A caisson for closing the entrance to a dry dock is of trapezoidal form, 16 m wide at the top and 12 m wide at the bottom, and 8 m deep. Find the total pressure and centre of pressure on the caisson if the water on the outside is 1 m below the top level of the caisson and the dock is empty.

Leave a Comment / Hydrostatic Forces on Surfaces / By
Trapezoidal Caisson Pressure Problem

Problem Statement

A caisson for closing the entrance to a dry dock is of trapezoidal form, 16 m wide at the top and 12 m wide at the bottom, and 8 m deep. Find the total pressure and centre of pressure on the caisson if the water on the outside is 1 m below the top level of the caisson and the dock is empty.

Given Data

  • Caisson Top Width = 16 m
  • Caisson Bottom Width, \( b = 12 \, \text{m} \)
  • Caisson Depth = 8 m
  • Water Depth, \( h_{water} = 8 - 1 = 7 \, \text{m} \)
  • Fluid is water, \( \rho = 1000 \, \text{kg/m}^3 \)

Solution

First, we must determine the properties of the submerged portion of the trapezoid.

1. Dimensions of Submerged Area

The caisson width changes by \(16 - 12 = 4\) m over a depth of 8 m. The rate of change is \(4 / 8 = 0.5\) m/m. Since the water is 1 m from the top, the width at the water surface (\(a\)) is:

$$ a = 16 - (1 \, \text{m} \times 0.5 \, \text{m/m}) $$ $$ a = 15.5 \, \text{m} $$

The wetted area \(A\) is a trapezoid of height 7 m with parallel sides 15.5 m and 12 m.

$$ A = \frac{a+b}{2} \times h_{water} $$ $$ A = \frac{15.5 + 12}{2} \times 7 $$ $$ A = 96.25 \, \text{m}^2 $$

2. Depth of Centroid (\(\bar{h}\))

The depth of the centroid of the wetted area from the free surface ('a') is given by:

$$ \bar{h} = \left(\frac{h_{water}}{3}\right) \left( \frac{a + 2b}{a + b} \right) $$ $$ \bar{h} = \left(\frac{7}{3}\right) \left( \frac{15.5 + 2(12)}{15.5 + 12} \right) $$ $$ \bar{h} = \left(\frac{7}{3}\right) \left( \frac{39.5}{27.5} \right) $$ $$ \bar{h} \approx 3.3515 \, \text{m} $$

3. Total Pressure (Force, \(F\))

$$ F = \rho g A \bar{h} $$ $$ F = 1000 \times 9.81 \times 96.25 \times 3.3515 $$ $$ F \approx 3164608 \, \text{N} $$ $$ F \approx 3.165 \, \text{MN} $$

4. Centre of Pressure (\(h^*\))

The moment of inertia about the centroid (\(I_G\)) for a trapezoid is:

$$ I_G = \frac{h_{water}^3}{36} \left( \frac{a^2 + 4ab + b^2}{a+b} \right) $$ $$ I_G = \frac{7^3}{36} \left( \frac{15.5^2 + 4(15.5)(12) + 12^2}{15.5+12} \right) $$ $$ I_G = 9.528 \times \left( \frac{240.25 + 744 + 144}{27.5} \right) $$ $$ I_G = 9.528 \times \left( \frac{1128.25}{27.5} \right) $$ $$ I_G \approx 390.96 \, \text{m}^4 $$

Now, we find the depth of the centre of pressure using the parallel axis theorem.

$$ h^* = \frac{I_G}{A\bar{h}} + \bar{h} $$ $$ h^* = \frac{390.96}{96.25 \times 3.3515} + 3.3515 $$ $$ h^* = \frac{390.96}{322.59} + 3.3515 $$ $$ h^* \approx 1.212 + 3.3515 $$ $$ h^* \approx 4.564 \, \text{m} $$
Final Results:

Total Pressure (Force): \( F \approx 3.165 \, \text{MN} \).

Depth of Centre of Pressure: \( h^* \approx 4.564 \, \text{m} \) below the free water surface.

Explanation of Concepts

Wetted Area: The first step is to determine the actual area of the caisson that is in contact with the water. Since the water is below the top edge, we must calculate the width of the trapezoid at the water's surface to find the dimensions of the submerged part.

Centroid of a Trapezoid: Unlike a simple rectangle, the centroid of a trapezoid is not at half its height. Its vertical position depends on the lengths of the two parallel sides. The formula \( \bar{h} = \frac{h}{3} \frac{a+2b}{a+b} \) calculates the distance of the centroid from the side with length 'a'. In this case, 'a' is the water surface width, so this formula directly gives the centroid's depth.

Centre of Pressure: The centre of pressure is where the resultant hydrostatic force acts. For a complex shape like a trapezoid, calculating the moment of inertia about the centroid (\(I_G\)) is a key step. This value reflects how the area is distributed. The final position (\(h^*\)) is always deeper than the centroid (\(\bar{h}\)) for a vertically submerged surface.

Related Posts

Leave a Comment

Your email address will not be published. Required fields are marked *

Scroll to Top