Fluid Statics for the FE Exam
Fluid statics deals with fluids at rest. Since there is no motion, there are no shear stresses, and the only stress is normal (pressure). Fluid statics problems appear frequently on the FE exam because they are fundamental and test your understanding of pressure and force concepts.
Pressure Fundamentals
Pressure is defined as force per unit area acting perpendicular to a surface:
P = F / A
Key properties of pressure in a static fluid:
- Pressure acts equally in all directions at a point (Pascal's law)
- Pressure increases with depth
- Pressure is the same at all points on a horizontal plane in a connected static fluid
- Pressure is independent of the shape of the container
Pressure Types
- Absolute pressure: Measured relative to a perfect vacuum. Always positive.
- Gauge pressure: Measured relative to atmospheric pressure. Can be positive (above atmospheric) or negative (below atmospheric, also called vacuum pressure).
- Relationship: P_abs = P_gauge + P_atm
Standard atmospheric pressure: P_atm = 101.325 kPa = 14.696 psi = 760 mmHg = 29.92 inHg = 1 atm
Pressure Variation with Depth
In an incompressible fluid at rest:
P = P_0 + gamma * h = P_0 + rho * g * h
Where P_0 is the pressure at the reference surface, gamma is the specific weight, rho is the density, g is gravitational acceleration, and h is the depth below the reference surface.
For multiple fluid layers:
P = P_0 + gamma_1 * h_1 + gamma_2 * h_2 + gamma_3 * h_3 + ...
Manometers
Manometers use columns of liquid to measure pressure or pressure differences. Analysis technique: start at one point with known pressure and work through each fluid column:
- Going down: add gamma * h
- Going up: subtract gamma * h
For a U-tube manometer measuring gauge pressure of fluid A using manometer fluid B:
P_A = gamma_B * h_B - gamma_A * h_A
Where h_B and h_A are the heights of the respective fluid columns.
Differential Manometers
Differential manometers measure the pressure difference between two points. The same step-by-step approach applies: start at one point, traverse through all fluid columns, and arrive at the other point.
Hydrostatic Forces on Plane Surfaces
Horizontal Surfaces
The force on a horizontal submerged surface is simply:
F = P * A = gamma * h * A
Where h is the depth to the surface and A is the surface area. The force acts uniformly over the entire surface.
Inclined and Vertical Surfaces
For an inclined or vertical surface, the pressure varies over the surface. The resultant force is:
F = gamma * h_c * A
Where h_c is the vertical depth to the centroid of the submerged surface. The force acts at the center of pressure, which is below the centroid. The depth to the center of pressure is:
y_cp = y_c + I_c / (y_c * A)
Where y_c is the distance from the liquid surface to the centroid measured along the inclined plane, and I_c is the second moment of area about the centroidal axis of the surface.
Hydrostatic Forces on Curved Surfaces
For curved surfaces, the analysis is more complex. The approach is to find the horizontal and vertical components separately:
- Horizontal component: Equals the force on the vertical projection of the curved surface. Use the plane surface formula with the projected area.
- Vertical component: Equals the weight of the fluid directly above (or below) the curved surface.
- Resultant force: F = sqrt(F_H^2 + F_V^2), acting at an angle theta = arctan(F_V / F_H) from horizontal.
Buoyancy
Archimedes' principle states that the buoyant force on a body immersed in a fluid equals the weight of the fluid displaced by the body:
F_B = gamma_fluid * V_displaced
For a body submerged in a single fluid, V_displaced is the volume of the body. For a floating body, V_displaced is the volume below the waterline.
Floating Body Equilibrium
A floating body is in equilibrium when the buoyant force equals the body's weight:
gamma_fluid * V_submerged = gamma_body * V_total
The fraction submerged equals the ratio of the body's specific gravity to the fluid's specific gravity:
V_submerged / V_total = SG_body / SG_fluid
Stability of Floating Bodies
A floating body is stable if, when tilted slightly, it returns to its upright position. Stability depends on the relative positions of:
- Center of gravity (G): The point through which the body's weight acts.
- Center of buoyancy (B): The centroid of the displaced fluid volume.
- Metacenter (M): The point about which the body tilts.
The metacentric height GM determines stability:
- GM positive (M above G): Stable. The body returns to upright.
- GM negative (M below G): Unstable. The body tends to capsize.
- GM = 0: Neutrally stable.
GM = BM - BG, where BM = I_waterplane / V_displaced and BG is the distance between the center of buoyancy and the center of gravity.
Practice Tips
- Master manometer analysis by practicing the step-by-step approach
- Know the centroid locations and moments of inertia for common shapes (rectangle, triangle, circle)
- Practice buoyancy problems involving objects floating in liquids with different specific gravities
- Always distinguish between gauge and absolute pressure
Fluid statics is one of the more intuitive topics in fluid mechanics. The concepts are physical and visual, and the problems follow consistent patterns. With practice, these become reliable points on the FE exam.