Determine the total force and location of centre of pressure on one face of the plate shown in the Figure below, immersed in a liquid of specific gravity 0.9.

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Hollow Circular Plate Problem

Problem Statement

Determine the total force and location of centre of pressure on one face of the plate shown in the Figure below, immersed in a liquid of specific gravity 0.9.

Given Data

  • External Diameter, \( D = 2 \, \text{m} \)
  • Internal Diameter, \( d = 1 \, \text{m} \)
  • Depth of top edge = 2 m
  • Liquid S.G. = 0.9, \( \rho = 900 \, \text{kg/m}^3 \)

Diagram of the Plate

Diagram of a submerged hollow circular plate

Solution

(i) Total Force (\(F\))

First, calculate the area of the hollow plate.

$$ A = \frac{\pi}{4} (D^2 - d^2) $$ $$ A = \frac{\pi}{4} (2^2 - 1^2) $$ $$ A = \frac{\pi}{4} (3) \approx 2.356 \, \text{m}^2 $$

Next, find the depth of the plate's centroid (\(\bar{h}\)).

$$ \bar{h} = (\text{Depth of top edge}) + \frac{D}{2} $$ $$ \bar{h} = 2 + \frac{2}{2} $$ $$ \bar{h} = 3 \, \text{m} $$

Now, calculate the total force on the plate.

$$ F = \rho g A \bar{h} $$ $$ F = 900 \times 9.81 \times 2.356 \times 3 $$ $$ F \approx 62412 \, \text{N} $$

(ii) Location of Centre of Pressure (\(h^*\))

The vertical depth of the centre of pressure is given by:

$$ h^* = \frac{I_G}{A \bar{h}} + \bar{h} $$

First, calculate the moment of inertia (\(I_G\)) for the hollow circular plate.

$$ I_G = \frac{\pi}{64} (D^4 - d^4) $$ $$ I_G = \frac{\pi}{64} (2^4 - 1^4) $$ $$ I_G = \frac{\pi}{64} (15) \approx 0.736 \, \text{m}^4 $$

Now, substitute all values into the formula for \(h^*\).

$$ h^* = \frac{0.736}{2.356 \times 3} + 3 $$ $$ h^* = \frac{0.736}{7.068} + 3 $$ $$ h^* \approx 0.104 + 3 $$ $$ h^* \approx 3.104 \, \text{m} $$
Final Results:

Total Force: \( F \approx 62.41 \, \text{kN} \).

Location of Centre of Pressure: \( h^* \approx 3.104 \, \text{m} \) from the free surface.

Explanation of Concepts

Area of a Hollow Shape: To find the area of the plate that is in contact with the liquid, we calculate the area of the outer circle and subtract the area of the inner circle.

Moment of Inertia for a Hollow Shape: Similarly, the moment of inertia about the centroid (\(I_G\)) is found by calculating the moment of inertia for the outer circle and subtracting the moment of inertia for the inner circle.

Centre of Pressure: The centre of pressure is the point where the total hydrostatic force acts. Because fluid pressure increases with depth, this point is always located below the centroid for a submerged vertical surface. The calculation confirms this, showing the centre of pressure is about 10.4 cm below the plate's geometric center.

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