What is metacentric height GM?

GM is the primary measure of initial stability for a floating body. When GM > 0, a heeled vessel experiences a restoring moment returning it to upright — this is called positive stability. When GM < 0 the vessel will capsize. Typical cargo ships target GM = 0.5–1.5 m. GM = KB + BM − KG, where KB is the height of the center of buoyancy, BM is the metacentric radius, and KG is the height of the center of gravity.

How is the center of buoyancy KB calculated?

For a rectangular box hull at draft d, the center of buoyancy is located at KB = d/2, the midpoint of the submerged volume. Real hull forms with flared sections or bilge curves require numerical integration of the sectional area curve, but d/2 is the standard first approximation used in early design stages.

What does the GZ righting lever curve show?

The GZ curve (static stability curve) plots the righting lever GZ against heel angle θ. For small angles, GZ ≈ GM·sin θ. The area under the curve up to the angle of vanishing stability gives a measure of dynamic stability. IMO regulations specify minimum area and GZmax requirements for passenger and cargo vessels.

How does beam width B affect stability?

The metacentric radius BM = I/V where I = LB³/12 for a rectangular waterplane. BM scales with B², so doubling the beam quadruples BM. This is why wide-beamed vessels like catamarans and barges are inherently more stable. However, a very high GM can lead to a 'stiff' ship with rapid rolling that is uncomfortable and structurally demanding.

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Naval Architecture

Buoyancy & Stability Simulator

Q: How is the center of buoyancy KB calculated?

For a rectangular box hull at draft d, the center of buoyancy is located at KB = d/2, the midpoint of the submerged volume. Real hull forms with flared sections or bilge curves require numerical integration of the sectional area curve, but d/2 is the standard first approximation used in early design stages.

Q: What does the GZ righting lever curve show?

The GZ curve (static stability curve) plots the righting lever GZ against heel angle θ. For small angles, GZ ≈ GM·sin θ. The area under the curve up to the angle of vanishing stability gives a measure of dynamic stability. IMO regulations specify minimum area and GZmax requirements for passenger and cargo vessels.

Q: How does beam width B affect stability?

The metacentric radius BM = I/V where I = LB³/12 for a rectangular waterplane. BM scales with B², so doubling the beam quadruples BM. This is why wide-beamed vessels like catamarans and barges are inherently more stable. However, a very high GM can lead to a 'stiff' ship with rapid rolling that is uncomfortable and structurally demanding.

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

Hull shape

Length L (m)

Beam B (m)

Depth D (m)

Displacement m (ton)

Fluid density ρ_f (kg/m³)

kg/m³

Center of gravity KG (m)

Heel angle θ (°)

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Draft d (m)

—

KB (m)

—

BM (m)

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GM (m)

● Stable

Theory

Draft: $d = \dfrac{m \times 10^3}{\rho_f \cdot L \cdot B}$
Center of buoyancy: $KB = d/2$
Metacentric radius: $BM = \dfrac{LB^3/12}{V}$
Metacentric height: $GM = KB + BM - KG$
Righting lever: $GZ \approx GM \cdot \sin\theta$

Drag the heel angle slider to animate the hull / Switch to "GZ Curve" tab for the static stability curve

What is Metacentric Height?

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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.

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.

Related Engineering Fields

The concepts you master through floating stability calculations are applied in various engineering fields beyond naval architecture. The first to mention is "Stability & Control" in aerospace engineering. The aircraft's "neutral point" is analogous to a ship's metacenter (M). The fore-and-aft relationship between the center of gravity (G) and the neutral point determines the aircraft's longitudinal static stability. Although the restoring force comes from aerodynamic forces instead of buoyancy, the core concept—that stability is determined by the positional relationship between the center of gravity and the center of the restoring force—is shared.

Another field is robotics, particularly posture control for bipedal or mobility robots. To prevent tipping, you must constantly evaluate the relationship between the robot's center of gravity projection and its support polygon (the area formed by its feet). This is very similar to considering the relationship between a floating body's center of gravity and center of buoyancy in a plane (2D). Taking it further, there is also a connection to buckling analysis in structural engineering. In evaluating the phenomenon where a slender column collapses (buckles) under axial force, you can find a mathematical analogy with GZ curve calculation at large heel angles, as both involve considering an "apparent righting moment" corresponding to the deflected shape.

For Further Learning

Once you're comfortable with the basic principles using this simulator, as a next step, learn about the influence of "non-rectangular cross-sections". Specifically, investigate how the movement of the center of buoyancy (B) and the formula for metacentric radius (BM) change for floating bodies with trapezoidal or rounded cross-sections. At the core of BM calculation lies the "second moment of area" of the waterplane. Understanding this concept will show you that the formula $$BM = \frac{I_{WL}}{\nabla}$$ (where $I_{WL}$: second moment of area of the waterplane, $\nabla$: displaced volume) is a universal one applicable to any shape.

To delve deeper, we recommend moving on to "Large-Angle Stability" and "Dynamic Stability". The area under the GZ curve represents the ability of the righting force to absorb energy from wind and waves as the ship heels (dynamic stability). For instance, a ship whose GZ curve peaks early and then descends might not withstand large waves, even if its GM appears large. In the professional world, these advanced assessments are essential to meet the stringent stability criteria set by the IMO (International Maritime Organization). Your advanced learning begins by playing with the tool and observing how changing each parameter affects the GZ curve's "shape" and "area".