CELE Hydraulics & Fluid Mechanics — Hydrostatic Pressure and Forces on SurfacesRevision Notes

Formulas

Exam Tips

• Memorize γ_water = 9.81 kN/m³; γ_seawater ≈ 10.05 kN/m³; γ_mercury = 13.6 × 9.81 = 133.4 kN/m³.
• When the problem says 'gauge pressure,' the free surface is at p = 0 — measure depth h from that surface directly.
• For problems with multiple fluid layers, compute the pressure step-by-step at each interface: p₂ = p₁ + γ₂h₂.
• In board exams, g = 9.81 m/s² is standard; some problems use g = 9.80 or 10 m/s² — read the problem data carefully.

Key Points

• Gauge pressure increases linearly with depth: p = γh, where γ = unit weight of fluid (9.81 kN/m³ for water at standard conditions).
• Absolute pressure = gauge pressure + atmospheric pressure (p_atm ≈ 101.325 kPa at sea level).
• Pascal's Law: pressure at a point in a static fluid acts equally in all directions and is transmitted undiminished throughout the fluid.
• Pressure is constant on any horizontal plane within a connected body of the same fluid — the fundamental basis for manometry.
• Pressure head h = p/γ expresses pressure as an equivalent height of fluid.
• For a fluid of specific gravity s, γ = s × 9.81 kN/m³.
• The hydrostatic pressure equation p = γh is valid for incompressible fluids (liquids) — applicable to nearly all civil engineering problems.
• Pressure is always perpendicular (normal) to any surface in contact with a static fluid.

Definitions

Section Title

Pressure Variation with Depth and Pascal's Law

Common Mistakes

• Using absolute pressure instead of gauge pressure in hydrostatic force formulas — remember: atmospheric pressure cancels out for submerged surfaces open to atmosphere on both sides.
• Forgetting to multiply by the correct specific gravity when the fluid is not water (e.g., seawater, oil, mercury).
• Mixing units: kPa vs Pa vs N/m² — always carry units explicitly and convert early.
• Assuming pressure only acts downward — pressure in a static fluid acts equally in ALL directions (Pascal's Law).
• Using depth to a corner or edge instead of depth to the CENTROID when computing total force.

Formulas

Exam Tips

• Always start at the known pressure side (often p_A known or = 0 for open-ended tube) and walk systematically to the unknown side.
• Sketch the manometer and label every fluid column with its height and fluid type before writing the equation.
• For mercury U-tube with water: p_A (kPa) = 13.6 × 9.81 × x(m) − 9.81 × h_water. Remember the factor of 13.6.
• Board exam shortcut: for mercury-water manometer, p_A ≈ 133.4x − 9.81h (all in consistent units).

Key Points

• Manometers measure pressure by equating fluid column heights at a common datum (usually the lower meniscus in a U-tube).
• Walking rule: moving DOWN the tube ADD γh; moving UP the tube SUBTRACT γh. The total sum from one end to the other equals zero (open end = atmospheric = 0 gauge).
• Simple (piezometer) manometer: open tube filled with the process fluid — reads gauge pressure directly as h.
• U-tube manometer: uses a gage fluid (usually mercury, s = 13.6) to measure higher pressures. The deflection x of gage fluid gives pressure difference.
• Differential manometer: connected between two points to measure pressure difference directly — no need to know absolute pressures at either point.
• For connected vessels at the same elevation with the same fluid, pressures must balance at the connecting level.
• The gage fluid must be immiscible with the process fluid and have a higher density (for most applications).

Definitions

Section Title

Manometry

Common Mistakes

• Forgetting the direction rule — ADDING when going down, SUBTRACTING when going up. A wrong sign flips the answer.
• Omitting the fluid column ABOVE the gage fluid on either side of the manometer.
• Using x (mercury deflection) as h (water column) — they represent different fluids at different γ values.
• Not converting mm to m before computing γh products.

Formulas

Exam Tips

• Memorize the 'two-thirds rule': for a rectangular gate with its TOP edge at the free surface, center of pressure = 2/3 of the gate height from the top. This is a direct board exam shortcut.
• For deeply submerged surfaces (large ȳ), the eccentricity I_g/(ȳA) → 0 and y_p ≈ ȳ — the center of pressure approaches the centroid.
• Always draw a clear diagram: mark the free surface, the surface dimensions, depth to top edge, centroid location, and center of pressure. This prevents nearly all setup errors.
• For gates hinged at one edge with a stop at the other, take moments about the hinge to find the stop reaction. Use y_p to locate F, then compute moments correctly.
• Check: Is the answer for y_p greater than ȳ? If yes, you're likely correct. If y_p < ȳ, you have a sign or formula error.
• Board exam shortcut for rectangle with top at depth d, height H: y_p = d + H/2 + H²/[12(d + H/2)] — derive this once and remember the form.

Key Points

• The total hydrostatic force on any plane surface equals the product of unit weight, depth to centroid, and area: F = γ·ȟ·A.
• This force acts not at the centroid but at the CENTER OF PRESSURE, which is always BELOW the centroid (for a surface with positive depth).
• The center of pressure depth (along the inclined plane): y_p = ȳ + I_g/(ȳ·A), where ȳ is measured along the inclined plane from the free-surface line.
• For a VERTICAL surface: ȳ = ȟ (depth to centroid equals distance along plane to centroid).
• For an INCLINED surface at angle θ from horizontal: ȟ = ȳ·sin θ — the depth and the along-plane distance differ.
• I_g = centroidal moment of inertia of the plane area about its horizontal centroidal axis.
• The eccentricity (distance below centroid to center of pressure) = I_g/(ȳ·A). As depth increases, eccentricity decreases — for very deep surfaces, force approaches the centroid.
• For a rectangle (width b, height h): I_g = bh³/12. For a circle (diameter d): I_g = πd⁴/64. For a triangle (base b, height h): I_g = bh³/36.
• The resultant force on any plane surface always passes through the center of pressure.
• For a surface touching the free surface (top edge at waterline), center of pressure is at 2/3 depth for a rectangle.
• This topic applies to: sluice gates, dam faces, tank panels, manhole covers, bulkheads.

Definitions

Section Title

Hydrostatic Force on Plane Surfaces

Common Mistakes

• Using the depth to the top edge or bottom edge instead of the depth to the CENTROID for computing F = γȟA.
• Confusing ȳ (along-plane distance) with ȟ (vertical depth) for inclined surfaces — they are equal only for vertical surfaces.
• Forgetting to add eccentricity e to ȳ to get y_p — some examinees report the centroid location as the center of pressure.
• Using the wrong I_g formula — rectangle is bh³/12 where h is the dimension parallel to the pressure gradient (vertical for vertical surfaces).
• For submerged (not surface-piercing) surfaces: ȳ ≠ h/2 — you must add the submergence depth to h/2.
• Taking moments about the wrong point when finding hinge reactions — always use y_p for moment arm, not ȟ.

Formulas

Exam Tips

• Always draw the free body diagram, label the free surface, shade the fluid volume above the curved surface, and identify the vertical projection rectangle.
• For a quarter-circle gate of radius R retaining water to the full height: V_above = (πR²/4)·w (quarter-circle area × width). F_H uses A_v = R·w with ȟ_v = R/2.
• The direction of F_V (up or down) depends on which side the fluid is on — ask: 'Is there fluid (real or imaginary) above this surface?' If yes, F_V is downward (or upward for imaginary).
• Circular-arc gate board shortcut: if asked for the moment about the center of curvature, it is ZERO — F passes through that point. This simplifies hinge reaction problems enormously.
• Resultant direction check: arctan(F_V/F_H). Verify the angle is physically reasonable (between 0° and 90° for quarter-circle).

Key Points

• For curved surfaces, the pressure direction varies point-to-point, so direct integration is impractical. Instead, RESOLVE into horizontal and vertical components.
• Horizontal component F_H: equals the force on the VERTICAL PROJECTION of the curved surface. Use the plane-surface method: F_H = γ·ȟ_v·A_v, where A_v is the vertically projected area and ȟ_v is its centroid depth.
• Vertical component F_V: equals the weight of the real or imaginary fluid column above the curved surface up to the free surface. F_V = γ·V_above.
• If fluid is ABOVE the surface: F_V acts DOWNWARD (weight of real fluid above pushes down).
• If fluid is BELOW the surface (e.g., bottom of a curved gate convex upward): F_V acts UPWARD (buoyancy concept — weight of imaginary fluid above).
• Resultant: F = √(F_H² + F_V²), inclined at angle α = arctan(F_V/F_H) from horizontal.
• For a CIRCULAR-ARC (cylindrical) curved surface: all pressure force vectors are radial (perpendicular to the surface and pointing toward the center of curvature). Therefore the RESULTANT PASSES THROUGH THE CENTER OF CURVATURE.
• This 'passes through center' property is a major board exam shortcut for circular-arc gates (Tainter gates, drum gates).
• The location of F_H uses the same y_p formula as plane surfaces; the location of F_V is at the centroid of the volume above.
• Three-dimensional (non-cylindrical) curved surfaces: same component method applies; V_above may require integration or geometric decomposition.

Definitions

Section Title

Hydrostatic Force on Curved Surfaces

Common Mistakes

• Treating F_V as the weight of fluid DIRECTLY above the FLAT horizontal projection — V_above must be the actual volume above the CURVED surface, which may include triangular or circular-sector portions.
• Forgetting to check the direction of F_V — downward if fluid is above the surface, upward if fluid is below.
• Using the curved-surface area (arc length × width) instead of the VERTICAL PROJECTION area for F_H.
• Not accounting for the volume of a triangular or sector region when computing V_above for a quarter-circle gate.
• Forgetting that the resultant on a circular arc always points toward the center — this is a useful check and shortcut.
• Ignoring atmospheric pressure on both sides (it cancels for open systems — only apply if one side is pressurized above atmospheric).

Formulas

Exam Tips

• Write the four I_g values on your scratch paper at the start of every board exam session: rectangle bh³/12, circle πd⁴/64, triangle bh³/36. These three cover >95% of problems.
• Double-check: I_g/(ȳA) must have units of m⁴/(m·m²) = m. If your units don't work out to meters, recheck the formula.
• For a surface-piercing rectangle (top at free surface): y_p = ȳ + I_g/(ȳA) = H/2 + (bH³/12)/[(H/2)(bH)] = H/2 + H/6 = 2H/3. Memorize this classic result.

Key Points

• Rectangle (b wide, h tall — h is the dimension in the pressure direction): I_g = bh³/12; centroid at h/2 from either edge; A = bh.
• Circle (diameter d): I_g = πd⁴/64; centroid at center; A = πd²/4.
• Triangle (base b, height h — vertex at top in typical gate problems): I_g = bh³/36; centroid at h/3 from base (2h/3 from vertex); A = bh/2.
• Semicircle (diameter d, flat side up): I_g = 0.1098r⁴ ≈ 0.11r⁴; centroid at 4r/(3π) from flat side; A = πr²/2.
• Parallel Axis Theorem (not needed for center-of-pressure formula — I_g is already the CENTROIDAL value): I = I_g + Ad², used only when shifting from centroid to another axis.
• For compound shapes: decompose into rectangles/triangles, find I_g for each, use parallel axis theorem to combine about a common axis if needed.
• The term I_g/(ȳA) is the eccentricity e — always has units of length (m).

Definitions

Section Title

Moment of Inertia Reference and Area Properties

Common Mistakes

• Using I about the base edge (bh³/3) instead of the centroidal I_g (bh³/12) — the formula y_p = ȳ + I_g/(ȳA) uses CENTROIDAL I only.
• For a triangle, confusing the centroid at h/3 from BASE vs h/3 from vertex — the centroid is at h/3 from the base (closer to base).
• Forgetting that b and h in I_g formulas refer to specific orientations — b is perpendicular to depth, h is in the depth direction.