A rectangular plane surface is 2 m wide and 3 m deep. It lies in a vertical plane in water. Determine the total pressure and position of the centre of pressure on the plane surface when its upper edge is horizontal and (a) coincides with the water surface, and (b) is 2.5 m below the free water surface.

Methodology Rationale

The method used here is fundamental in fluid mechanics for analyzing forces on submerged objects. The key idea is that the pressure exerted by a fluid is not uniform over a vertical surface; it increases linearly with depth (P = ρgh). Therefore, we cannot simply multiply pressure by area to find the force.

• Total Pressure (F): To find the total force, we use the pressure at the centroid (center of gravity, ħ) of the surface. This gives the average pressure acting on the area, and multiplying it by the total area (A) gives the resultant hydrostatic force.
• Centre of Pressure (h*): This is the point where the resultant force (F) effectively acts. Because pressure is greater at the bottom of the surface than at the top, this point is always located below the centroid (ħ). The formula for h* accounts for this non-uniform pressure distribution using the moment of inertia (I_G), which describes how the area’s shape is distributed relative to its centroid.

Summary of Calculated Steps

In this problem, we first calculated the properties for a plane at the water’s surface, finding a total force of 88.3 kN acting at a depth of 2.0 m. When the same plane was submerged 2.5 m deeper, the pressure at every point on the surface increased. This resulted in a significantly larger total force of 235.4 kN. The center of pressure also moved deeper to 4.1875 m, but it moved closer to the centroid (4.0 m), as the relative difference in pressure between the top and bottom of the plane became less significant compared to the overall pressure head.

Applications in Engineering

This type of analysis is not just academic; it is critical in the design and safety assessment of numerous real-world engineering structures. Knowing the magnitude of the hydrostatic force and its exact point of application is essential to prevent structural failure.

• Dam Design: Engineers must calculate the total pressure on the face of a dam to ensure it is thick and strong enough to withstand the immense force of the reservoir water. The location of the center of pressure helps determine where the dam experiences the most stress.
• Sluice Gates & Canal Locks: The gates used to control water flow in rivers and canals must be designed to handle the hydrostatic force. The operating mechanisms (hoists, motors) must be powerful enough to overcome this force.
• Water Tanks: The walls of large water storage tanks (both rectangular and cylindrical) are designed based on these principles to prevent them from bursting.
• Shipbuilding: The hull of a ship is a submerged surface. Naval architects perform these calculations to ensure the hull’s integrity and stability in the water.
• Submarines & Underwater Structures: For deep-sea structures, where pressure is enormous, these calculations are vital for survival.