Stability Basics: K, B, G, M, and GM
Every vessel stability question on the USCG exam builds on five reference points arranged vertically in the hull. Understanding what each point represents, where it is, and what moves it is the foundation of all stability analysis.
The lowest structural point of the hull. All vertical stability measurements are taken upward from K. KG = height of G above keel. KB = height of B above keel.
K is always zero — the baseline reference. You cannot move K. Every other measurement in stability (KB, KG, KM) is measured from K upward.
The geometric center of the underwater displaced volume. Buoyancy acts vertically upward through B. As the vessel heels, B shifts toward the low side because more hull volume is submerged there.
B moves with heel — toward the low side. This shift is what creates the righting force. KB is typically around 50-55% of draft for box-shaped hulls.
The point through which all vessel weight acts downward. G is determined entirely by the loading condition — where every weight is located. G does NOT move when the vessel heels; it is fixed by the loading.
G is the one point the captain can control. Lower G = more stable. G rises when weight is added high, G falls when weight is added low. Free surface effect raises G virtually.
The point where the vertical line through the shifted B (when heeled at small angles) intersects the vessel centerline. M is fixed by hull geometry and displacement — the captain cannot change it through loading.
M is above G in a stable vessel. M is approximately fixed for small angles (up to ~15 degrees). BM = I/V, where I is the waterplane moment of inertia and V is displacement volume.
The vertical distance from G to M. GM = KM - KG = KB + BM - KG. Positive GM: M above G, vessel is stable. Negative GM: G above M, vessel is unstable and will capsize.
GM > 0 is the minimum requirement for stability. High GM = stiff (snappy roll, uncomfortable, cargo stress). Low positive GM = tender (slow roll, may not meet criteria). Exam asks what raises or lowers G.
The Master GM Formula
Where:
- KB — height of center of buoyancy above keel (from hydrostatic tables at current draft)
- BM — metacentric radius = I/V (waterplane inertia / displaced volume; from tables)
- KG — height of center of gravity above keel (calculated from loading condition)
- KM — KB + BM (height of metacenter above keel; read directly from tables)
In practice:
- Read KM from hydrostatic tables at the current mean draft
- Calculate KG from the loading: sum of (weight x height) / total displacement
- Subtract total FSC (free surface corrections) to get effective GM
- Compare effective GM to minimum required GM in stability letter
Stiff vs Tender Vessels
A high positive GM creates a powerful righting moment at small angles of heel. The vessel snaps back to upright very quickly — a short roll period. While stable, this creates problems.
- + Excellent initial stability; hard to capsize
- + Returns quickly to upright after a wave
- - Very uncomfortable for crew and passengers
- - High accelerations can damage cargo, electronics
- - Structural stress on masts, rigging, superstructure
- - Crew fatigue reduces safety — counterproductive
A small positive GM means the vessel heels easily and returns slowly — a long, lazy roll period. Comfortable in calm water but potentially dangerous if GM approaches zero.
- + Comfortable ride for passengers and crew
- + Reduced structural loads from rolling
- - Marginal stability — small loading change can cause negative GM
- - May not meet minimum regulatory GM criteria
- - Free surface effect has a much larger proportional impact
- - In heavy weather, slow return to upright allows wave-on-wave heeling
The Righting Arm (GZ) Curve and Range of Stability
The GZ curve (statical stability curve) is the complete picture of a vessel's stability across all angles of heel from 0 to 180 degrees. While GM tells you about initial stability at small angles, the GZ curve shows what happens when the vessel heels significantly — whether it will survive a knockdown or capsize.
| Angle Range | Righting Arm GZ | What It Means |
|---|---|---|
| 0 degrees | GZ = 0 | Vessel upright. No righting moment needed. The curve starts here. |
| 5-15 degrees | GZ increases linearly | Initial stability region. Slope = GM / 57.3 (radians conversion). Higher GM = steeper slope. |
| ~30-40 degrees | Peak GZ (maximum righting arm) | IMO requires peak GZ at or beyond 30 degrees. Peak is the maximum righting moment the vessel can generate. |
| 40-60 degrees | GZ decreasing | Vessel still has positive righting arm but less than the peak. Still self-righting if heeled to this range. |
| AVS (angle of vanishing stability) | GZ = 0 | Curve crosses zero. Beyond this angle the vessel has no righting moment and will capsize. IMO requires AVS at 60+ degrees minimum. |
| Beyond AVS | GZ negative | Capsizing moment — vessel cannot self-right. Energy required to capsize is the area under the GZ curve from 0 to AVS. |
IMO Intact Stability Criteria (Reg. A.749)
Area Under GZ Curve
- 0 to 30 deg:Minimum 0.055 m·rad
- 0 to 40 deg:Minimum 0.090 m·rad
- 30 to 40 deg:Minimum 0.030 m·rad additional
GZ Values and Angles
- Peak GZ:Min 0.20 m at angle ≥ 30 degrees
- Peak angle:Should not be less than 25 degrees
- Initial GM:Minimum 0.15 m
Free Surface Effect and Free Surface Correction
Free surface effect (FSE) is one of the most frequently tested stability topics on the USCG exam. It describes the loss of effective GM caused by liquid sloshing in partially filled tanks — and it can be the difference between a stable vessel and a capsizing one.
How Free Surface Effect Works
When a vessel heels, liquid in a partially filled tank shifts to the low side. This shifting mass acts as if the vessel's center of gravity has risen — even though G physically stays where it is. We call this the "virtual rise in G" and quantify it as the free surface correction (FSC).
The effect is entirely about the freedom of liquid to move. A completely full tank has no free surface — the liquid cannot shift. A completely empty tank has no liquid to shift. A tank at any partial fill level has a free surface and will exhibit FSE. The worst case is approximately 50% fill, where the free surface is largest relative to the volume of liquid.
Free Surface Correction Formula
- i = second moment of area of free surface about its centroid (m&sup4;)
- ρL = density of tank liquid (kg/m³)
- V = volume of vessel displacement (m³)
- ρSW = density of seawater (1025 kg/m³)
In practice: FSC values for each tank at each fill level are tabulated in the vessel stability booklet. Sum FSC for all partially filled tanks and subtract from solid GM.
Why Tank Width Matters Enormously
The second moment of area (i) of a rectangular free surface is proportional to the cube of the tank breadth (i = L × b³ / 12). This means doubling a tank's width multiplies the free surface correction by 8. Wide tanks are dramatically worse than narrow tanks.
| Scenario | Effect on FSC | Exam Implication |
|---|---|---|
| Double tank breadth | Multiply FSC by 8 | Wide tanks are the biggest FSE danger |
| Add longitudinal centerline bulkhead (splits tank) | Reduce FSC by factor of 4 | Longitudinal subdivision is the most effective FSE mitigation |
| Fill tank completely | FSC = 0 | Full or empty: no free surface effect |
| Use heavy fuel oil vs. diesel | FSC increases with liquid density | Ballast water (higher density) has more FSE than diesel |
| Two tanks same total volume, same width — double length vs single | Same FSC (i proportional to length also) | Only breadth reduction helps; length does not |
List vs Angle of Loll: Causes, Distinctions, and Corrective Action
The distinction between list and angle of loll is one of the highest-priority stability topics on the USCG exam — and one of the most dangerous errors a captain can make operationally. The corrective actions are opposite, and applying the wrong correction to loll can cause immediate capsize.
| Aspect | LIST | ANGLE OF LOLL |
|---|---|---|
| Cause | Off-center G (transverse) — weight loaded asymmetrically, fuel burned from one side, cargo shifted | Negative GM — G has risen above M; buoyancy cannot create a righting moment at zero heel |
| GM Value | Positive GM (vessel is stable, just heeled) | Negative or zero GM (vessel is fundamentally unstable) |
| Behavior | Vessel heels to one side and stays there; rolls around that angle | Vessel falls to one side or the other; may alternate sides suddenly in a seaway |
| Danger Level | Moderate — vessel is stable but uncomfortable and at reduced stability margin | Severe — vessel may capsize with little warning, especially in a seaway |
| Correct Action | Shift weight toward the high side, or remove weight from the low side, to bring G back to centerline | Lower G immediately: add low ballast, flood double-bottom tanks, pump out high tanks, remove high weights |
| WRONG Action | Moving weight to the low side makes list worse | Moving weight to the HIGH side can cause sudden catastrophic capsize — the vessel lolls to the other side violently |
| How to Tell the Difference | Vessel rolls symmetrically around the heeled angle; GM calculations show positive GM | Vessel resists righting; rolls sluggishly; may alternate between port and starboard loll |
Correcting a List (Step by Step)
- Identify the cause: which weight is off-center?
- Shift weight from the low side toward the centerline or high side
- Alternatives: pump fuel/ballast to equalize port and starboard tanks
- Remove weight from the low side (offload to dock if needed)
- Avoid departure until list is corrected — regulations may prohibit it
The vessel is stable throughout this process. You have positive GM and are simply re-centering G.
Correcting Angle of Loll (Step by Step)
- Do NOT shift weight to the high side — the vessel will loll to the other side violently or capsize
- Flood double-bottom tanks or low ballast tanks to lower G
- Pump out high-position tanks (reduce high weight)
- Shift high deck weights to lowest available position
- Reduce free surface effect: top off or completely drain partially filled tanks
- Do this gradually and carefully — sudden flooding can make the situation worse
The vessel has negative or zero GM. The only fix is to lower G below M.
Worked Example: Recognizing Loll vs List on the Exam
Scenario A: A vessel has a list of 8 degrees to port. All fuel tanks are full. Passengers have been loading on the port side. GM calculations show GM = +0.35 m.
Diagnosis: This is a list. GM is positive. The cause is the off-center loading of passengers on the port side. Correct action: Move passengers to starboard, redistribute boarding to center the G.
Scenario B: A vessel heels 12 degrees to starboard in calm water. When a small wave hits, it momentarily heels to port and then back to starboard. The roll is very sluggish. Free surface corrections have not been applied to three partially filled tanks.
Diagnosis: This is angle of loll. The alternating behavior, sluggish roll, and uncorrected free surface effect creating negative GM are the tell-tale signs. Correct action: Lower G immediately. Fill the partially filled tanks completely or drain them. Add low ballast. Do NOT move weights to the high side.
Loading and Weight Distribution: Shifting, Adding, and Removing Weights
The exam frequently presents scenarios where a weight is shifted, added, or removed and asks for the effect on G, GM, trim, and/or draft. Mastering the cause-and-effect relationships between loading decisions and stability parameters is essential for both the exam and safe vessel operations.
Effect of Loading Changes on G and GM
| Loading Action | Effect on G | Effect on GM | Stability |
|---|---|---|---|
| Add ballast to double-bottom tanks | Lowers G | Increases GM | Better |
| Load heavy cargo in lowest hold | Lowers G | Increases GM | Better |
| Fill low-position fuel tanks | Lowers G | Increases GM | Better |
| Empty high-position tanks completely | Eliminates free surface, lowers G | Increases GM | Better |
| Load heavy gear on flybridge / top deck | Raises G | Decreases GM | Worse |
| Passengers on upper deck / crowding rail | Raises effective G | Decreases GM | Worse |
| Ice or water accumulation on deck | Raises G (progressive) | Decreases GM — dangerous | Worse |
| Partially fill tanks (~50%) | Free surface raises effective G | Decreases GM | Worse |
| Raise anchor chain from chain locker | Raises G while handling | Temporarily decreases GM | Worse |
| Remove heavy mast or equipment | Lowers G | Increases GM | Better |
Shifting a Weight (Already Aboard)
When weight is shifted (moved from one location to another aboard the same vessel):
- • Displacement does not change — total weight aboard is the same
- • Mean draft does not change — same displacement in same density water
- • G moves toward the new position of the weight
- • Transverse shift: G moves to port or starboard (affects list)
- • Vertical shift: G moves up or down (affects GM)
- • Longitudinal shift: G moves forward or aft (affects trim, not GM)
w = weight shifted, d = distance, W = displacement
Adding or Removing a Weight
When weight is added to or removed from the vessel:
- • Displacement changes — more or less total weight
- • Mean draft changes — vessel sinks deeper or rises
- • G moves toward the added weight (or away from removed weight)
- • KM changes with the new draft (from hydrostatic tables)
- • Net effect on GM depends on both the KG change and the KM change
W = original displacement, w = added weight, kg = height of added weight
Combined Effects on Draft, Trim, and GM
| Action | Mean Draft | Trim | GM | Stability |
|---|---|---|---|---|
| Shift weight from low to high (same ship) | No change | Depends on longitudinal shift | Decreases (G rises) | Worse |
| Shift weight from aft to forward | No change | Trims by head | No change | No transverse effect |
| Add weight to centerline at F | Increases (parallel sinkage) | No change | Depends on KG vs. current G | If added low: improves. High: worsens. |
| Remove topside weight | Decreases | Depends on longitudinal position | Increases (G lowers) | Improves |
| Fill double-bottom ballast tank | Increases | Depends on tank position | Increases (G lowers significantly) | Significantly improves |
| Consume fuel from low tank | Decreases | Depends on tank position | Decreases (G rises as low weight removed) | Worsens slightly |
Trim and Draft Calculations
Trim is the longitudinal inclination of the vessel — the difference between forward and aft draft. USCG exam questions on trim involve calculating draft changes when weights are added, removed, or shifted. The calculations use hydrostatic data from the stability booklet.
Trim Formula Reference
| Term | Formula | Notes |
|---|---|---|
| Trim | Trim = Aft Draft - Forward Draft | Positive trim = trimmed by stern (normal). Negative = trimmed by head (bow down). |
| Change in Trim | Delta T = (w x d) / MCTC | w = weight added/shifted (tonnes), d = distance from center of flotation F (m), MCTC from hydrostatic tables |
| Forward Draft Change | Delta F = Delta T x (l_F / LBP) | l_F = distance from F to forward draft mark. Positive if trimming by head. |
| Aft Draft Change | Delta A = Delta T x (l_A / LBP) | l_A = distance from F to aft draft mark. Positive if trimming by stern. |
| Mean Draft Change | Delta d = w / TPC | TPC = tonnes per centimeter immersion from hydrostatic tables. Pure parallel sinkage when weight added at F. |
| GM Formula | GM = KB + BM - KG | KM = KB + BM found in hydrostatic tables at given draft. KG calculated from loading condition. |
| Free Surface Correction | FSC = (i x rho_L) / (V x rho_SW) | i = second moment of area of free surface; rho = density; V = displacement volume. From stability booklet in practice. |
| Effective GM | GM_eff = GM_solid - FSC | Subtract total FSC for all partially filled tanks from the solid GM to get effective metacentric height. |
Worked Example: Trim Calculation
Given:
- Vessel displacement: 400 tonnes
- MCTC: 5.0 tonne-metres/cm
- LBP: 50 metres
- Distance from F to aft draft mark: 25 m
- Distance from F to forward draft mark: 25 m
- Weight added: 20 tonnes, 15 metres forward of F
- Original drafts: Fwd 2.50 m, Aft 2.80 m (trim = 0.30 m by stern)
Solution:
- Trimming moment = 20 t × 15 m = 300 t·m (forward of F = trims by head)
- Change in trim = 300 / 5.0 = 60 cm = 0.60 m by the head
- Aft draft change = 0.60 × (25/50) = +0.30 m (rises by 0.30 m since trim is by head at bow)
- Forward draft change = 0.60 × (25/50) = +0.30 m (sinks by 0.30 m)
- Mean draft increase = 20 / (TPC in tonnes/cm) — requires TPC from tables
- New trim = original trim − change in trim = 0.30 m − 0.60 m = −0.30 m (by head)
Final answer: The vessel is now trimmed 0.30 m by the head (bow is lower than stern).
Trimmed by the Stern (Normal)
Aft draft is greater than forward draft. Most vessels are designed to operate with slight stern trim for better steering and propeller efficiency. A small trim by the stern (0.3 to 1.0 m on most vessels) is considered normal and desirable.
Caused by: weight concentrated aft of F (engines, fuel tanks, stores aft), stern-heavy cargo distribution.
Trimmed by the Head (Bow Down)
Forward draft is greater than aft draft. Usually undesirable — reduces rudder effectiveness, increases risk of bow diving in head seas, and can make steering difficult. Requires corrective action before departure.
Caused by: heavy cargo forward, forward fuel tanks full / aft tanks empty, anchor chain piled forward.
Load Lines: Plimsoll Marks and Seasonal Zones
Load lines (Plimsoll marks) are the maximum permitted loading depths for different sea conditions and seasons. They are a legal requirement under the International Load Line Convention and domestic CFR regulations, and they directly encode reserve buoyancy requirements. The USCG exam tests your ability to identify each mark and know when it applies.
Deepest permissible mark. Tropical zone, fresh water. More load allowed because fresh water is less dense than salt.
Fresh water loading at standard (summer) conditions. Vessel floats deeper than in salt water — less buoyancy per unit volume.
Tropical zone, salt water. Slightly more load allowed than summer due to calmer average sea conditions in tropical zones.
Primary reference mark. Summer season in temperate zones, salt water. All other marks are derived from S.
Winter temperate zones. Less load allowed — freeboard is greater (hull rides higher) to provide reserve buoyancy in heavier seas.
Most restrictive mark. Applies to vessels under 100m LOA in the North Atlantic winter zone. Highest freeboard required.
How Load Lines Are Assigned and Displayed
Physical Location
- • Located amidships on both port and starboard sides of the hull
- • The circle with horizontal line = Summer load line (S mark)
- • Marks are welded or stamped into the hull, not painted alone
- • Letters on the circle identify the assigning authority (AB = ABS, etc.)
- • TF mark is lowest; WNA mark is highest (hull must ride highest)
Legal Requirements
- • Load Line Certificate must be aboard and valid
- • Vessel must not depart with applicable mark submerged
- • Applies when leaving port — allowable to submerge mark underway if fuel burns off
- • Master is personally responsible for compliance
- • Penalties for overloading: detention, fine, criminal liability
Deadweight, Displacement, and Reserve Buoyancy
Three fundamental measures of a vessel's capacity and seaworthiness. The exam tests whether you can distinguish between them and understand what each tells you about the vessel's condition.
Total weight of vessel + all aboard. Equals weight of water displaced. Unit: long tons or metric tonnes.
Displacement = volume x water density. Saltwater: 64 lb/ft3 or 1.025 t/m3. Fresh water: 62.4 lb/ft3 or 1.000 t/m3.
Weight of the empty vessel: hull, machinery, fixed equipment. No cargo, fuel, stores, passengers, or crew.
Established by inclining experiment. The starting point for all loading calculations.
Maximum weight of everything the vessel can carry: cargo, fuel, stores, fresh water, crew, passengers. DWT = Loaded displacement - Lightship displacement.
DWT is the vessel capacity number. Do not confuse with displacement. A vessel at maximum DWT is at load draft.
The buoyancy provided by the intact watertight hull above the waterline. Greater freeboard = more reserve buoyancy = greater ability to absorb flooding or wave action without sinking.
Load lines ensure minimum reserve buoyancy. Lower freeboard = less reserve = more danger in a seaway. Overloading eliminates reserve buoyancy.
The vertical distance from the waterline to the main deck edge amidships. Freeboard = depth - draft.
Freeboard and reserve buoyancy are directly related. The Plimsoll marks ensure minimum freeboard for each zone and season.
The vertical distance from the waterline to the lowest point of the keel. Forward draft and aft draft may differ (trim). Mean draft is the average.
Read from draft marks on the hull. Forward and aft draft marks are at the perpendiculars. Amidships marks check hogging/sagging.
Archimedes' Principle Applied
Salt Water
Density: 64 lb/ft³ or 1.025 t/m³
Standard for most load line and stability calculations
Fresh Water
Density: 62.4 lb/ft³ or 1.000 t/m³
Vessel floats deeper; load line F mark applies
Dock Water
Density between 1.000 and 1.025 t/m³
Rivers, harbors, estuaries; use DWA correction
Stability Letter and What It Certifies
The stability letter (or stability booklet with approval letter) is a regulatory document that certifies a vessel meets minimum stability standards for its class, route, and service. It is a required document for inspected vessels under 46 CFR and is frequently tested on captain license exams.
What the Stability Letter Contains
- • Maximum number of passengers
- • Required minimum GM or maximum KG for each loading condition
- • Tank loading requirements (which tanks must be full or empty)
- • Maximum allowable free surface corrections
- • Operational restrictions (route limitations, sea state limits)
- • Lightship KG established by inclining experiment
- • Maximum cargo weight and stowage restrictions
Inclining Experiment
The inclining experiment is the official method for determining a vessel's lightship KG. A known weight is shifted a measured distance transversely, and the resulting angle of heel is measured with a pendulum (or inclinometer). The formula:
From GM and the hydrostatic tables (which give KM at the test draft), KG = KM − GM. This lightship KG is the foundation of all subsequent stability calculations.
Passenger Vessel Stability: Transverse Stability and CPR
Passenger vessels face unique stability challenges because the weight and distribution of passengers can change rapidly and unpredictably. Regulations impose additional stability requirements beyond basic GM minimums.
CFR Title 46 requires minimum metacentric heights for uninspected and inspected passenger vessels. The stability letter specifies the minimum required GM for the vessel in the departure condition.
Inspected passenger vessels must demonstrate adequate transverse stability with all passengers shifted to one side. The vessel must not heel excessively under this condition. The stability letter specifies maximum passenger count.
The vessel must not heel more than a specified maximum angle (typically 14 degrees) under a steady wind heeling moment applied with all passengers on the high side.
The GZ curve must maintain positive righting arms to a minimum angle. IMO requires at least 15 degrees beyond the point of maximum GZ for most passenger vessels.
For certain passenger vessels, the ratio of righting energy to wind heeling energy must exceed a minimum value. This ensures enough dynamic stability exists to resist capsizing in beam-on wind and waves.
For vessels with weather decks open to flooding, stability calculations must account for trapped water on deck — free communication of water to the sea. This is critical for open-deck passenger vessels.
Standard Passenger Weight
USCG regulations use a standard weight per passenger for stability calculations:
- • 160 lbs (72.6 kg) per person — standard for most small passenger vessels
- • Some regulations use 185 lbs for compliance with newer CFR requirements
- • Height assumption: passenger center of gravity at approximately 3.9 ft (1.2 m) above deck
Crowding scenarios tested on the exam:
- • All passengers shifted to one rail: vessel must not heel excessively
- • All passengers on upper deck + wind: combined criterion
- • Rapid passenger boarding: temporary G shift during loading
Exam Strategy: Highest-Yield Stability Topics
These eight areas account for the majority of stability questions on the USCG captain license exam. Master them and you will handle any stability question the exam throws at you.
The exam heavily tests this distinction. Loll = negative GM, vessel unstable, lower G immediately. List = positive GM, off-center G, shift weight. Moving weight to the high side corrects list but can capsize a vessel with loll.
Any question about tank fill levels and stability: free surface effect is worst around half-full. Full tank = zero FSE. Empty tank = zero FSE. Half-full = maximum FSE. In a seaway, top off or drain tanks completely.
Stiff vessel (high GM): short roll period, uncomfortable, can cause structural damage to cargo, reduces crew effectiveness. Tender vessel (low positive GM): long roll period, comfortable, but marginal stability — any increase in G can create negative GM.
The exam may show a GZ curve and ask to identify: initial slope (related to GM), angle of maximum GZ, area under the curve (dynamic stability), and angle of vanishing stability (AVS). Know that a vessel with a larger area under the GZ curve requires more energy to capsize.
TF-F-T-S-W-WNA from deepest to shallowest. Fresh water is deepest (least dense, more volume displaced per ton). WNA is shallowest (worst conditions, most freeboard required). The exam asks which mark applies in which zone and season.
Adding weight forward of F trims by the head (bow down). Adding weight aft of F trims by the stern. Shifting weight forward trims by the head. When asked about trim change, always identify whether the moment is forward or aft of F first.
Shifting a weight already aboard changes G position but not displacement or mean draft. Adding a weight changes both displacement and G position. The exam distinguishes between these calculations.
The stability letter must be aboard inspected vessels. It specifies operating conditions, maximum passengers, loading limits. An operator who departs in conditions that do not meet the approved stability data is in violation even if the vessel has not capsized.
Frequently Asked Exam Questions
These questions represent the types of stability and loading scenarios that appear on the USCG captain license exam. Review them carefully — each tests a distinct concept.
What is the difference between list and angle of loll, and how do you correct each?+
What is metacentric height (GM) and why does it matter?+
How does free surface effect work and what is the free surface correction?+
What do the Plimsoll load line marks mean and where are they found?+
How do you calculate the effect of shifting a weight on trim and draft?+
What is the stability letter and what does it certify?+
What is the righting arm (GZ) curve and what does it show?+
What is the difference between deadweight, displacement, and reserve buoyancy?+
Ready to Test Your Stability Knowledge?
Practice hundreds of USCG-style stability and loading questions — including worked trim calculations, GM scenarios, list vs loll identification, and load line questions — on the Nail the Test exam simulator.