Hydrostatic Force on a Curved Surface

X-Component of the Resultant Force

Y-Component of the Resultant Force

For a curved surface AB, as shown in the figure, the magnitude and the line of action of the resultant force F_R exerted on the surface can be best derived by splitting the force into its horizontal and vertical components.

The x-component of the resultant force F_Rx is the normal force acting on the vertical projection of the curved surface (i.e., surface AE). This force F_Rx passes through the center of pressure for the projected area AE.

The y-component of the resultant force F_Ry is the weight of the liquid directly above the curved surface (i.e., volume ABCD). Note that this volume can be either real or imaginary. In this case, the volume is real since the liquid actually occupies this volume. Another example will be given later to illustrate an imaginary volume. This force, F_Ry ,passes through the center of gravity of volume ABCD. If the gravitational acceleration is assumed to be constant and the fluid is incompressible, then the center of gravity is the same as the centroid of the fluid volume.

The magnitude of the resultant force is then determined by
     F_R = ( F²_Rx + F²_Ry )^0.5

By inspection, it is noted that pressure forces are always perpendicular to the surface AB (i.e., the normal stresses). Since all points on a circle have a normal passing through the center of a circle, the resultant force F_R has to pass through point E. The direction of the resultant force is given by

     θ = tan ^-1 (F_Ry / F_Rx)

For non-circular shapes, the resultant angle is not used and the horizontal and vertical gravity center may not align with the actual surface. Again, it is easiest to keep the horizontal and vertical components separate in all calculations.