Hydrostatic Pressure and Forces on Surfaces - Cheat Sheet
Your last-minute revision companion for hydrostatic pressure, manometry, and forces on plane and curved surfaces. Master the key formulas, definitions, and common pitfalls in 30 minutes.
Pressure Fundamentals
Formulas
p = γh
Meaning
p = gauge pressure (kPa), γ = unit weight of fluid (kN/m³), h = depth below free surface (m)
Watch out
This gives GAUGE pressure only — add atmospheric pressure (101.325 kPa) for absolute pressure. Do NOT confuse h (depth) with height above datum.
When to use
Calculate gauge pressure at any depth in a static fluid; h is always measured vertically downward from surface.
p_absolute = p_gauge + p_atm
Meaning
Absolute pressure = gauge pressure + atmospheric pressure (typically 101.325 kPa)
Watch out
The problem may ask for gauge or absolute — read carefully. For water at standard conditions, p_atm ≈ 101.325 kPa or 10.33 m of water head.
When to use
When you must express pressure in absolute terms; most engineering problems use gauge unless stated otherwise.
h_pressure = p / γ
Meaning
h_pressure = pressure head (m), p = pressure (kPa), γ = unit weight (kN/m³)
Watch out
Different fluids have different γ. For water γ = 9.81 kN/m³, for mercury γ = 133.1 kN/m³. Always use correct fluid unit weight.
When to use
Express pressure as an equivalent height of fluid column (e.g., 'pressure of 98.1 kPa = 10 m of water head').
Common values
Value
9.81 kN/m³
Symbol
γ_water
Quantity
Unit weight of water
Value
133.1 kN/m³
Symbol
γ_mercury
Quantity
Unit weight of mercury
Value
13.6
Symbol
s_mercury
Quantity
Specific gravity of mercury
Value
101.325 kPa or 10.33 m water head
Symbol
p_atm
Quantity
Standard atmospheric pressure
Important facts
- Pressure increases linearly with depth in a static, incompressible fluid.
- Pressure acts perpendicular (normal) to every surface.
- All points on the same horizontal plane in a connected fluid experience equal pressure.
- For water: γ = 9.81 kN/m³; for mercury: γ ≈ 133.1 kN/m³ (s = 13.6).
- Atmospheric pressure at sea level ≈ 101.325 kPa or 1 atm; often taken as 10.33 m of water head.
- Pressure is a scalar quantity (magnitude only, no direction).
Key definitions
Gauge pressure
Example
Water at 5 m depth: p = 9.81 × 5 = 49.05 kPa gauge.
Pressure measured above atmospheric pressure; the pressure a manometer or gauge reads directly.
Pressure head
Example
49.05 kPa in water = 49.05/9.81 = 5 m head.
The vertical height of a fluid column that would produce a given pressure; expressed in metres of that fluid.
Hydrostatic
Example
Water behind a dam exerts hydrostatic forces on the dam face.
Pressure and forces in a fluid AT REST (no flow, no acceleration).
Pascal's Law
Example
Pressure at a given depth is the same whether measured vertically, horizontally, or at any angle.
Pressure at a point acts equally in all directions; any external pressure applied to a confined fluid is transmitted undiminished throughout.
Diagrams to know
- Pressure variation with depth (linear graph: p vs h).
- Pressure distribution on a vertical or inclined submerged surface (triangular or trapezoidal distribution).
- Hydrostatic pressure acting perpendicular to surfaces in all directions.
Manometry
Formulas
Rule: Add γh going DOWN; subtract γh going UP.
Meaning
Walk along the manometer tube from one end to the other, tallying pressure changes as you ascend or descend.
Watch out
Sign error is the #1 mistake — carefully track whether you are moving up or down in each segment. Write the equation at BOTH ends and set equal.
When to use
Analyze any U-tube, differential, or open manometer to find pressure difference or unknown pressure.
p_A + γ_A × h_A = p_B + γ_B × h_B + γ_mercury × h_mercury
Meaning
General manometer balance: pressure + (unit weight × height) is equal along the fluid path.
Watch out
Always use the correct unit weight for each fluid segment. Do NOT mix up heights of different fluids.
When to use
When the manometer contains multiple fluid columns (water on one side, mercury in the tube, etc.).
ΔP = γ_gage × Δh
Meaning
For a differential manometer, pressure difference = (unit weight of gage fluid) × (deflection height).
Watch out
Δh is the deflection of the gage fluid level, NOT the total height of liquid in the tubes.
When to use
Quick calculation when using a single gage fluid (e.g., mercury) to measure pressure difference between two points.
Common values
Value
13.6 (relative to water)
Symbol
s_Hg
Quantity
Mercury density (specific gravity)
Value
13.6:1 (mercury column is 13.6 times smaller for same pressure)
Symbol
s_ratio
Quantity
Ratio of water to mercury manometer sensitivity
Important facts
- A piezometer measures only gauge pressure (not absolute); it reads zero when connected to atmosphere.
- U-tube manometers can measure both pressure and pressure difference.
- Mercury is often used as a gage fluid because its high density gives good resolution (small tube height for large pressure).
- When both limbs of a manometer are open to atmosphere, it reads the pressure difference between the two connection points.
- Always set up the manometer equation starting from one end and ending at the other, checking units and fluid type at each step.
Key definitions
Manometer
Example
A mercury U-tube manometer with a 250 mm deflection measuring water pressure.
A device (typically a U-tube) that measures pressure or pressure difference by balancing fluid columns.
Differential manometer
Example
Piezometer (simple tube) or U-tube with mercury to measure flow-induced pressure drop.
A manometer used to measure pressure difference between two points; often uses a denser fluid (mercury) for sensitivity.
Piezometer
Example
A tube inserted into a tank wall; water rises h = p / γ above the tank surface.
The simplest manometer: a tube connected to a point in the fluid; fluid rises to a height proportional to gauge pressure.
Diagrams to know
- Simple U-tube piezometer (one arm open, one closed at bottom).
- Differential U-tube manometer (both arms submerged in different pressure regions).
- Inverted manometer (for measuring low pressures or negative gauge pressures).
- Inclined manometer (for greater sensitivity; angle increases the scale of deflection).
Reactions or equations
p_1 + γ_water × h_1 = p_2 + γ_water × h_2 + γ_mercury × h_mercury
Note
Substitute values with consistent units (kPa and m, or kPa and mm). Solve for unknown pressure.
Conditions
Open manometer with water on both sides and mercury in the central tube.
Forces on Plane Surfaces
Formulas
F = γ × h̄ × A
Meaning
F = total hydrostatic force (kN), γ = unit weight (kN/m³), h̄ = depth of centroid below surface (m), A = area of surface (m²)
Watch out
h̄ must be the PERPENDICULAR distance to the centroid, not along the incline. For a vertical gate, h̄ equals the centroidal depth. For an inclined gate, h̄ = ȳ × sin(θ), where ȳ is measured along the plane.
When to use
Calculate the magnitude of the total hydrostatic force on any plane submerged or partially submerged surface.
y_p = ȳ + I_g / (ȳ × A)
Meaning
y_p = distance from free surface to center of pressure (measured along the plane), ȳ = distance to centroid (along plane), I_g = centroidal second moment of inertia about horizontal axis through centroid (m⁴)
Watch out
The center of pressure is ALWAYS BELOW the centroid (the term I_g/(ȳ×A) is always positive). Do NOT confuse y_p with h̄ for inclined surfaces. The extra term I_g/(ȳ×A) accounts for pressure variation across the surface.
When to use
Locate the point of application of the hydrostatic force (center of pressure) on any submerged plane surface.
I_g = (1/12) × b × h³ (rectangle), I_g = (1/64) × π × d⁴ (circle), I_g = (1/36) × b × h³ (triangle)
Meaning
Centroidal second moment of inertia for common cross-sections (b = width, h = height, d = diameter).
Watch out
Always use the centroidal moment of inertia (I_g about the centroidal axis), NOT the polar moment. For composite areas, sum individual I_g values, then use the parallel-axis theorem if needed.
When to use
When calculating the center of pressure for gates with standard geometric shapes.
h̄ = ȳ × sin(θ)
Meaning
For an inclined surface, h̄ (vertical depth of centroid) = ȳ (distance along incline) × sin(angle from horizontal).
Watch out
This relationship is essential for inclined gates. If θ = 90° (vertical), sin(90°) = 1, so h̄ = ȳ. For θ = 0° (horizontal), sin(0°) = 0, so h̄ = 0 (no pressure difference across a horizontal surface).
When to use
When the surface is inclined at angle θ from horizontal; converts between depth and distance along the plane.
Common values
Value
h/2
Symbol
ȳ_rect
Quantity
Rectangle centroid depth (top at surface, height h)
Value
2h/3
Symbol
y_p_rect
Quantity
Rectangle center of pressure (top at surface, height h)
Value
h/3 from base
Symbol
ȳ_tri
Quantity
Triangle centroid (base at bottom)
Important facts
- The hydrostatic force on a plane surface is the product of pressure at the centroid and the area.
- The center of pressure is always below the centroid because pressure increases with depth.
- For a vertical rectangle with top at the surface, the center of pressure is at 2/3 of the height.
- For fully submerged surfaces, the center of pressure is closer to the centroid than for partially submerged surfaces (smaller I_g/(ȳ×A) term).
- The force always acts perpendicular to the plane surface (inward).
- For composite surfaces (e.g., a trapezoid), split into simpler shapes and calculate I_g using the parallel-axis theorem.
Key definitions
Center of pressure
Example
A vertical rectangular gate 2 m × 3 m has its centroid at 1.5 m depth and center of pressure at 2.0 m depth.
The point on a submerged surface through which the resultant hydrostatic force acts; located below the centroid for surfaces under varying pressure.
Centroid
Example
For a rectangle, centroid is at (b/2, h/2); for a triangle at (b/3, h/3).
The geometric center of an area; the point at which the total area can be concentrated for moment calculations.
Second moment of inertia (I_g)
Example
For a rectangle with width b and height h: I_g = bh³/12 about the centroidal axis.
A measure of how an area is distributed about an axis; used to calculate the center of pressure and resistance to bending.
Plane surface
Example
A vertical rectangular dam face, an inclined sluice gate, or a triangular submerged panel.
A flat submerged boundary (gate, wall, etc.) against which hydrostatic pressure acts uniformly across its area.
Diagrams to know
- Pressure distribution diagram on a vertical gate (linear gradient from surface to bottom).
- Location of centroid vs. center of pressure on a vertical rectangular gate.
- Pressure prism (3D representation of pressure distribution on a plane surface).
- Inclined gate with pressure distribution perpendicular to the surface.
Reactions or equations
F = γ × h̄ × A; y_p = ȳ + I_g / (ȳ × A)
Note
These two equations must be used together: first find F (magnitude), then find y_p (location). Both are independent of surface shape orientation (they apply to vertical, inclined, horizontal, even rotated surfaces).
Conditions
Any plane surface (vertical, inclined, partially or fully submerged).
Forces on Curved Surfaces
Formulas
F_H = γ × h̄ × A_vert
Meaning
F_H = horizontal component of hydrostatic force (kN), h̄ = depth of centroid of vertical projection (m), A_vert = area of vertical projection (m²)
Watch out
Do NOT use the actual surface area A; use only the projected area A_vert. The vertical projection must be perpendicular to the direction of F_H. For a surface curving left-right, F_H acts horizontally; for one curving up-down, F_H acts vertically.
When to use
Calculate the horizontal component of force on any curved surface. Treat the VERTICAL PROJECTION as a plane surface.
F_V = γ × V
Meaning
F_V = vertical component of hydrostatic force (kN), γ = unit weight (kN/m³), V = volume of fluid directly above the curved surface (real or imaginary)
Watch out
V is the VOLUME of fluid above the surface, not the volume of the curved surface itself. If the surface curves upward (like a dome), the 'fluid above' is imaginary (a free body diagram). If it curves downward (like a bowl), the fluid above is real.
When to use
Calculate the vertical component of force on a curved surface. It equals the weight of the fluid body (real or virtual) above the surface.
F = √(F_H² + F_V²)
Meaning
F = magnitude of total hydrostatic force resultant on a curved surface.
Watch out
F acts at the center of the curved surface only for certain shapes (e.g., circular arc through the center of curvature). For general curved surfaces, the line of action may not pass through the geometric center; use moment equations if needed.
When to use
After calculating F_H and F_V separately, combine them to find the total force magnitude.
Direction of F: tan(α) = F_V / F_H
Meaning
α = angle of resultant force from horizontal; arctan(F_V/F_H) gives the angle.
Watch out
Check the quadrant: if both F_H and F_V point in positive directions, the resultant is in the first quadrant. Sketch the components to avoid sign errors.
When to use
Determine the direction (angle) of the resultant hydrostatic force on a curved surface.
Common values
Value
F_H = γ × (R/2) × R = γR²/2; F_V = γ × (π R²/4)
Symbol
F_quarter_circle
Quantity
Quarter-circle arc (radius R, water surface at top)
Important facts
- The horizontal component F_H acts perpendicular to the vertical projection and is independent of the curvature (same as for a plane surface).
- The vertical component F_V always equals the weight of fluid above (real or virtual); it does not depend on horizontal projection.
- For a surface with constant radius of curvature (e.g., circular arc), the resultant force passes through the center of curvature.
- A curved surface experiences both horizontal and vertical components unless the surface is entirely vertical (F_H only) or entirely horizontal (F_V only).
- The line of action of the resultant force is important for stability and moment calculations (e.g., against overturning).
Key definitions
Curved surface
Example
A quarter-circle or semi-circular spillway gate retaining water; a spherical tank bottom.
A non-planar submerged boundary (e.g., circular arc gate, dome, sphere, cone) against which hydrostatic pressure acts.
Vertical projection
Example
A cylindrical curved gate projects as a rectangle onto a vertical plane; a spherical cap projects as a circle.
The shadow or orthogonal projection of a curved surface onto a vertical plane; the area used to calculate F_H.
Free body diagram (for curved surfaces)
Example
For an upward-curving dome, imagine a fluid column above it even if empty; F_V = γ × (that imaginary volume).
An imaginary control volume enclosing the curved surface; used to account for the vertical force as the weight of fluid above (real or virtual).
Center of curvature
Example
A quarter-circle gate curving inward has its center of curvature at the corner; the resultant force points toward that corner.
For a circular-arc surface, the point about which the arc is centered; the resultant hydrostatic force passes through this point.
Diagrams to know
- Quarter-circle or semi-circular gate showing vertical and horizontal projections.
- Horizontal and vertical components of force on a curved surface (vector diagram).
- Free body diagram of a curved surface with imaginary fluid column above (for upward-curving surfaces).
- Resultant force direction for a circular arc passing through center of curvature.
Reactions or equations
F_H = γ × h̄_vert × A_vert; F_V = γ × V_above; F = √(F_H² + F_V²)
Note
These three equations must be applied in sequence. F_H and F_V are components; combine to get the resultant F. The direction is tan(α) = F_V/F_H.
Conditions
Any curved surface under hydrostatic pressure.
Special Cases & Applications
Formulas
For surface piercing (top at free surface): y_p = (2/3) × h (vertical rectangle)
Meaning
When the top edge of a vertical rectangular gate is at the water surface, the center of pressure is at 2/3 of the total depth.
Watch out
This is a special case of the general formula y_p = ȳ + I_g/(ȳ×A). It only applies to vertical rectangles; other shapes and orientations require the full formula.
When to use
Quick check for vertical rectangular gates with the top edge at the free surface.
For fully submerged surface: y_p ≈ ȳ + I_g/(ȳ×A) (small correction, approaches ȳ as depth increases)
Meaning
The center of pressure approaches the centroid for very deep or very large areas.
Watch out
Do NOT ignore the correction term I_g/(ȳ×A) for moderate depths; it becomes significant for shallow surfaces.
When to use
For deeply submerged surfaces (e.g., dam at 100 m depth), the pressure distribution is nearly uniform, and y_p ≈ ȳ.
Gate or structural stability: sum moments about hinge = 0 to find opening/closing conditions.
Meaning
For hinged gates, equate clockwise to counterclockwise moments about the hinge pin.
Watch out
Include weight of gate, distance from hinge to force application point (y_p), and any applied external forces. Moments must be taken about a consistent reference point (usually the hinge).
When to use
Determine if a gate will open (uplift force) or remain closed under hydrostatic loading, or find the force needed to hold it closed.
Important facts
- For a hinged gate, the hydrostatic force acts at the center of pressure, not at the centroid; this can cause significant moments.
- Gate behavior (remain closed vs. open) depends on the balance of moments about the hinge.
- External loads (weights, cables, locks) must be included in moment balance.
- A gate experiences maximum stress at the hinge due to the cantilever moment.
- For stability analysis, check both the magnitude of force and its line of action (moment arm).
Key definitions
Hinge or pin support
Example
A spillway gate hinged at its top edge; hydrostatic force causes a moment that either opens or closes the gate.
A support that allows a gate to rotate about a fixed point; the reaction acts at the hinge, and moments are summed about it.
Buoyant force
Example
A floating gate or pontoon displaces a volume of water equal to its weight.
The net vertical force on a fully or partially submerged object; equals the weight of fluid displaced (Archimedes' principle).
Diagrams to know
- Hinged gate with hydrostatic force at center of pressure.
- Moment diagram showing clockwise (closing) vs. counterclockwise (opening) moments about hinge.
- Free body diagram of a submerged gate with all forces and moments labeled.
Must remember
- Gauge pressure p = γ × h; it is ZERO at the free surface and increases linearly with depth. Always distinguish gauge (above atmospheric) from absolute (includes atmospheric).
- For ANY plane surface, the total hydrostatic force magnitude is F = γ × h̄ × A, where h̄ is the depth of the CENTROID (not the center of pressure).
- The center of pressure y_p = ȳ + I_g/(ȳ×A) is ALWAYS BELOW the centroid for submerged surfaces; the force acts here, not at the centroid.
- For a vertical rectangular gate with top at the water surface: centroid is at h/2, center of pressure is at 2h/3 (a quick landmark).
- For curved surfaces, resolve into TWO components: F_H (horizontal, using vertical projection) and F_V (vertical, = weight of fluid above). Then F = √(F_H² + F_V²).
- In manometry, walk along the tube, ADDING γ×h going DOWN and SUBTRACTING γ×h going UP; set the two ends equal. Sign errors are the #1 mistake.
- For inclined plane surfaces, h̄ (vertical depth of centroid) = ȳ (distance along plane) × sin(θ); do NOT confuse these.
- The vertical component of force on a curved surface, F_V = γ × V, equals the WEIGHT of the fluid above (real or imaginary); it is independent of the horizontal projection.
- Pressure acts PERPENDICULAR to every surface; for a plane, the force is normal to the plane; for a curved surface, the force has both normal and tangential components that resolve into F_H and F_V.
- For stability analysis of gates or dams, always use the center of pressure (not centroid) as the line of action when summing moments about a hinge; neglecting this is a common exam error.
Last minute tips
- CHECK YOUR DEPTHS: Depth h in p = γh is always measured VERTICALLY downward from the free surface, not along an incline. For an inclined gate, first find h̄ = ȳ × sin(θ), then calculate F = γ × h̄ × A. This is in ~40% of exam questions.
- CENTER OF PRESSURE IS NOT CENTROID: Always remember y_p = ȳ + I_g/(ȳ×A) is below the centroid. Forgetting this term causes wrong moment calculations and is worth 5-10 marks. Quick check: for a vertical rectangle with top at surface, y_p should be 2h/3, not h/2.
- MANOMETER WALK: Draw the manometer and trace your finger along the tube from one end to the other. Write down pressure + γ×(height up or down). This physical act prevents sign errors. Test yourself on one practice problem before the exam.
- CURVED SURFACES HAVE TWO FORCES: Always split into F_H (using vertical projection) and F_V (weight of fluid above). Many students forget F_V entirely and get zero credit. Sketch a free body diagram of the curved surface with both components labeled.
- UNIT CONSISTENCY: Use consistent units (kPa and m, or kPa and mm, but NOT mixed). A common error is calculating depth in metres, then second moment in mm⁴, leading to nonsense results. Check units after every calculation.
Comparison tables
Plane vs. Curved Surfaces — Force Calculation
| Aspect | Plane Surface | Curved Surface |
|---|---|---|
| Horizontal component (F_H) | F_H = γ × h̄ × A (full area of plane) | F_H = γ × h̄ × A_vert (vertical projection only) |
| Vertical component (F_V) | F_V = 0 (no vertical component; pressure acts only perpendicular to plane) | F_V = γ × V (weight of real or imaginary fluid above) |
| Location of force (F_H) | At center of pressure: y_p = ȳ + I_g/(ȳ×A) | Same formula applied to the vertical projection |
| Location of force (F_V) | N/A | At the centroid of the volume above (real or virtual) |
| Resultant magnitude | F = F_H (acts perpendicular to plane) | F = √(F_H² + F_V²) |
| Common example | Vertical dam, inclined sluice gate, submerged tank wall | Spillway arch, circular-arc gate, dome roof, spherical tank |
Manometer Types — When to Use
| Manometer Type | Setup | Measures | Advantage | Limitation |
|---|---|---|---|---|
| Piezometer (simple tube) | Tube connected to pressure point, open to atmosphere | Gauge pressure only at that point | Simplest, no calculation needed; h = p/γ | Cannot measure negative (vacuum) pressures; long tube for high pressures |
| U-tube (open both ends) | Two limbs connected at bottom, connected to two pressure points | Pressure difference between two points | Works for both positive and negative pressure differences | Both ends must be accessible; symmetric design may be bulky |
| Differential (with gage fluid) | U-tube with denser fluid (mercury) in bottom, water above | Large pressure differences with compact display | High sensitivity; mercury (s=13.6) gives 13.6× smaller deflection than water | Must account for multiple fluid columns; mercury is toxic |
| Inverted (vacuum at top) | Tube sealed at top with vacuum, open at bottom to pressure point | Negative gauge pressure (suction) | Measures vacuums and low positive pressures | Requires vacuum seal; limited to ~10 m of water equivalent |
Center of Pressure — Key Differences from Centroid
| Characteristic | Centroid (ȳ) | Center of Pressure (y_p) |
|---|---|---|
| Definition | Geometric center of the area; balance point if area had uniform density | Point where total hydrostatic force acts; accounts for varying pressure with depth |
| Calculation | By geometry (ȳ = ∫y dA / A) | y_p = ȳ + I_g/(ȳ×A) |
| Location relative to centroid | Reference point | Always BELOW centroid (for submerged surfaces) |
| Depth from surface (vertical gate) | ȳ = h/2 | y_p = 2h/3 (for surface-piercing rectangle) |
| Physical meaning | Where you could concentrate the entire area for moment calculations (in statics of rigid bodies) | Where the pressure force acts; critical for determining overturning moments and gate behavior |
| Depends on | Shape only (independent of pressure) | Shape AND depth of submersion (pressure variation) |
Unit Weights of Common Fluids
| Fluid | Unit Weight (kN/m³) | Specific Gravity (s) | Notes |
|---|---|---|---|
| Water (fresh, 4°C) | 9.81 | 1.0 | Standard reference; used in most problems unless stated otherwise |
| Mercury | 133.1 | 13.6 | Very dense; used in manometers for high sensitivity and compact design |
| Oil (typical) | ~8.84 (s ≈ 0.9) | ~0.9 | Lighter than water; common in hydraulic systems |
| Seawater | ~10.05 (s ≈ 1.025) | ~1.025 | Slightly denser than fresh water; used for coastal/marine problems |
Diagrams
Hydrostatic Force Calculation Decision Tree
A decision tree guiding whether to use plane or curved surface methods, and the sequence of calculations for each. Essential for exam time-management and avoiding wrong approach.
Hydrostatic Pressure and Forces — Complete Concept Map
A comprehensive mind map showing all major topics, formulas, and connections. Use this to see how concepts relate and ensure you have covered every topic before the exam.
Manometer Problem-Solving Flowchart
Step-by-step flowchart for solving any manometer problem. Key decision points include identifying the manometer type and catching sign/unit errors early.
Pressure and Force State Transitions
A state diagram showing how pressure develops from the free surface downward and how it leads to force and moment considerations. Useful for understanding the causal chain in hydrostatic problems.