Fluid Statics and Pressure Measurement

Introduction

Fluid statics examines fluids at rest and the forces produced by them. Although a stationary fluid has no shear stress, it can exert substantial normal forces on tanks, retaining walls, dams, gates, foundations, pipelines, and building services. Correct pressure evaluation is therefore fundamental to construction technology, hydraulic engineering, water-supply systems, drainage, fire protection, and geotechnical works.

This chapter develops the hydrostatic pressure law, distinguishes absolute and gauge pressure, explains pressure-measuring instruments, and determines forces and lines of action on plane and curved surfaces. It also introduces buoyancy, flotation, and stability, which are essential when assessing tanks, pontoons, submerged structures, pipelines, and temporary works.

Learning objectives

After studying this chapter, you should be able to:

• explain how pressure varies in a fluid at rest;
• convert between absolute, gauge, and vacuum pressure;
• analyze piezometers, manometers, and pressure gauges;
• calculate hydrostatic forces and centers of pressure on plane surfaces;
• determine resultant forces on curved surfaces;
• apply Archimedes’ principle to buoyancy and flotation;
• assess the stability of submerged and floating bodies;
• identify assumptions, measurement errors, and practical limitations.

1. Pressure in a fluid at rest

Pressure is the normal force per unit area. At a point in a static fluid it is isotropic: its magnitude is the same in every direction. This follows from considering an infinitesimally small fluid element. If pressure differed by direction, an unbalanced moment or force would cause the element to rotate or translate.

Here p is pressure in pascals (Pa), F_n is normal force in newtons (N), and A is area in square metres (m²). One kilopascal equals 1,000 Pa. The definition applies to a sufficiently small area over which pressure is uniform; for a finite surface, force is obtained by integration.

For a fluid of density ρ in a gravitational field, the vertical equilibrium equation is

where z is vertical elevation measured upward and g is gravitational acceleration, approximately 9.81 m/s² near Earth’s surface. If density is constant, integration gives

or, using depth h below a free surface exposed to pressure p₀,

This equation assumes a fluid at rest, constant density, uniform gravitational acceleration, and negligible capillary effects. It shows that pressure increases linearly with depth and is independent of container shape. The quantity ρgh is the hydrostatic pressure increase relative to the free surface.

For water, taking ρ = 1000 kg/m³,

Thus each metre of water depth produces approximately 9.81 kPa of pressure increase. For seawater, a typical design value is ρ ≈ 1025 kg/m³, although the actual value depends on temperature and salinity.

Pressure head

Pressure can be expressed as an equivalent fluid-column height, called pressure head:

Pressure head has units of metres of the selected fluid. It is useful in hydraulic calculations, but the fluid must always be stated; 10 m of water is not equivalent to 10 m of mercury in physical column height.

Pressure surfaces and connected fluids

In a single homogeneous fluid at rest, points at the same elevation have the same pressure. In connected fluids of different densities, pressure remains continuous across an interface, but pressure gradients differ because dp/dz depends on density. A free surface is horizontal when the only body force is gravity.

2. Absolute, gauge, and vacuum pressure

Absolute pressure is measured relative to a perfect vacuum. Gauge pressure is measured relative to local atmospheric pressure:

where p_abs is absolute pressure, p_g is gauge pressure, and p_atm is atmospheric pressure. Vacuum pressure is used when the pressure is below atmospheric pressure:

A gauge reading of zero does not indicate zero molecular pressure; it indicates equality with the surrounding atmosphere. Absolute pressure is required in thermodynamics, cavitation assessment, and gas-law calculations. Gauge pressure is usually sufficient for forces on surfaces exposed to the same atmosphere on their opposite side, because atmospheric contributions cancel.

Atmospheric pressure varies with altitude and weather. The standard sea-level value is approximately 101.325 kPa absolute, but site-specific measurements should be used for precision work.

Worked example 1: pressure at depth

A water tank is open to the atmosphere. Find the gauge and absolute pressure 4.5 m below the free surface, taking atmospheric pressure as 101.3 kPa.

Using ρ = 1000 kg/m³ and g = 9.81 m/s²,

Therefore,

The result is valid because the tank is open and the water density is treated as constant.

3. Pressure-measuring instruments

Piezometer

A piezometer is a vertical transparent tube connected to a liquid system. The liquid rises until its hydrostatic column balances the pressure. For a liquid connected to a reservoir open to atmosphere,

A piezometer is simple and accurate for positive pressures in liquids, but it is unsuitable for gases, vacuum pressures, very high pressures, or situations where the required tube height is excessive.

U-tube manometer

A U-tube manometer uses a manometric liquid, often mercury or another dense liquid, to compare pressures. The analysis follows a pressure walk: pressure increases when moving downward through a fluid by ρg∆h and decreases when moving upward.

For a simple manometer measuring the pressure of a liquid relative to atmosphere, with manometric-liquid density ρ_m and connected-fluid density ρ, a common arrangement gives

The exact signs and heights depend on the geometry. The reliable procedure is to begin at a known pressure, move through each column, and equate the pressure at the same level in the continuous manometric liquid. This avoids memorizing geometry-specific formulas.

A differential manometer measures the pressure difference between two points. If the connecting fluid has density ρ and the manometric liquid has density ρ_m, the pressure difference includes both the manometric-column contribution and the elevation difference of the tapping points. Elevation must never be ignored when the pressure taps are at different levels.

Inclined manometer

An inclined tube magnifies the movement of the liquid meniscus. If the inclined length is L and the tube angle to the horizontal is θ, the vertical head is

It is useful for small pressure differences, but it is more sensitive to reading errors, temperature effects, and meniscus interpretation.

Bourdon pressure gauge

A Bourdon gauge contains a curved elastic tube. Internal pressure tends to straighten the tube, and mechanical linkage converts the movement into a pointer reading. It is robust and widely used in construction equipment and water systems. Calibration, zero error, hysteresis, vibration, and mechanical damage must be considered.

Pressure transducers

Electronic transducers convert pressure into an electrical signal. Strain-gauge, piezoresistive, capacitive, and piezoelectric devices are common. Digital sensors permit data logging and remote monitoring, but require correct range selection, calibration, electrical protection, temperature compensation, and appropriate installation. A sensor should not be used beyond its rated pressure or chemical compatibility.

4. Hydrostatic force on plane surfaces

A plane submerged surface experiences a distributed pressure. The resultant force equals the integral of pressure over the area:

If both sides are exposed to the same atmosphere, atmospheric pressure cancels and gauge pressure may be used. For a plane surface of area A whose centroid is at vertical depth h_c below the free surface,

This is valid for a constant-density liquid and a planar surface. The force acts normal to the surface through the center of pressure, not generally through the centroid.

Center of pressure

The center of pressure is the point through which the resultant hydrostatic force acts. Taking moments about the free-surface line parallel to the surface width gives

where y_p is the vertical depth of the center of pressure, y_c is the centroid depth, and I_G is the second moment of area about the horizontal centroidal axis parallel to the free surface. This form applies to a plane surface in a liquid with a horizontal free surface and vertical depth coordinates.

For an inclined plane, if θ is the angle between the plane and the free surface, and y is distance measured along the plane from the free-surface intersection,

The vertical depth is h_p = y_p sin θ. The center of pressure lies below the centroid for a fully submerged surface because pressure increases with depth. For a horizontal plane at one depth, pressure is uniform and the center of pressure coincides with the centroid.

The second moment of area depends on geometry. For a rectangle of width b and height d, about its horizontal centroidal axis,

Worked example 2: rectangular inspection hatch

A vertical rectangular hatch is 1.2 m wide and 1.5 m high. Its top edge is 0.8 m below the water surface. Determine the resultant force and center-of-pressure depth.

Area:

Centroid depth:

Resultant force:

For the rectangle,

Thus,

The resultant acts normal to the hatch at approximately 1.67 m below the free surface. Hinge reactions would additionally require force and moment equilibrium of the hatch.

5. Hydrostatic forces on curved surfaces

For a curved surface, the resultant force is conveniently resolved into horizontal and vertical components.

• The horizontal component equals the hydrostatic force on the vertical projection of the curved surface:

where A_v is the vertical projected area and h_c is its centroid depth.

• The vertical component equals the weight of the imaginary fluid directly above the curved surface up to the free surface, including its direction. For a submerged surface, this is often called the fluid-volume method:

where V is the relevant imaginary fluid volume. The line of action passes through the centroid of that volume. The resultant magnitude is

and its inclination to the horizontal is

The signs of the components depend on whether the fluid is above or below the surface and on the chosen coordinate directions. Atmospheric pressure should be included only when it does not cancel with pressure on the opposite side.

This method is particularly useful for curved tank walls, sluice gates, pipe bends, dam faces, and submerged shells. It is not correct to apply the plane-surface formula directly to a curved surface.

6. Buoyancy and flotation

Archimedes’ principle states that a body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced:

where ρ_f is fluid density and V_d is displaced volume. The buoyant force acts through the center of buoyancy, which is the centroid of the displaced fluid volume.

For a fully submerged body of constant volume in a uniform fluid, the buoyant force is independent of depth, provided the fluid density remains constant and compressibility is negligible. Pressure itself increases with depth, but the net difference between the lower and upper pressure forces remains equal to the displaced-fluid weight.

A body floats when its weight equals buoyancy:

Consequently, the submerged fraction of a uniform floating body is related to the density ratio:

where ρ_b is the average density of the body, including enclosed air spaces. This relation assumes static equilibrium and a single uniform surrounding fluid.

Stability of submerged bodies

For a fully submerged body, small angular stability depends on the relative positions of the center of gravity G and center of buoyancy B. If the center of gravity is below the center of buoyancy, the body tends to return to its original orientation. If G is above B, it tends to overturn. Coincidence gives neutral stability in the idealized case.

Stability of floating bodies

When a floating body tilts slightly, the displaced volume changes shape and the center of buoyancy shifts. The new buoyant force intersects the original vertical axis at the metacenter M. The metacentric height is

where BM is the distance from the center of buoyancy to the metacenter and BG is the distance between buoyancy and gravity centers. For a floating body,

where I_W is the second moment of the water-plane area about the axis of heel and V_d is displaced volume.

Static initial stability requires

If GM < 0, a small tilt produces an overturning tendency. A large positive GM improves initial stability but may cause abrupt motions and high structural loads. Stability assessment must also consider free-surface effects in partially filled tanks, wind, waves, mooring forces, cargo shifts, and large-angle behavior.

The approximate restoring moment for a small heel angle φ is

For small angles, sin φ ≈ φ when φ is expressed in radians. This is an initial-stability approximation, not a complete analysis of capsizing resistance.

7. Applied case: stability of a construction pontoon

Background

Consider an illustrative steel pontoon used to support a small crane during work beside a bridge. The pontoon is rectangular, 12 m long, 4 m wide, and 1.5 m deep. Its loaded mass is 120 tonnes. The center of gravity is 0.9 m above the bottom. The pontoon floats in fresh water, and the water-plane dimensions remain approximately 12 m by 4 m for the operating condition.

Problem

Determine the operating draft and assess initial transverse stability. Assume a prismatic displacement volume and neglect waves, mooring forces, structural deformation, and free-surface corrections.

Analysis and solution

The loaded weight is

The required displaced volume is

The water-plane area is

Therefore the draft is

This exceeds the stated pontoon depth of 1.5 m, so the proposed loading is physically impossible under the assumptions. The pontoon would be submerged before displacing 120 m³. Its maximum prismatic displacement at the top edge is only

corresponding to a mass of 72 tonnes in fresh water, before allowing freeboard or safety margin.

Outcome and engineering decision

The pontoon is under-capacity by a substantial margin. The crane and cargo must be reduced, the pontoon dimensions increased, or additional pontoons connected. A proper design would then calculate freeboard, trim, longitudinal and transverse stability, local deck strength, connection forces, wave effects, wind loads, crane overturning moments, and regulatory load cases.

Learning points

The example illustrates that buoyancy is not merely a force calculation. It is a feasibility constraint. A design that satisfies equilibrium algebraically may still fail because the required draft exceeds available depth or safe freeboard. Construction decisions must combine hydrostatics with structural, operational, and safety requirements.

8. Practical limitations, safety, and best practice

Pressure instruments should be selected according to range, accuracy class, temperature, chemical compatibility, vibration, and transient pressure. A gauge should normally operate within the central portion of its range rather than continuously at its limit. Isolation valves and snubbers may protect instruments from surges. Pressure lines should be routed to avoid trapped air, sediment accumulation, freezing, and thermal expansion.

Manometer readings require attention to meniscus shape, parallax, density variation, and zero position. Mercury is toxic and must be handled under applicable environmental and occupational regulations; safer alternatives may be appropriate. Electronic sensors require calibration traceability and protection from moisture and electromagnetic interference.

Hydrostatic calculations often fail because engineers use the wrong reference pressure, omit elevation differences, confuse mass density with specific weight, or use a centroid instead of a center of pressure. Every calculation should state the fluid, density, datum, pressure reference, geometry, and assumptions.

For temporary works and construction sites, stored water pressure can release energy suddenly. Tanks, cofferdams, gates, hoses, and pipelines require rated components, controlled filling and emptying, inspection, and safe exclusion zones. Pressure testing should follow project specifications and relevant codes. Instrument data used for compliance or public safety should be retained with calibration records.

Computational fluid dynamics and digital monitoring can extend hydrostatic analysis to variable-density fluids, transient loading, and complex geometry. However, software does not remove the need to verify boundary conditions, mesh quality, material properties, sensor calibration, and conservation of mass and force. Simple hand calculations remain essential for checking model output.

Chapter Summary

Fluid statics is governed by the balance of pressure forces and body forces. In a constant-density fluid at rest, pressure increases linearly with depth according to p = p₀ + ρgh. Absolute pressure is referenced to vacuum, while gauge pressure is referenced to atmosphere. Piezometers, manometers, Bourdon gauges, and electronic transducers measure pressure in different operating ranges and with different limitations.

The resultant force on a plane submerged surface equals ρg h_c A and acts at the center of pressure, generally below the centroid. Curved-surface forces are resolved into horizontal and vertical components using projected area and imaginary-fluid-volume methods. Archimedes’ principle states that buoyancy equals the weight of displaced fluid. Floating equilibrium requires weight to equal buoyancy, while initial stability depends on the relative positions of the center of gravity, center of buoyancy, and metacenter. Sound engineering practice combines these equations with explicit assumptions, measurement quality, safety controls, and regulatory requirements.

Key Terms

Pressure: Normal force per unit area.

Hydrostatic pressure: Pressure in a fluid at rest caused by fluid weight and applied pressure.

Gauge pressure: Pressure measured relative to atmospheric pressure.

Absolute pressure: Pressure measured relative to a perfect vacuum.

Vacuum pressure: Amount by which absolute pressure is below atmospheric pressure.

Pressure head: Equivalent height of a specified fluid column.

Specific weight: Weight per unit volume, equal to ρg.

Piezometer: Vertical liquid column used to measure positive pressure in a liquid.

Manometer: Instrument that determines pressure from liquid-column differences.

Hydrostatic force: Resultant force produced by fluid pressure on a surface.

Center of pressure: Point through which the resultant hydrostatic force acts.

Second moment of area: Geometric quantity used to determine the distribution of area about an axis.

Buoyancy: Upward force exerted by a fluid on an immersed body.

Center of buoyancy: Centroid of the displaced-fluid volume.

Flotation: Condition in which a body is supported by buoyancy at the fluid surface.

Metacenter: Point used to assess the initial stability of a floating body.

Metacentric height: Distance between the metacenter and center of gravity.

Freeboard: Vertical distance from a floating body’s waterline to its upper edge.

Differential pressure: Difference between pressures at two locations.

Specific gravity: Ratio of a substance’s density to a reference density.

Review Questions

Short Answer Questions

1. Why is pressure at a point in a static fluid isotropic?
2. Distinguish absolute pressure from gauge pressure.
3. State the assumptions underlying p = p₀ + ρgh.
4. Why is the center of pressure below the centroid of a vertical submerged rectangle?
5. State Archimedes’ principle.

Long Answer Questions

1. Derive the hydrostatic pressure relation for a constant-density fluid and explain the meaning of each term.
2. Explain the operation, advantages, and limitations of piezometers, U-tube manometers, and Bourdon gauges.
3. Derive the expression for the center of pressure on a vertical plane surface.
4. Explain how horizontal and vertical components of force are determined on a curved surface.
5. Discuss the stability of submerged and floating bodies using the center of gravity, center of buoyancy, and metacenter.

Analytical / Application Questions

1. A vertical rectangular floodgate is 2 m wide and 3 m high, with its top edge 1 m below the water surface. Calculate the resultant force and center-of-pressure depth.
2. A differential manometer connects two water pipelines whose tapping points are at different elevations. Develop a pressure-walk procedure and identify the data required before calculating the pressure difference.
3. A floating work platform is modified by adding a heavy crane at one side. Analyze how the added mass affects draft, center of gravity, metacentric height, heel, and operational safety.

Multiple Choice Questions with Answers

Question 1

For a constant-density fluid at rest, how does pressure change with depth?

A. It decreases linearly

B. It increases linearly

C. It remains constant

D. It increases exponentially

Correct Answer: B. It increases linearly

Question 2

A pressure gauge reads 250 kPa gauge when atmospheric pressure is 101 kPa. What is the approximate absolute pressure?

A. 149 kPa

B. 250 kPa

C. 351 kPa

D. 25.6 kPa

Correct Answer: C. 351 kPa

Question 3

The resultant hydrostatic force on a plane surface is calculated using which centroid depth?

A. The shallowest point depth

B. The deepest point depth

C. The centroid depth

D. The center-of-pressure depth

Correct Answer: C. The centroid depth

Question 4

For a vertical submerged plane surface, the center of pressure is generally located:

A. Above the centroid

B. At the free surface

C. Below the centroid

D. Outside the fluid

Correct Answer: C. Below the centroid

Question 5

A piezometer is generally unsuitable for measuring:

A. Positive pressure in water

B. Small liquid pressure heads

C. Gas pressure

D. Pressure in an open liquid system

Correct Answer: C. Gas pressure

Question 6

The buoyant force on a submerged body equals:

A. The body’s mass

B. The weight of displaced fluid

C. The pressure at the free surface

D. The weight of the container

Correct Answer: B. The weight of displaced fluid

Question 7

A floating body is in vertical equilibrium when:

A. Its weight exceeds buoyancy

B. Buoyancy exceeds its weight

C. Its weight equals buoyancy

D. Its center of gravity is at the free surface

Correct Answer: C. Its weight equals buoyancy

Question 8

For initial stability of a floating body, which condition is required?

A. GM < 0

B. GM = −1

C. GM > 0

D. BM = 0

Correct Answer: C. GM > 0

Question 9

Water has density 1000 kg/m³. What gauge pressure is produced by a 3 m water column? Take g = 9.81 m/s².

A. 2.94 kPa

B. 9.81 kPa

C. 29.43 kPa

D. 300 kPa

Correct Answer: C. 29.43 kPa

Question 10

The horizontal component of force on a curved submerged surface equals the force on its:

A. Horizontal projection

B. Vertical projection

C. Curved perimeter only

D. Centroidal axis

Correct Answer: B. Vertical projection