Adjust hull dimensions, mass, and center of gravity to compute metacentric height GM and righting lever GZ in real time. Find the boundary between stable and capsizing — interactively.
Hull Parameters
Length L (m)
m
Beam B (m)
m
Depth D (m)
m
Displacement m (ton)
ton
Fluid density ρ_f (kg/m³)
kg/m³
Center of gravity KG (m)
m
Heel angle θ (°)
°
● Stable
Results
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Draft d (m)
—
KB (m)
—
BM (m)
—
GM (m)
Ship
Use the heel-angle slider to tilt the hull / open the GZ Curve tab to inspect the righting lever.
What exactly is "metacentric height" and why is it so important for ships? I see it's a key output in the simulator.
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Basically, the metacentric height (GM) is a single number that tells you how stable a floating object is against tipping over. Think of it as the "restoring force lever arm" when the ship heels. A positive GM means it's stable and will try to return upright. In the simulator, you can see GM change in real time as you adjust the Center of Gravity (KG) slider.
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Wait, really? So if I lower the KG by moving the slider down, the GM gets bigger? Does that always mean more stability?
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Exactly! Lowering the center of gravity (KG) increases GM, which generally means a stronger righting moment. But it's a trade-off. For instance, a very high GM (like on a heavy, shallow barge) makes a ship very "stiff"—it snaps back to upright so quickly it can be uncomfortable and cause high stresses. Try it: set KG very low and then use the heel angle slider. You'll see the righting lever (GZ) grow rapidly.
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Okay, so GM is the initial stability. But the simulator also draws a whole "GZ Curve". What's that for?
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Great question! The GZ curve shows the full story. GM only predicts stability at small angles. The curve plots the righting lever (GZ) against the heel angle. A common case is a cruise ship: it might have a moderate GM, but its wide hull shape creates a GZ curve that remains positive up to 60-70 degrees, meaning it can survive a severe storm. Change the Beam (B) parameter and watch how the curve's shape and maximum value change dramatically.
Physical Model & Key Equations
The simulator calculates how deep the hull sinks (draft) based on Archimedes' principle: the weight of the displaced water equals the ship's weight.
$$d = \frac{m \times 10^3}{\rho_f \cdot L \cdot B}$$
Where d is draft (m), m is displacement (ton), ρ_f is fluid density (kg/m³), and L & B are hull length and beam (m). The factor $10^3$ converts tons to kilograms.
The core of stability analysis is finding the Metacentric Height (GM). It's the vertical distance between the Center of Gravity (G) and the Metacenter (M), a theoretical pivot point for small angles of heel.
$$GM = KB + BM - KG$$
KB is the height of the Center of Buoyancy (approx. draft/2). BM is the Metacentric Radius ($I / V$), where $I$ is the waterplane's moment of inertia and $V$ is the displaced volume. KG is the Center of Gravity height you control. A positive GM means stability. The Righting Lever GZ ≈ GM sin θ is the horizontal distance creating the restoring moment.
Frequently Asked Questions
A negative GM indicates that the center of gravity (G) is above the metacenter (M), meaning the ship is unstable. If it tilts slightly, no restoring force acts, and it will capsize. In this simulator, try lowering the center of gravity or increasing the ship's beam to make GM positive.
The horizontal axis is the heel angle (degrees), and the vertical axis is GZ (righting lever length). In the range where GZ is positive, the ship tends to return to its upright position; when it becomes negative, the ship capsizes. The higher the maximum GZ and the larger the angle at which GZ becomes zero (angle of vanishing stability), the more stable the ship is considered.
Even if you change the displacement (tons), unless you also change the ship's dimensions (length, beam) or fluid density, the balance between buoyancy and weight remains the same, so the draft may appear unchanged. After changing the values, please click the "Run Calculation" button to recalculate.
This tool is a simplified learning tool based on a rectangular box model. Actual ships are affected by complex hull shapes and water surface conditions, so dedicated CAD/CAE software is required for design. However, it is useful for intuitively understanding the basic concepts of GM and GZ.
Real-World Applications
Ship Design & Classification: Naval architects use these exact calculations to design hulls that meet strict safety rules. For instance, a container ship must have a GZ curve with a minimum area under it to ensure it has enough "energy" to recover from a large wave. The simulator's parameters mirror early-stage design choices.
Loading & Cargo Management: On an operational ship, the captain and officers must calculate stability before loading. Raising the KG by stacking heavy containers too high can reduce GM to dangerous levels. This is why you can adjust KG in the simulator—it's a critical operational variable.
Offshore Structures: Platforms like semi-submersible oil rigs rely entirely on buoyancy and stability. Engineers analyze their GM and GZ curves at various drafts (simulated by changing displacement 'm') to ensure stability during tow-out and under storm conditions.
CAE & Simulation Software: Tools like NX CAE or ANSYS AQWA perform advanced stability analysis using the same fundamental principles. They solve these equations for complex, non-rectangular hull shapes, validating designs before physical scale models are even built. This simulator teaches the core physics behind those multi-million dollar software packages.
Common Misconceptions and Points to Note
When starting with this simulator, there are several points that are easy to misunderstand, especially for CAE beginners. First and foremost, understand that a larger GM is not always better. While a larger GM does increase righting ability, it also makes the ship's motion more abrupt and "stiff," leading to a poor ride. For example, a small fishing vessel with a GM exceeding 3m will pitch and roll sharply with a short period when hitting waves, contributing to crew fatigue and cargo damage. In practice, you need to balance stability with habitability and operability.
Next, always keep in mind that the tool's model is an extremely simplified "rectangular prism" shape. Actual ships have curved surfaces, and the shape near the waterline (waterline flare) significantly influences the GZ curve's form. While you change the "Beam B" in this tool, actual hull design often employs a "tumblehome" shape (where the top is narrower) to finely tune righting characteristics at an angle of heel.
Finally, note that the simulator's "Vertical Center of Gravity KG" is just a single value. On a real ship, the center of gravity constantly changes due to fuel, ballast water, and cargo movement. Furthermore, free surface effect (the effective rise in the center of gravity due to liquid sloshing in tanks) cannot be accurately assessed by simply increasing the calculated KG. It's important to understand the limitations of applying the principles learned with this tool to more complex real-world problems.
Enter Hull Length (m) in vL and Beam (m) in vB to define vessel geometry; Depth (m) in vD sets the hull profile.
Input Draft (m) in sLNum to calculate submerged volume and center of buoyancy position (KB).
The simulator computes Metacentric Radius (BM) from second moment of area and displays metacentric height (GM = KB + BM - KG) in real time; observe the GZ stability curve update as you adjust parameters.
Worked Example
A general cargo vessel: Length 120 m, Beam 18 m, Depth 10 m, operating at draft d = 7.2 m. The simulator calculates KB = 3.6 m (half draft), BM = 156 m³/(120×18×7.2) = 0.76 m, assuming KG (center of gravity) = 4.8 m, yielding GM = 3.6 + 0.76 − 4.8 = −0.44 m. Negative GM indicates instability; reducing cargo height or ballasting to lower KG to 4.2 m improves GM to 0.16 m, restoring adequate stability margin per IMO requirements (minimum 0.15 m for seagoing vessels).
Practical Notes
Validate KB using KB ≈ d/2 for rectangular hulls; finer ship forms require integration of immersed cross-sections.
Monitor GM continuously during loading operations; even small upward shifts in cargo stowage can eliminate positive stability in narrow-margin vessels.
Cross-check GZ curve zero-crossing angle against heel angle at maximum GZ; vessels with GZ peak below 20° heel indicate reduced dynamic stability and vulnerability in heavy seas.