3.2 Hydrostatic Forces on Submerged Surfaces

Hydrostatic Forces and Buoyancy

Hydrostatic forces describe how fluids at rest push against surfaces and objects. These concepts are central to designing anything that interacts with fluids, from dams and retaining walls to ships and submarines. This section covers how to calculate forces on plane and curved surfaces, then moves into buoyancy and floating stability.

Hydrostatic Force on Plane Surfaces

The total hydrostatic force on a submerged plane surface equals the pressure at the surface's centroid multiplied by the surface area:

$F_h = p_c \cdot A$

The pressure at the centroid depends on how deep the centroid sits below the free surface:

$p_c = \gamma h_c$

where $\gamma$ is the specific weight of the fluid (weight per unit volume, e.g., 9810 N/m³ for water) and $h_c$ is the vertical depth from the free surface to the centroid.

To find the total force on a submerged plane surface:

1. Locate the centroid of the surface and measure its vertical depth $h_c$ below the free surface.
2. Calculate the pressure at the centroid: $p_c = \gamma h_c$.
3. Multiply by the surface area: $F_h = p_c \cdot A$.

Center of pressure ($y_p$) is the point where the resultant force actually acts. Because pressure increases with depth, the center of pressure is always below the centroid. For a vertical surface:

$y_p = \frac{I_c}{A \, h_c} + h_c$

where $I_c$ is the second moment of area (moment of inertia) of the surface about its centroidal axis. For common shapes: $I_c = \frac{bh^3}{12}$ for a rectangle, $I_c = \frac{\pi r^4}{4}$ for a circle.

For an inclined surface tilted at angle $\theta$ from the horizontal:

$y_p = \frac{I_c \sin^2\theta}{A \, h_c} + h_c$

Notice that as $\theta$ decreases (surface becomes more horizontal), the correction term shrinks and the center of pressure moves closer to the centroid. Inclined surfaces show up in sloped dam faces, tank walls, and ramps.

Hydrostatic Force on Curved Surfaces

You can't use the simple $F_h = p_c A$ formula for curved surfaces because the pressure acts in different directions at each point along the curve. Instead, the approach is to break the resultant force into horizontal and vertical components.

• Horizontal component $F_H$: Equals the hydrostatic force on the vertical projection of the curved surface. Think of it as the "shadow" the curve casts on a vertical plane. You calculate this projection's area and centroid depth, then use the plane-surface method above.
• Vertical component $F_V$: Equals the weight of the fluid directly above the curved surface, extending up to the free surface. If there's no actual fluid above the surface (e.g., the curved surface faces upward into air), you use the weight of the imaginary fluid column that would occupy that space, and the force acts downward on the surface.

To find the resultant force, combine the two components:

$F_R = \sqrt{F_H^2 + F_V^2}$

The direction of the resultant is:

$\tan \alpha = \frac{F_V}{F_H}$

where $\alpha$ is the angle from horizontal. The line of action passes through the point where the horizontal and vertical component lines of action intersect. This method works for cylindrical gates, spherical tank walls, and any curved geometry.

Buoyancy and Archimedes' Principle

Buoyancy is the net upward force a fluid exerts on a submerged or partially submerged object. It exists because pressure increases with depth, so the fluid pushes harder on the bottom of an object than on the top.

Archimedes' principle states that the buoyant force equals the weight of the fluid displaced by the object:

$F_b = \rho g V_d$

• $\rho$ = fluid density (kg/m³), e.g., 1000 kg/m³ for freshwater, 1025 kg/m³ for seawater
• $g$ = gravitational acceleration (9.81 m/s²)
• $V_d$ = volume of fluid displaced (m³)

This applies to both fully and partially submerged objects. An iceberg floating in seawater displaces a volume of water whose weight equals the iceberg's total weight. Since ice is slightly less dense than seawater, about 90% of the iceberg sits below the surface.

Buoyant Force and Object Stability

Three outcomes are possible when an object is placed in a fluid:

• Floating: The object is less dense than the fluid overall. It settles at a depth where the buoyant force exactly equals its weight ($F_b = W$). It only displaces a fraction of its total volume.
• Neutrally buoyant: The object's average density equals the fluid density. It neither rises nor sinks (like a properly ballasted submarine).
• Sinking: The object is denser than the fluid. The buoyant force at full submersion is still less than the object's weight, so it sinks.

Stability of floating objects depends on what happens when the object tilts slightly. Three points matter:

• Center of gravity (G): Where the object's weight effectively acts.
• Center of buoyancy (B): The centroid of the displaced fluid volume. When the object tilts, B shifts toward the side that's more deeply submerged.
• Metacenter (M): The point where the shifted line of action of the buoyant force crosses the object's original centerline.

The metacentric height $GM$ determines stability:

• Positive $GM$ (M above G): The object is stable. When tilted, the buoyant force creates a restoring moment that pushes it back upright. Wider, lower vessels tend to have large positive $GM$ values.
• Zero $GM$ (M coincides with G): Neutral stability. The object stays at whatever angle it's tilted to.
• Negative $GM$ (M below G): The object is unstable. Any small tilt produces a moment that increases the tilt, leading to capsizing.

For a simple rectangular cross-section floating in water, the metacentric height can be calculated as:

$GM = \frac{I_{waterplane}}{V_d} - BG$

where $I_{waterplane}$ is the second moment of area of the waterplane (the cross-section at the water surface) and $BG$ is the distance between the center of buoyancy and the center of gravity. Ship designers use this relationship to ensure vessels remain stable under expected loading conditions.