Hydrostatic forces act on submerged surfaces due to fluid pressure. In static fluids, pressure increases linearly with depth according to the equation P equals rho g h, where rho is fluid density, g is gravitational acceleration, and h is depth. Pascal's principle states that pressure is transmitted equally in all directions. This varying pressure distribution creates resultant forces on any submerged surface, which is fundamental to understanding hydrostatic force analysis.
When analyzing pressure distribution on submerged plane surfaces, we observe that pressure varies linearly from top to bottom of the surface. The pressure at any point depends on the vertical depth below the free surface, expressed as P equals gamma h, where gamma is the specific weight of the fluid. This linear pressure distribution can be visualized as a pressure prism, creating a three-dimensional representation that helps us understand how forces are distributed across the submerged surface.
The total hydrostatic force on a submerged surface is calculated using the formula F equals gamma h c A, where gamma is the specific weight, h c is the depth to the centroid, and A is the surface area. This formula comes from integrating pressure over the entire surface area. For our example of a rectangular gate at 45 degrees with centroidal depth of 3 meters and area of 4 square meters, using a specific weight of 9800 Newtons per cubic meter, the total force equals 117,600 Newtons acting perpendicular to the surface.
The center of pressure is the point where the resultant hydrostatic force acts on the surface. It is not the same as the centroid of the surface. The center of pressure is calculated using the formula y c p equals y c plus I c divided by y c A, where I c is the second moment of area about the centroidal axis. For inclined surfaces, the center of pressure is always located below the centroid. In our rectangular gate example, with a centroidal depth of 3 meters and second moment of area of 1.33 square meters to the fourth power, the center of pressure is located at 3.11 meters depth.
Hydrostatic force analysis has critical engineering applications. For a vertical rectangular dam gate with width 3 meters and height 4 meters, the centroidal depth is 2 meters and area is 12 square meters, resulting in a total force of 235,200 Newtons with center of pressure at 2.17 meters depth. For a circular submarine window with diameter 1 meter at 50 meters depth, the area is 0.785 square meters, producing a force of 384,650 Newtons. Since the window is circular, the center of pressure coincides with the centroid. These calculations are essential for structural design and safety analysis.