Hydrostatics: Forces on Surfaces

Hydrostatics: Forces on Surfaces

Calculating hydrostatic forces on plane and curved submerged surfaces, including magnitude and location (Center of Pressure).

Concept Overview

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.

Force on Plane Surfaces

Concept Overview

Consider a plane surface of arbitrary shape with area

A

, inclined at an angle

\theta

to the free surface.

Hydrostatic Force on a Rectangular Plane

Depth to Top Edge ($h_1$): 2.0 m

Incline Angle ($\theta$): 90°

Gate Height ($h$): 3.0 m

Gate Width ($b$): 2.0 m

Total Force (F):0.0 kN

Centroid Depth (h̄):0.00 m

Centroid Distance (ȳ):0.00 m

Center of Pressure (y_p):0.00 m

Notice that the Center of Pressure (y_p, red dot) is always slightly deeper than the Centroid (ȳ, blue dot). This eccentricity decreases as the depth increases.

Magnitude of Hydrostatic Force

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

Magnitude of Hydrostatic Force

Calculating hydrostatic forces on plane and curved submerged surfaces, including magnitude and location (Center of Pressure).

F = P_{cg} A = \gamma \bar{h} A

Variables

Symbol	Description	Unit
$\bar{h}$	Vertical distance from the free surface to the Centroid ($C$) of the area.	-
$\gamma$	Specific weight of the fluid.	-
$A$	Total area of the surface.	-

Center of Pressure (Location)

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 ( $y_p$ ):

Center of Pressure (Location)

y_p = \bar{y} + \frac{I_{xc}}{\bar{y} A}

Variables

Symbol	Description	Unit
$y_p$	Distance from the free surface to $CP$ along the incline.	-
$\bar{y}$	Distance from the free surface to the Centroid ($C$) along the incline ($\bar{h} = \bar{y} \sin\theta$).	-
$I_{xc}$	Moment of Inertia of the area about the centroidal axis parallel to the free surface.	-

Eccentricity ( $e$ ): The distance between Centroid and Center of Pressure.

e = y_p - \bar{y} = \frac{I_{xc}}{\bar{y}A}

Common Moments of Inertia

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 $r$ ): $I_{xc} = 0.1098 r^4$

Parallel Axis Theorem

Finding the moment of inertia about an axis parallel to the centroidal axis.

Parallel Axis Theorem

When calculating the moment of inertia (

I

) for shapes not centered on the axis of rotation, the parallel axis theorem is used:

Parallel Axis Theorem

Finding the moment of inertia about an axis parallel to the centroidal axis.

I = I_{xc} + A d^2

Variables

Symbol	Description	Unit
$I_{xc}$	Moment of inertia about the centroidal axis.	-
$A$	Area of the shape.	-
$d$	Perpendicular distance between the centroidal axis and the parallel axis.	-

Force on Curved Surfaces

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

Horizontal Component ( $F_H$ )

The horizontal force on a curved surface is equal to the hydrostatic force on the vertical projection of the curved surface.

Horizontal Component ($F_H$)

F_H = \gamma \bar{h}_{proj} A_{proj}

Acts at the center of pressure of the projected vertical area.

Vertical Component ( $F_V$ )

The vertical force is equal to the weight of the fluid volume (real or imaginary) directly above the curved surface extending to the free surface.

Vertical Component ($F_V$)

F_V = \gamma V_{above}

Acts through the centroid of the volume.

Note

Resultant Force ( $R$ ):

R = \sqrt{F_H^2 + F_V^2}

Direction ( $\theta$ ):

\tan\theta = \frac{F_V}{F_H}

The Pressure Prism

Pressure Prism Method

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).

Hoop Tension in Thin-Walled Cylinders

The circumferential stress induced in thin-walled pipes or tanks due to internal fluid pressure.

Thin-Walled Cylindrical Tanks

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.

T = \frac{P D}{2}

$T$ = Hoop tension force per unit length along the cylinder
$P$ = Internal fluid pressure
$D$ = Inside diameter

The circumferential (hoop) stress (

\sigma_c

) in a wall of thickness

t

\sigma_c = \frac{P D}{2 t}

Stability of Gravity Dams

Evaluating the safety of retaining structures subjected to hydrostatic forces.

Modes of Failure

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

Overturning: The dam rotating about its downstream "toe". The Factor of Safety against overturning ( $FSO$ ) is the ratio of Righting Moments (RM) to Overturning Moments (OM) about the toe.
$FSO = \frac{\sum RM}{\sum OM} \ge 1.5$
Sliding: The dam being pushed horizontally downstream. The Factor of Safety against sliding ( $FSS$ ) relates frictional resistance to the driving hydrostatic force.
$FSS = \frac{\mu \sum F_V}{\sum F_H} \ge 1.5$
Where $\mu$ is the coefficient of friction and $\sum F_V$ is the net vertical force (Weight - Uplift).
Bearing Capacity: The pressure under the base exceeding the allowable soil/rock bearing pressure, which can cause foundation settlement or crushing at the toe.

Key Takeaways

Centroid vs CP: The force acts at the Center of Pressure ( $CP$ ), which is always deeper than the Centroid ( $C$ ).
Parallel Axis Theorem: Use $I = I_{xc} + A d^2$ to find moments of inertia about parallel axes.
Curved Surfaces: Decompose into Horizontal (projection) and Vertical (weight of fluid) components.
Pressure Prism: For rectangular surfaces, calculate force as the volume of the pressure prism and locate it at the volume's centroid.

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Concept Overview

Force on Plane Surfaces

Concept Overview

Hydrostatic Force on a Rectangular Plane

Magnitude of Hydrostatic Force

Magnitude of Hydrostatic Force

Center of Pressure (Location)

Center of Pressure (Location)

Common Moments of Inertia

Parallel Axis Theorem

Parallel Axis Theorem

Parallel Axis Theorem

Force on Curved Surfaces

Horizontal Component (FHF_HFH​)

Horizontal Component ($F_H$)

Vertical Component (FVF_VFV​)

Vertical Component ($F_V$)

Note

The Pressure Prism

Pressure Prism Method

Hoop Tension in Thin-Walled Cylinders

Thin-Walled Cylindrical Tanks

Stability of Gravity Dams

Modes of Failure

Horizontal Component ( $F_H$ )

Vertical Component ( $F_V$ )