Fluid Statics

NCEES FE Civil spec: Fluid Mechanics, C. Fluid statics. FE Reference Handbook: Fluid Mechanics chapter, static fluids and forces on submerged surfaces.

Fluid statics deals with fluids at rest, where pressure depends only on depth. Civil engineers use it for dams, gates, tanks, retaining walls holding water, and buoyancy of floating structures. On the FE Civil exam the classic problem is the resultant force and its location on a submerged gate.

What the NCEES spec covers

Section C, "Fluid statics", covers hydrostatic pressure, gauge and absolute pressure, manometers, forces on plane and curved submerged surfaces, center of pressure, and buoyancy and stability of floating bodies.

Key concepts

Pressure increases linearly with depth and acts equally in all directions at a point. It does not depend on the shape of the container.
Gauge vs absolute. Absolute pressure equals gauge pressure plus atmospheric pressure (about 101.3 kPa or 14.7 psi).
Force on a plane surface equals the pressure at the centroid times the area.
Center of pressure is always below the centroid for an inclined or vertical surface, because pressure is larger lower down.
Curved surfaces. The horizontal component equals the force on the vertical projection. The vertical component equals the weight of fluid above the surface.
Buoyancy equals the weight of fluid displaced and acts through the centroid of the displaced volume.

Core formulas

p = p_{0} + γ h p_{abs} = p_{gauge} + p_{atm}

F_{R} = γ h_{c} A = p_{c} A y_{p} = y_{c} + \frac{I _{c}}{y _{c} A}

F_{B} = γ_{f} V_{displaced}

Here y is measured along the plane of the surface from the free surface, and $I_{c}$ is the area moment of inertia of the surface about its own centroidal axis (for a rectangle, $b h^{3} /12$ ).

Where it is in the FE Reference Handbook

The Fluid Mechanics chapter covers characteristics of a static liquid: pressure and depth, manometers, forces on submerged plane surfaces and buoyancy. The centroid and moment of inertia of the gate shape come from the plane-area table in the Statics chapter.

Worked example

A vertical rectangular gate is 2 m wide and 3 m tall. Its top edge is 1.5 m below the water surface, and the other side is open to air. Find the resultant hydrostatic force and the depth of the center of pressure.

Step 1: centroid depth and area.

h_{c} = 1.5 + \frac{3}{2} = 3.0 m A = 2 (3) = 6 m^{2}

Step 2: resultant force.

F_{R} = 9.81 (3.0) (6) = 176.6 kN

Step 3: center of pressure. For a vertical gate, y equals depth.

I_{c} = \frac{2 ( 3 ) ^{3}}{12} = 4.5 m^{4} y_{p} = 3.0 + \frac{4.5}{3.0 ( 6 )} = 3.25 m

The resultant acts 3.25 m below the surface, 0.25 m below the centroid. If the gate were hinged at its top edge, the moment about the hinge would be 176.6 × (3.25 − 1.5) = 309 kN·m.

Common mistakes

Using the depth to the bottom of the gate, or the gate height, instead of the centroid depth.
Measuring y vertically on an inclined gate. The y in the center of pressure formula runs along the incline.
Using I about the base instead of about the centroid.
Adding atmospheric pressure when it acts on both sides and cancels.
For buoyancy, using the total volume of a floating body instead of the submerged volume.

See the hydrostatic force on a gate example and the area moment of inertia lesson.

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

What is the approximate gauge pressure 5 m below a water surface open to the atmosphere? Use ρ = 1,000 kg/m³ and g = 9.81 m/s². See the worked solution or practice the Fluids & Hydraulics deck
A vertical plane gate has submerged area 4 m² with its centroid 2 m below a water surface. The other side is exposed to atmospheric pressure. What is the net hydrostatic force? Use water specific weight γ = 9.81 kN/m³. See the worked solution or practice the Fluids & Hydraulics deck
A solid sphere of diameter 0.10 m and mass 2.0 kg hangs on a light string, fully submerged in a water current flowing horizontally at 1.5 m/s. Take the sphere's drag coefficient as 0.47 and the water density as 1000 kg/m³. Including the buoyant force on the sphere, the steady-state angle the string makes with the vertical is most nearly: See the worked solution or practice the Fluids & Hydraulics deck

Related Calculators and Examples

Frequently Asked Questions

How do you calculate hydrostatic force on a submerged gate?

Multiply the pressure at the centroid of the gate by its area: F = gamma h_c A, where h_c is the vertical depth of the centroid below the free surface.

Why is the center of pressure below the centroid?

Pressure increases with depth, so the lower part of the gate carries more load. The resultant therefore acts below the centroid, by I_c / (y_c A) measured along the plane.

What is the buoyant force on a submerged object?

It equals the weight of the fluid displaced, gamma times the submerged volume, and acts upward through the centroid of the displaced volume.

Back to all Fluid Mechanics topics