages during cargo/ballast operations are also checked.

Setting Standards: Foster a culture where stability calculations are performed diligently and conservatively.

Decision Making: Make informed decisions based on stability calculations, especially when planning non-standard loading conditions or when dealing with situations that might affect stability (e.g., heavy lifting, icing, water on deck).

Awareness of Limitations: Understand the limitations of theoretical calculations and the assumptions made (e.g., cargo VCG estimation). Always err on the side of safety.

A thorough understanding and diligent application of the principles of transverse stability are non-negotiable for the safe operation of a bulk carrier. The GM and GZ curve are not just numbers or graphs; they are vital indicators of the vessel’s ability to remain upright and survive the perils of the sea. The Master’s commitment to ensuring adequate stability at all times is a cornerstone of their professional duty.