GM: why ships self-right¶

A ship tilted by a wave, a gust of wind, or a shifting load must develop a force that pushes it back upright. Whether that force exists — and how strong it is — reduces to a single number: the metacentric height, GM. This lesson unpacks what GM means physically, how the equation \(GM = KB + BM - KG\) is derived from first principles, and how perform_analysis in src/analysis.c computes and judges it.

The mental model 🧠¶

GM is the length of the lever that fights to stand the ship back up. When the hull heels, gravity pulling down through G and buoyancy pushing up through the shifted B no longer line up — they are offset by a horizontal gap, and that gap times the ship's weight is a righting moment that twists the vessel upright. A tall GM means a stiff ship that snaps back hard; a small GM means a tender ship that lolls and rolls slowly.

It all rides on one subtraction: GM = KB + BM − KG. KB + BM is how high the metacentre M sits — fixed by the hull and how deep it floats. KG is how high the centre of gravity sits — fixed by how you load. As long as M stays above G the ship self-rights. Stack cargo too high and KG climbs until it passes M: GM goes negative, the lever now pushes the wrong way, and the ship capsizes. That sign flip is the line between a voyage and a casualty, which is why every IMO check in CargoForge keys off r.gm.

What this actually means (plain English)¶

No jargon — here's what the ideas in this lesson actually mean, and why they matter.

GM (metacentric height) = "a single number that tells you whether a tilted ship will push itself back upright or keep tipping over" — if r.gm (computed in perform_analysis) is positive, the restoring force wins; if it is zero or negative, the ship is in trouble.
KB, BM, KG = "three heights measured from the ship's keel, whose difference gives you GM" — KB is where buoyancy acts, KG is where weight acts, and BM is the geometry bonus from hull width; perform_analysis computes each one before adding them with r.gm = r.kb + r.bm - r.kg.
Righting lever GZ = "the horizontal gap between where buoyancy pushes up and where gravity pulls down" — the larger this gap at a given heel angle, the harder the ship fights to come back; gz_at_angle computes it for the full GZ curve beyond small angles.
Free-surface correction = "a penalty on GM because sloshing liquid in a half-full tank acts like raising the centre of gravity" — calculate_virtual_kg_rise sums this penalty across all partially filled tanks and subtracts it to give gm_corrected, the value every subsequent check uses.
Too tender / too stiff = "two opposite loading mistakes" — a tender ship (low GM, below 0.3 m) rolls slowly but has almost no reserve against capsize; a stiff ship (GM above 3.0 m) snaps back so violently it damages cargo and crew; print_loading_plan flags both conditions by name.
IMO criteria = "six regulatory minimums a loaded ship must clear before sailing" — perform_analysis sets imo_compliant = 1 only when gm_corrected and the GZ curve areas all exceed every threshold simultaneously; failing even one means the loading plan must change.

Why it matters: A loading plan that looks fine on paper can produce a negative GM when heavy cargo is stowed too high or tanks are left half-full — the ship becomes a capsizing risk before it leaves port. perform_analysis exists precisely to catch this before it happens.

The three heights¶

Every stability calculation pivots on three vertical distances, all measured from the keel (the bottom of the hull) upward.

Symbol	Name	What it marks
\(K\)	Keel	Reference origin — always 0
\(B\)	Centre of Buoyancy	Centroid of the submerged volume; pushes upward
\(G\)	Centre of Gravity	Centroid of the entire ship weight; pulls downward
\(M\)	Transverse Metacentre	The point about which the ship rotates at small angles

So:

\(KB\) — how high the buoyancy force acts
\(KG\) — how high the weight force acts
\(KM = KB + BM\) — height of the metacentre above the keel

And:

\[GM = KB + BM - KG = KM - KG\]

waterline

GZ

θ

K B G M

buoyancy ↑ weight ↓

M is above G, so the couple rights the ship. GM = GZ / sin θ > 0

Why GM tells you whether the ship self-rights¶

Imagine tilting the ship slightly to starboard by angle \(\theta\). The submerged volume shifts toward the low side, moving \(B\) to starboard. The weight still acts through \(G\) on the centreline. These two forces — buoyancy up through the new \(B\), gravity down through \(G\) — form a couple.

The horizontal separation between where those forces act is called the righting lever \(GZ\):

\[GZ(\theta) \approx GM \cdot \sin\theta \quad \text{(for small angles)}\]

If \(GM > 0\): \(G\) is below \(M\). The couple acts to push the ship back upright. The ship is stable.
If \(GM = 0\): no restoring force at small angles. Neutral — marginally dangerous.
If \(GM < 0\): \(G\) is above \(M\). The couple acts to push the ship further over. Capsizing.

Negative GM

A ship with \(GM < 0\) will not return to upright after a small disturbance. It will find a new equilibrium (the "loll angle") or capsize. Causes include top-heavy loading, free liquid in tanks, or ice accumulation on topsides.

How the code computes it¶

perform_analysis in src/analysis.c works in two modes depending on whether a hydrostatic table CSV has been loaded.

Box-hull fallback (no CSV table)¶

When ship->hydro is NULL, the code uses analytic formulas for a rectangular barge:

/* from src/analysis.c — box-hull path */
r.draft = displaced_vol / (ship->length * ship->width * BLOCK_COEFF);
r.kb = KB_FACTOR * r.draft;

float inertia_t = (ship->length * powf(ship->width, 3) / 12.0f) * WATERPLANE_COEFF;
r.bm = inertia_t / displaced_vol;

The constants are defined at the top of the file:

#define BLOCK_COEFF      0.75f   /* Cb: underwater hull vol / bounding box */
#define WATERPLANE_COEFF 0.85f   /* Cw: waterplane area / L*B              */
#define KB_FACTOR        0.53f   /* KB ~ 0.53*T for typical cargo ships     */

Why these values?

BLOCK_COEFF = 0.75 means the hull displaces 75% of its bounding box volume — typical for a loaded cargo vessel (a box barge = 1.0, a streamlined yacht ≈ 0.35).
KB_FACTOR = 0.53 is the empirical rule-of-thumb \(KB \approx 0.53T\) valid for cargo-ship hull forms; the true value depends on the hull's underwater shape.
WATERPLANE_COEFF = 0.85 scales the waterplane area below a full rectangle, giving a more realistic second moment of area \(I_T\).

\(BM\) comes from the transverse second moment of the waterplane area divided by the displaced volume:

\[BM = \frac{I_T}{\nabla} = \frac{C_w \cdot L \cdot B^3 / 12}{\nabla}\]

This is exact for a wall-sided hull and a good approximation for ships of moderate form.

Table-based hydrostatics (CSV loaded)¶

When a hydrostatic table is present (ship->hydro->loaded == 1), the code interpolates \(KB\) and \(BM\) directly from the table at the computed draft — these values already account for the ship's actual hull shape:

/* from src/analysis.c — table-based path */
r.draft = hydro_draft_from_displacement(ship->hydro, displacement_t);

HydroEntry he;
hydro_interpolate(ship->hydro, r.draft, &he);

r.kb = he.kb;
r.bm = he.bm;

KG — the centre of gravity¶

\(KG\) is computed in the single-pass cargo accumulation loop by summing vertical weight moments, then dividing:

/* from src/analysis.c — KG computation */
float vertical_moment = ship->lightship_weight * ship->lightship_kg;

for (int i = 0; i < ship->cargo_count; i++) {
    const Cargo *c = &ship->cargo[i];
    if (c->pos_x < 0) continue;           /* skip unplaced items */

    float cz = c->pos_z + c->dimensions[2] / 2.0f;   /* centroid height */
    vertical_moment += c->weight * cz;
}

r.kg = vertical_moment / displacement_kg;

Each item's vertical centroid is its pos_z (base of the item, from the hull origin) plus half its height. Unplaced items (pos_x < 0) are excluded — they are not yet aboard.

Assembling GM¶

With \(KB\), \(BM\), and \(KG\) known, the formula is a single line in src/analysis.c:

/* GM before free surface correction */
r.gm = r.kb + r.bm - r.kg;

Free-surface correction¶

Liquid in a partially filled tank sloshes when the ship heels. This shifts the effective centre of gravity upward — a virtual rise in \(KG\) — reducing \(GM\). The correction is calculated from the tank geometry:

\[\text{FSM} = \rho \cdot \frac{l \cdot b^3}{12} \qquad \text{(per tank)}\]

\[\text{FSC} = \frac{\sum \text{FSM}}{\Delta} \qquad \text{(virtual KG rise, metres)}\]

where \(l\) and \(b\) are the tank's length and breadth, \(\rho\) is the liquid density, and \(\Delta\) is the ship's displacement in tonnes. Empty and 100%-full tanks contribute zero (no free surface).

In src/analysis.c:

r.free_surface_correction = 0.0f;
if (ship->tanks && ship->tanks->count > 0) {
    r.free_surface_correction = calculate_virtual_kg_rise(ship->tanks, displacement_t);
}
r.gm_corrected = r.gm - r.free_surface_correction;

Every subsequent calculation — heel, GZ curve, IMO criteria — uses gm_corrected, not the raw gm. This is why the output report prints both:

  GM (transverse)       : 1.43 m
  Free surface corr     : -0.089 m
  GM (corrected)        : 1.34 m

Always correct for free surface

Forgetting the FSC is a real-world loading error. Ballast tanks are typically kept either empty or full — a half-filled double-bottom tank can reduce GM by 0.1–0.3 m, which can be the margin between pass and fail on IMO criteria.

The GZ curve — stability beyond small angles¶

\(GM\) is only a linearisation valid for small heels (roughly below 10–15°). At larger angles, the righting lever \(GZ(\theta)\) is computed via the wall-sided formula:

\[GZ(\theta) = \sin\theta \cdot \left[ GM + \frac{BM}{2} \tan^2\theta \right]\]

/* from src/analysis.c */
static float gz_at_angle(float gm, float bm, float theta_deg) {
    float theta = theta_deg * (float)M_PI / 180.0f;
    float tan_theta = tanf(theta);
    return sinf(theta) * (gm + bm * tan_theta * tan_theta / 2.0f);
}

The \(BM \tan^2\theta / 2\) term accounts for the increasing waterplane inertia effect as the ship heels — this is what makes the GZ curve rise above the small-angle linear approximation at moderate angles.

IMO criteria: the regulatory floor¶

International regulations (IMO Resolution MSC.267/85, Part A, Chapter 2.2) define six minimum thresholds that every cargo vessel must meet. CargoForge-C checks all six using gm_corrected:

/* from src/analysis.c */
r.imo_compliant = (
    gm_effective   >= IMO_GM_MIN &&        /* >= 0.15 m */
    r.gz_at_30     >= IMO_GZ_AT_30_MIN &&  /* >= 0.20 m */
    r.gz_max_angle >= IMO_GZ_MAX_ANGLE &&  /* >= 25 deg  */
    r.area_0_30    >= IMO_AREA_0_30_MIN && /* >= 0.055 m·rad */
    r.area_0_40    >= IMO_AREA_0_40_MIN && /* >= 0.090 m·rad */
    r.area_30_40   >= IMO_AREA_30_40_MIN   /* >= 0.030 m·rad */
) ? 1 : 0;

Criterion	Threshold	Why it matters
\(GM \geq 0.15\) m	`IMO_GM_MIN`	Minimum self-righting ability at small angles
\(GZ_{30°} \geq 0.20\) m	`IMO_GZ_AT_30_MIN`	Adequate lever at a realistic wave-heel angle
Angle of max GZ \(\geq 25°\)	`IMO_GZ_MAX_ANGLE`	Righting arm must peak late enough to resist large rolls
Area 0–30° \(\geq 0.055\) m·rad	`IMO_AREA_0_30_MIN`	Energy absorbed in early heeling
Area 0–40° \(\geq 0.090\) m·rad	`IMO_AREA_0_40_MIN`	Total energy reserve up to steep heel
Area 30–40° \(\geq 0.030\) m·rad	`IMO_AREA_30_40_MIN`	Reserve between 30° and 40° — guards against "dead ship" scenarios

The area criteria capture energy, not just force: a wider, flatter GZ curve survives a wave better than a tall, narrow one with the same peak.

What "too tender" and "too stiff" mean¶

print_loading_plan in src/analysis.c classifies the corrected GM for the human-readable report:

if (gm_display < 0.3f)       gm_str = "CRITICAL - Too tender";
else if (gm_display > 3.0f)  gm_str = "WARNING - Too stiff";
else if (gm_display >= 0.5f && gm_display <= 2.5f) gm_str = "Optimal";
else                          gm_str = "Acceptable";

Too tender (\(GM < 0.3\) m): slow, wallowing roll with little reserve. A heavy sea could capsize the ship.
Optimal (\(0.5 \leq GM \leq 2.5\) m): comfortable roll period, good reserve.
Too stiff (\(GM > 3.0\) m): short, violent snap-roll. Structurally hard on cargo, crew, and fastenings — and counterintuitively dangerous if a high sea coincides with the ship's roll period (resonance).

Stiffness is often a bigger practical concern for cargo operators than tenderness: heavy cargo concentrated low makes a stiff ship, which produces acceleration loads that damage cargo and crew.

Recap¶

\(GM = KB + BM - KG\): height of the metacentre above the keel, minus height of the centre of gravity.
\(GM > 0\) means the ship self-rights; \(GM \leq 0\) means it does not.
perform_analysis computes \(KB\) and \(BM\) either analytically (box-hull constants) or by interpolating a hydrostatic CSV table, and \(KG\) from a single-pass weight-moment sum over all placed cargo.
The free-surface correction reduces \(GM\) by a virtual KG rise from sloshing liquid; all IMO criteria use the corrected value gm_corrected.
The GZ curve extends stability assessment beyond small angles using the wall-sided formula; gz_at_angle implements it in four lines of C.
IMO MSC.267/85 sets six thresholds; imo_compliant = 1 only when all six pass simultaneously.

Check yourself¶

If GM is exactly zero, is the ship already unstable?

It's right on the boundary — no restoring force develops at all, so any small disturbance can tip it further with nothing pushing back. Strictly negative GM is where it actively capsizes without help.

Loading cargo higher raises KG. What happens to GM, and by how much?

GM shrinks one-for-one with KG — every extra metre KG rises subtracts a metre from GM = KB + BM - KG, since KB and BM are hull-geometry properties that don't change just because cargo moved higher.

Next: Trim and heel.