4.2 Stability of Floating Bodies

Stability and Equilibrium Concepts

Stability analysis tells you whether a floating body will right itself after being disturbed or tip over. This is foundational for designing anything that floats, from cargo ships to offshore platforms. The core tool here is metacentric height (GM), which quantifies a floating body's resistance to tilting.

Stability and Equilibrium of Floating Bodies

Stability is a floating body's tendency to return to its original upright position after being disturbed by wind, waves, or changes in loading. A stable body "fights back" against tilting; an unstable one keeps going over.

Equilibrium is the state where the body is at rest with zero net force and zero net moment. For a floating body, this requires two conditions:

• The body's weight equals the buoyant force
• The center of gravity (G) and center of buoyancy (B) are vertically aligned

Both conditions must hold simultaneously. If G and B aren't aligned vertically, the body experiences a moment that causes it to rotate.

Types of Equilibrium

The three types of equilibrium are distinguished by the relative positions of the metacenter (M) and the center of gravity (G):

• Stable equilibrium: M is above G, giving a positive metacentric height ($GM > 0$). When tilted, the body generates a restoring moment that brings it back upright. Think of a wide-hulled boat with heavy ballast low in the hull.
• Unstable equilibrium: M is below G, giving a negative metacentric height ($GM < 0$). When tilted even slightly, the body generates an overturning moment that pushes it further from its original position. A tall, narrow vessel with top-heavy loading behaves this way.
• Neutral equilibrium: M coincides with G, so $GM = 0$. When tilted, the body simply stays in its new position. There's no restoring moment and no overturning moment. A perfectly symmetrical object with uniform mass distribution can exhibit this behavior.

Factors Affecting Stability and Stability Analysis

Key Factors in Floating Body Stability

Center of gravity (G) is the point where the body's total weight effectively acts. It depends entirely on how mass is distributed within the body. Lowering G improves stability, which is why ships carry heavy cargo and ballast as low as possible.

Center of buoyancy (B) is the centroid of the displaced fluid volume, and it's where the buoyant force effectively acts. Here's the critical detail: B shifts position when the body tilts, because the shape of the submerged volume changes. This shift is what creates (or fails to create) a restoring moment.

Metacenter (M) is defined as the point where the vertical line through the new center of buoyancy (B') intersects the original vertical centerline when the body is given a small angular displacement. The vertical distance from G to M is the metacentric height (GM), and it's the single most important parameter for stability.

Shape and dimensions play a major role. Wider, shallower bodies tend to be more stable than narrow, deep ones. A flat barge is inherently more stable than a narrow canoe because the wider waterplane area produces a larger metacentric radius (more on this below).

Stability Analysis: Calculating GM

The standard approach to stability analysis follows these steps:

1. Calculate the metacentric height using:

$GM = KB + BM - KG$

where:

• $KB$ = vertical distance from the keel (K, the lowest point of the hull) to the center of buoyancy (B)
• $BM$ = metacentric radius, calculated as $BM = \frac{I}{V}$, where $I$ is the second moment of area (moment of inertia) of the waterplane about the longitudinal axis, and $V$ is the total displaced volume of fluid
• $KG$ = vertical distance from the keel (K) to the center of gravity (G)

Notice that $BM = \frac{I}{V}$ is where shape matters most. A wider waterplane gives a larger $I$, which increases $BM$, which increases $GM$. That's the mathematical reason wide boats are more stable.

1. Interpret the result:

   • $GM > 0$: Stable equilibrium. The body will return upright after a small tilt.
   • $GM < 0$: Unstable equilibrium. The body will capsize when disturbed.
   • $GM = 0$: Neutral equilibrium. The body remains at whatever angle it's displaced to.

2. Evaluate loading conditions. Adding, removing, or shifting weight changes the position of G and therefore changes $KG$. For example, stacking containers high on a cargo ship raises G, reduces GM, and can push the vessel toward instability. Proper ballast and even cargo distribution are essential to maintaining a safe positive GM.

3. Account for external forces. Wind, waves, and currents apply tilting moments to the body. The restoring moment from a positive GM must be large enough to counteract these. A sailboat heeling under wind load stays upright because the shift in B creates a restoring moment that balances the wind's overturning moment. If GM is too small, even moderate external forces can capsize the vessel.