Hydrostatics: Forces on Surfaces

When a surface is submerged in a static fluid, the fluid exerts a pressure force normal to the surface. Since pressure increases with depth, the distribution of force is non-uniform.

Consider a plane surface of arbitrary shape with area $A$, inclined at an angle $\theta$ to the free surface.

The total hydrostatic force on a plane surface is the product of the pressure at the centroid and the area.

The point of application of the resultant force is called the Center of Pressure ($CP$). It is always below the centroid because pressure increases with depth.

Location along the inclined plane ($y_p$):

The distance between the Centroid and the Center of Pressure is called the eccentricity ($e$).

Common centroidal moments of inertia used in center-of-pressure calculations include:

• Rectangle ($b \times h$): $I_{xc} = \frac{bh^3}{12}$
• Triangle ($b \times h$): $I_{xc} = \frac{bh^3}{36}$
• Circle (radius $r$): $I_{xc} = \frac{\pi r^4}{4}$
• Semicircle (radius $r$): $I_{xc} = 0.1098 r^4$

When calculating the moment of inertia ($I$) for shapes not centered on the axis of rotation, the parallel axis theorem is used.

For curved surfaces, it is easier to calculate the horizontal and vertical components of the force separately.

For curved surfaces, the resultant hydrostatic force and its direction are found by combining the horizontal and vertical components.

For rectangular plane surfaces, the hydrostatic force can be visualized and calculated using a 3D Pressure Prism. The base of the prism is the area of the surface, and the altitude (height) at any point is the pressure ($P = \gamma h$) at that depth.

• Magnitude: The total force ($F$) equals the volume of the pressure prism.
• Location: The Center of Pressure ($CP$) passes through the centroid of this 3D prism volume.

This geometric approach is often simpler than using moments of inertia, especially for surfaces starting at the free surface (where the prism is a wedge and the force acts at $2/3$ depth).

When a cylinder of diameter $D$ contains fluid at an internal pressure $P$, the fluid pushes outwards, creating tension in the walls. The force tending to split the cylinder in half longitudinally is resisted by the tensile force ($T$) in the pipe wall.

The tension in the walls induces a circumferential (hoop) stress in the material.

Gravity dams resist the hydrostatic force of water solely through their own massive weight. They are checked against three primary failure modes: Overturning, Sliding, and Bearing Capacity.

Overturning: The dam rotating about its downstream "toe". The Factor of Safety against overturning ($FSO$) is the ratio of Righting Moments ($\sum RM$) to Overturning Moments ($\sum OM$) about the toe.

Sliding: The dam being pushed horizontally downstream. The Factor of Safety against sliding ($FSS$) relates frictional resistance to the driving hydrostatic force.

Bearing Capacity: The pressure under the base exceeding the allowable soil/rock bearing pressure, which can cause foundation settlement or crushing at the toe.