Questions: Hydrostatic Force on Vertical Submerged Surfaces

Question 1

A rectangular gate 2 m tall and 1 m wide is submerged vertically with its top edge exactly at the water surface. Where does the resultant hydrostatic force act on the gate?

• A. At the centroid of the gate — 1 m from the top
• B. At the water surface, since that is where pressure acts first
• C. At 4/3 m from the top (2/3 of the gate height from the top)
• D. At the bottom of the gate, since that is where pressure is highest

Answer: (C)

For a rectangular gate with its top at the free surface, the center of pressure is located at 2/3 of the total gate height from the top — here, 2/3 × 2 m = 4/3 m from the top. This follows from y_cp = ȳ + I_c/(ȳ·A): with top at surface, ȳ = 1 m (centroid depth), I_c = wd³/12 = (1)(2³)/12 = 2/3 m⁴, A = 2 m², so the shift is (2/3)/(1×2) = 1/3 m below the centroid, placing the center of pressure at 1 + 1/3 = 4/3 m from the top. The force acts below the centroid because deeper portions carry more pressure.

Question 2

A dam engineer computes the total hydrostatic force on a vertical gate correctly using F = ρg·h̄·A but then assumes the force acts at the centroid to determine the moment on the gate hinge. What error will result?

• A. The engineer will overestimate the moment, leading to an over-designed (but safe) hinge
• B. The engineer will underestimate the overturning moment, potentially under-designing the structural support
• C. No error — for practical engineering purposes, the centroid and center of pressure are close enough to treat as equivalent
• D. The engineer will compute the wrong total force, not just the wrong moment

Answer: (B)

The center of pressure is always below the centroid, so the force arm from any reference point near the gate top is larger than the centroid location implies. Placing the force at the centroid underestimates the moment arm and therefore underestimates the overturning moment the hinge must resist. This is a potentially dangerous unconservative error: the hinge or structural support would be under-designed for the actual load. The center of pressure can be significantly below the centroid for tall gates with their top near the surface.

Question 3

The total hydrostatic force on a vertical submerged surface equals the pressure at the geometric centroid of the surface multiplied by the surface area.

• T. True
• F. False

Answer: True

Correct. Integrating the linearly varying pressure p(h) = ρgh over the surface area gives F = ρg·h̄·A, where h̄ is the depth of the centroid. This is equivalent to using the average pressure over the surface, and for a linearly varying distribution the average equals the value at the midpoint — the centroid. The result is sometimes surprising because students expect to need the full pressure distribution, but the integration simplifies to centroid pressure times area.

Question 4

As the water depth above a submerged vertical gate increases, the center of pressure moves further below the centroid of the gate.

• T. True
• F. False

Answer: False

The opposite is true. The shift below the centroid is given by I_c/(ȳ·A). As water depth above the gate increases, ȳ (the centroid depth) grows, making the denominator larger and the shift smaller. The center of pressure migrates toward the centroid as submersion depth increases, but never reaches it for a fully submerged surface. For a gate deeply submerged, the pressure variation across the gate is small relative to the mean pressure, so the resultant acts close to — but always below — the centroid.

Question 5

Why does the resultant hydrostatic force on a vertical surface act below the centroid rather than at it, and why does this distinction matter for structural engineering?

Model answer: Hydrostatic pressure increases linearly with depth (p = ρgh), so the lower portions of a vertical surface experience higher pressure per unit area than the upper portions. When the pressure distribution is integrated to find the resultant, the deeper (higher-pressure) regions contribute disproportionately more force, pulling the effective point of action downward from the geometric center. The center of pressure is located at y_cp = ȳ + I_c/(ȳA), always below the centroid by the term I_c/(ȳA). For structural design, this matters because the force creates a moment about any support point; placing the force at the wrong location underestimates that moment, potentially causing structural failure in dams, lock gates, or tank walls.

The key structural insight is that knowing only the magnitude of the hydrostatic force is insufficient for design — you must also know where it acts. A gate hinge, a dam buttress, or an underwater seal must be sized for the actual moment the force creates. Treating the force as acting at the centroid systematically underestimates this moment because the true location is always further from the top of the structure.