Fluid Pressure over Curved Plate of Constant Width

Pressure acting normal to a submerged curved plate varies continuously because the plate's surface orientation changes at each point. As depth increases, pressure magnitude increases due to the weight of water above. The changing geometry means the normal direction shifts, causing both magnitude and directional variation across the plate surface.

The resultant force is found by separately calculating horizontal and vertical force components. This involves representing the distributed loading as equivalent loading in two dimensions: accounting for water weight through the block's centroid, pressure along the vertical side as a trapezoidal distribution, and pressure along the horizontal side as a rectangular distribution. Summing these three forces yields the resultant.

The centroid is the point through which distributed forces act. The water's weight force acts through the block's centroid. The trapezoidal pressure force along the vertical side acts through the trapezoid's centroid, while the rectangular pressure force along the horizontal side acts through the rectangle's centroid. These centroid locations are essential for accurate force calculations.

The center of pressure location is determined by applying the principle of moments. This involves calculating the moment of each force component about a reference axis, then dividing the total moment by the resultant force magnitude. This method ensures the single resultant force produces the same rotational effect as the distributed pressure loading.

The pressure distribution along the vertical side is trapezoidal, varying from zero at the surface to maximum at the bottom, creating a force magnitude equal to the trapezoid's area. The pressure distribution along the horizontal side is constant, creating a rectangular distribution with force magnitude equal to the rectangle's area. These different distributions require distinct calculation approaches.

Two-dimensional equivalent loading simplifies complex three-dimensional pressure calculations by breaking the problem into manageable components. This method separates the water's weight effect from pressure distributions on vertical and horizontal surfaces, allowing each force to be calculated independently using geometric properties like area and centroid location before combining them into a resultant.

The magnitude of force due to water weight equals the product of specific weight, plate thickness, and block area. This force acts through the block's centroid. By multiplying these three parameters, engineers can quickly determine the gravitational force component without integrating pressure across the entire submerged volume.