Questions: Hydrostatic Forces on Submerged Surfaces

Question 1

A rectangular gate has its centroid at depth ȳ = 4 m below the free surface. The hydrostatic resultant force equals ρg × 4 m × A. At what depth does this resultant force actually act?

• A. At exactly 4 m depth — the force acts at the centroid
• B. At a depth greater than 4 m — the center of pressure lies below the centroid
• C. At a depth less than 4 m — the lower pressure near the top shifts the action point upward
• D. At the free surface, because that is where the pressure gradient originates

Answer: (B)

The magnitude formula F = ρg·ȳ·A uses the centroid depth to compute the force, but the resultant acts at the center of pressure, which is always below the centroid. The offset is I_c/(ȳ·A), where I_c is the second moment of area about the centroidal axis. The deeper parts of the gate experience higher pressure, biasing the net moment toward the deep end. Using the centroid depth for magnitude is a shortcut that works because pressure varies linearly and the centroid is the area-weighted mean; it does not mean the force acts there.

Question 2

A submerged gate is initially at shallow depth (ȳ = 2 m). It is then lowered to ȳ = 20 m, with area and orientation unchanged. How does the location of the center of pressure change relative to the centroid?

• A. The offset grows larger because higher absolute pressure increases the moment imbalance
• B. The offset shrinks because at great depth, pressure variation across the gate is small compared to the mean pressure, so the load is nearly uniform
• C. The center of pressure moves above the centroid at great depth
• D. The offset remains constant because I_c and A are unchanged

Answer: (B)

The offset of the center of pressure below the centroid is I_c/(ȳ·A). As ȳ increases with I_c and A fixed, the offset decreases. Physically: at shallow depth, pressure changes significantly from top to bottom of the gate (relative to the mean pressure), strongly biasing the moment. At great depth, the same absolute pressure change across the gate is a tiny fraction of the mean pressure, so the loading is nearly uniform — and a nearly uniform load acts at its centroid. The center of pressure converges toward the centroid as depth increases.

Question 3

The resultant hydrostatic force on a submerged flat surface acts at the centroid of the surface.

• T. True
• F. False

Answer: False

The resultant acts at the center of pressure, which lies below the centroid (for any surface that is not under perfectly uniform pressure, i.e., any surface with finite vertical extent). The centroid depth is used only to compute the magnitude of the resultant force, not its point of application. Applying the force at the centroid rather than the center of pressure would give the wrong moment — a critical error in designing gates, dams, and retaining walls where the moment arm determines structural loads.

Question 4

For a curved submerged surface, the horizontal component of the hydrostatic resultant equals the hydrostatic force on the vertical projection of that curved surface.

• T. True
• F. False

Answer: True

Because pressure is a scalar acting normal to the surface, integrating pressure vectors over a curved surface requires decomposition. Horizontal equilibrium on the fluid volume bounded by the curved surface and a vertical projection plane shows that the net horizontal force on the curve must equal the force on its vertical projection — a flat vertical plate at the same depth. This projected-area approach lets you use the familiar plane-surface formula (F = ρg·ȳ·A_projected) for the horizontal component without doing a curved surface integral.

Question 5

Explain physically why the center of pressure on a submerged inclined plane lies below the centroid, and describe what happens to this offset as the surface is submerged to greater and greater depth.

Model answer: Hydrostatic pressure increases linearly with depth, so deeper parts of a submerged surface experience greater pressure than shallower parts. The pressure distribution is non-uniform: the load per unit area is larger at the bottom than at the top. When computing the moment of this distributed load about the centroid axis, the high-pressure deep region contributes a larger moment than the low-pressure shallow region, pulling the effective point of action (center of pressure) downward, below the centroid. As depth increases, the mean pressure ρg·ȳ grows large relative to the pressure variation across the surface (which depends only on surface height, not ȳ). The variation becomes an increasingly small fraction of the mean, making the load nearly uniform — and a uniform load acts at the centroid. The offset I_c/(ȳ·A) therefore shrinks toward zero as ȳ increases.

This depth-dependence has an important engineering implication: deep submerged gates and hull sections can be analyzed with less concern about center-of-pressure offset than shallow ones, where the pressure gradient across the surface is a large fraction of the mean pressure.