Centre of Pressure Calculator

Determine the location where the total pressure force acts on a submerged vertical surface.

Hydrostatic Force and Centre of Pressure Calculator

What Principle Is Used for Calculating the Centre of Pressure?

The calculation of the centre of pressure relies on the Principle of Moments. This principle states that the moment of the resultant force about any axis is equal to the sum of the moments of the individual pressure forces about the same axis. In fluid statics, the fluid exerts a pressure that varies with depth. To find the single point where the total force from this pressure field can be considered to act, we integrate the pressure over the area and then find the “balance point” of that force distribution. This point is the centre of pressure.

Essentially, we are finding the weighted average location of the pressure. The “weight” at any given point is the magnitude of the pressure at that depth. Because pressure increases linearly with depth in a uniform fluid, the centre of pressure will always be located below the geometric centroid of the submerged area. This concept is fundamental in engineering design for structures like dams, gates, and ship hulls.

Centre of Pressure Formula and Explanation

For a vertically submerged rectangular plate, the principle of moments leads to a specific formula for the depth of the centre of pressure (h_cp) from the fluid surface.

The formula is derived by equating the moment from the resultant force (F_R * h_cp) to the integral of the moments from the differential forces (∫ h * dF). For a rectangular plate, this simplifies to:

h_cp = (I_c / (y_c * A)) + y_c

Where the variables are defined as:

Variables for Centre of Pressure Calculation

Variable	Meaning	Unit (Metric)	Typical Range
h_cp	Depth of the Centre of Pressure	meters (m)	h1 to (h1+Height)
I_c	Second moment of area about the centroid	meters⁴ (m⁴)	> 0
y_c	Depth of the Centroid of the area	meters (m)	> 0
A	Total Area of the submerged surface	meters² (m²)	> 0

Practical Examples

Example 1: Metric Units

Consider a rectangular gate in a freshwater reservoir.

• Inputs:
  • Fluid Density (water): 1000 kg/m³
  • Top Edge Depth: 2 m
  • Plate Height: 4 m
  • Plate Width: 5 m
• Results:
  • Centroid Depth (y_c): 2 + (4 / 2) = 4 m
  • Total Hydrostatic Force (F_R): ~784,800 N
  • Centre of Pressure (h_cp): ~4.33 m

Example 2: Imperial Units

Imagine a metal plate submerged in oil.

• Inputs:
  • Fluid Density (oil): 57.4 lb/ft³
  • Top Edge Depth: 3 ft
  • Plate Height: 5 ft
  • Plate Width: 6 ft
• Results:
  • Centroid Depth (y_c): 3 + (5 / 2) = 5.5 ft
  • Total Hydrostatic Force (F_R): ~9,520 lbf
  • Centre of Pressure (h_cp): ~5.88 ft

How to Use This Centre of Pressure Calculator

Follow these steps to accurately determine the centre of pressure:

1. Select Unit System: Choose between Metric (meters, kilograms) and Imperial (feet, pounds) units. The labels and default values will update automatically.
2. Enter Fluid Density: Input the density of the fluid the object is submerged in. The default is 1000 kg/m³ for water.
3. Define Plate Position: Enter the ‘Top Edge Depth’, which is the distance from the fluid surface to the top of your plate.
4. Define Plate Dimensions: Input the ‘Plate Height’ and ‘Plate Width’. The calculator assumes a rectangular shape.
5. Calculate: Click the “Calculate” button to see the results. The primary result is the Centre of Pressure depth, with intermediate values for Total Force, Centroid Depth, and Moment also displayed. The diagram will update to reflect your inputs.

Key Factors That Affect the Centre of Pressure

• Submersion Depth: The deeper the object is submerged, the greater the pressure, and the farther the centre of pressure will be from the centroid.
• Object Geometry: The shape of the submerged area is critical. The formula used here is for a rectangle; other shapes like triangles or circles have different second moments of area (I_c), which changes the result.
• Fluid Density: A denser fluid exerts more force at any given depth, which increases the total force and moment, though it doesn’t change the geometric position of the centre of pressure relative to the centroid.
• Angle of Inclination: This calculator assumes a vertical surface. If the surface is inclined, the calculations for depth and the moment arm become more complex.
• Free Surface Location: All depth measurements are relative to the free surface of the fluid, where pressure is considered zero (gauge pressure).
• Uniform Gravity: The calculations assume a constant gravitational field, which is a safe assumption for all terrestrial engineering applications.

Understanding is also crucial in many contexts.

Frequently Asked Questions (FAQ)

Why is the centre of pressure below the centroid?

Because pressure increases with depth, the lower portion of the submerged surface experiences greater force than the upper portion. This “bottom-heavy” distribution of force shifts the effective point of action (the centre of pressure) downwards, below the geometric center (the centroid).

What happens if the top edge of the plate is at the surface?

If the top edge depth is zero, the pressure distribution is a simple triangle. For a rectangle in this case, the centre of pressure will be exactly two-thirds of the way down the plate from the surface.

Does the width of the plate affect the centre of pressure’s depth?

No. For a vertically submerged rectangle, the width influences the total force and moment, but it cancels out of the final equation for the depth (h_cp). The depth is determined by the plate’s height and its position relative to the fluid surface.

Can the centre of pressure be outside the body?

Yes, particularly in aerodynamics. For certain shapes and flow conditions, the net moment can be so significant that the theoretical point of action lies outside the physical boundaries of the object.

What is the difference between centre of pressure and centre of gravity?

The centre of pressure is where the total aerodynamic or hydrostatic force acts, while the centre of gravity is where the object’s weight acts. The relationship between these two points is critical for the stability of objects like airplanes and rockets.

Does this calculation apply to moving fluids?

No, this calculator is for fluid statics (stationary fluids). When a fluid is in motion, the principles of apply, and other forces like drag come into play, which can shift the centre of pressure.

How is the second moment of area (I_c) calculated?

For a rectangle, I_c = (width * height³) / 12. This value represents the shape’s resistance to bending and is crucial for determining how pressure is distributed relative to the centroid.

What is the ‘Principle of Moments’?

The Principle of Moments, also known as Varignon’s theorem, states that the moment of a resultant force about a point is equal to the sum of the moments of its components about the same point. This is the core concept used to locate the centre of pressure.