Hydrostatics: Forces on Surfaces

Hydrostatics: Forces on Surfaces

$Learning Objectives$

Determine the magnitude and line of action of hydrostatic forces on horizontal, vertical, and inclined plane surfaces.
Use centroid depth and area second moments correctly in center-of-pressure calculations.
Interpret hydrostatic loading through pressure diagrams and pressure prisms.
Analyze moments on hinged gates and support forces using static equilibrium.
Resolve hydrostatic force on curved surfaces into horizontal and vertical components and locate their lines of action.
Apply local internal pressure to thin-walled cylindrical members using the thin-wall idealization.
Formulate introductory overturning, sliding, uplift, and bearing-equilibrium checks for gravity structures without assuming uncited acceptance limits.

$Hydrostatic Resultant$

The hydrostatic resultant is the single force that is statically equivalent to the distributed normal pressure acting over a submerged surface.

$Pressure Loading on a Submerged Surface$

Static-fluid pressure acts normal to the wetted surface and generally varies with vertical depth. When atmospheric pressure acts on both sides of a gate, atmospheric contributions cancel and the net loading can be evaluated using gage pressure. If the pressures on the two sides have different references or boundary values, use the net pressure distribution rather than automatically using $\gamma h$ alone.

$Interactive Hydrostatic-Force Exploration$

Use the simulation to observe how depth and geometry change the distributed pressure, resultant magnitude, and center of pressure. Compare the displayed force with the analytical centroid-depth relation.

Hydrostatic Force on an Inclined Rectangular Gate

Learning objective: See how depth, inclination, area, and fluid density change the hydrostatic resultant and center of pressure.

The angle is measured from the horizontal free surface, and the top-edge depth is a vertical distance.

Vertical depth to top edge (

h_1

)2.0 m

Gate angle (

\theta

)90°

Length along gate3.0 m

Gate width2.0 m

Fluid density (

\rho

)1,000 kg/m³

Resultant force: 206.0 kN
Centroid vertical depth, $\bar h$: 3.500 m
Center-of-pressure vertical depth, $h_p$: 3.714 m
Center of pressure from top edge: 1.714 m

This model assumes a static, constant-density liquid and uniform pressure at the free surface. It reports both the distance along the plane and the vertical depth, which must not be interchanged in center-of-pressure calculations.

Resultant Force on a Plane Surface

Computes the resultant when pressure varies linearly with depth in a constant-density liquid.

F=p_cA=\gamma\bar hA

Variables

Symbol	Description	Unit
$F$	Hydrostatic resultant force	N
$p_c$	Net pressure at the area centroid	Pa
$A$	Wetted plane area	$m^2$
$\gamma$	Liquid specific weight	$N/m^3$
$\bar h$	Vertical depth of the area centroid below the zero-gage free surface	m

Hydrostatic loading on a plane

Distributed pressure, centroid depth, and resultant direction.

$Centroid Depth Is Vertical Depth$

The force magnitude uses the vertical centroid depth $\bar h$ , regardless of whether the plane is vertical or inclined. For a plane inclined at angle $\theta$ above horizontal, an along-plane coordinate $\bar y$ is related by $\bar h=\bar y\sin\theta$ when $\bar y$ is measured from the free-surface intersection.

$Center of Pressure$

The center of pressure is the point through which the hydrostatic resultant acts on a surface.

Center of Pressure along an Inclined Plane

Locates the resultant along a plane when gage pressure is generated by liquid depth from a zero-gage free surface.

y_p=\bar y+\frac{I_G}{A\bar y}

Variables

Symbol	Description	Unit
$y_p$	Along-plane distance from the free-surface intersection to the center of pressure	m
$\bar y$	Along-plane distance from the free-surface intersection to the area centroid	m
$I_G$	Centroidal area second moment about an in-plane axis parallel to the free-surface line	$m^4$
$A$	Wetted plane area	$m^2$

Vertical Depth of Center of Pressure

Expresses center-of-pressure depth directly in vertical coordinates for an inclined plane.

h_p=\bar h+\frac{I_G\sin^2\theta}{A\bar h}

Variables

Symbol	Description	Unit
$h_p$	Vertical depth of the center of pressure	m
$\bar h$	Vertical centroid depth	m
$I_G$	Centroidal area second moment about the appropriate in-plane axis	$m^4$
$\theta$	Plane angle above horizontal	degrees or rad

$The Center of Pressure Is Not Always Below the Centroid$

For a vertical or inclined plane with pressure increasing with depth, the center of pressure lies deeper than the centroid. A horizontal plane at one elevation has uniform pressure, so its resultant acts through the area centroid. A nonzero uniform pressure added to a depth-varying distribution also shifts the resultant toward the centroid, so use the actual net pressure distribution when the free-surface pressure does not cancel.

$e$

For an inclined plane under a zero-gage free-surface pressure distribution, eccentricity is the along-plane distance from the area centroid to the center of pressure.

Center-of-Pressure Eccentricity

Measures the pressure-gradient-induced offset from centroid to center of pressure.

e=y_p-\bar y=\frac{I_G}{A\bar y}

Variables

Symbol	Description	Unit
$e$	Along-plane eccentricity	m
$I_G$	Centroidal area second moment	$m^4$
$A$	Area	$m^2$
$\bar y$	Along-plane centroid distance from the free-surface intersection	m

$I_G$

The area second moment quantifies how area is distributed about a selected axis. In center-of-pressure analysis, the required centroidal axis lies in the plane of the surface and is parallel to the free-surface intersection line.

$Common Centroidal Area Second Moments$

Rectangle of width $b$ parallel to the axis and depth dimension $d$ perpendicular to it: $I_G=bd^3/12$ .
Triangle of base $b$ parallel to the axis and altitude $d$ perpendicular to it: $I_G=bd^3/36$ .
Circle of radius $R$ about any centroidal diameter: $I_G=\pi R^4/4$ .
For composite areas or shifted axes, use the parallel-axis theorem with consistent axis orientation.

Parallel-Axis Theorem

Transfers an area second moment from a centroidal axis to a parallel axis.

I=I_G+Ad^2

Variables

Symbol	Description	Unit
$I$	Area second moment about the shifted axis	$m^4$
$I_G$	Area second moment about the parallel centroidal axis	$m^4$
$A$	Area	$m^2$
$d$	Perpendicular separation between the parallel axes	m

$Pressure Prism$

A pressure prism is a geometric representation in which the base is the loaded surface and the local prism height is proportional to pressure. Its volume represents resultant force, and the centroid of that volume locates the resultant line of action.

$Pressure Diagrams and Pressure Prisms$

For a vertical rectangle beginning at a zero-gage free surface, pressure varies linearly from zero to $\gamma h$ , producing a triangular pressure diagram. The resultant of that triangle acts at two-thirds of the depth measured from the zero-pressure edge. For a surface whose top is already submerged, the pressure diagram is trapezoidal and can be decomposed into uniform and triangular parts for a transparent force-and-moment calculation.

Hydrostatic pressure prism

Pressure distribution grows with depth and produces a resultant.

$Interactive 3D Pressure Prism & Gate Kinematics$

Explore the 3D volume of the pressure prism and normal pressure vectors in real time. Switch between rectangular, inclined, and curved radial Tainter gates to observe the resultant line of action, centroid offset, and trunnion pin reaction forces.

Hydrostatic Pressure Prisms on Submerged Gates 3D

Spatial pressure volume, resultant force vector, center of pressure offset, and radial Tainter gate trunnion pin mechanics.

Loading 3D Hydrostatic Pressure Simulation…

3D Pressure Prism VolumeNormal Pressure Vectors

Gate & Hydrostatic Controls

Incline Angle (

\theta

)60°

Water Surcharge (

h_{\text{top}}

)1.20 m

Gate Slanted Length (

L

)2.40 m

Gate Width (

W

)1.80 m

Resultant Force

F_R

94.9 kN

Centroid Depth

\bar{h}

2.24 m

CP Depth

h_{cp}

2.40 m

Eccentricity

e

18.6 cm

Top Pressure

p_{\text{top}}

11.8 kPa

Bottom Pressure

p_{\text{bot}}

32.2 kPa

$Hinged-Gate Moment Balance$

A hinged gate is analyzed by replacing the distributed fluid pressure with its resultant at the center of pressure and then enforcing rotational equilibrium about the hinge.

Static Moment Equilibrium about a Gate Hinge

Relates hydrostatic, weight, cable, and other applied moments about the hinge.

\sum M_{\text{hinge}}=0

Variables

Symbol	Description	Unit
$M$	Moment of an applied force about the hinge	N·m

Hinged-gate moment balance

Hydrostatic force and resistance balanced about a hinge.

$Why Moments Are Taken about the Hinge$

Unknown hinge reaction forces have zero moment arm about the hinge. This makes the hinge a convenient moment center for solving cable tension, required holding force, or the condition for impending opening.

$F_H$

The horizontal component of hydrostatic force on a curved surface equals the hydrostatic resultant on the surface's vertical projection.

Horizontal Component on a Curved Surface

Uses the vertical projected area of the curved surface.

F_H=\gamma\bar h_{\text{proj}}A_{\text{proj}}

Variables

Symbol	Description	Unit
$F_H$	Horizontal force component	N
$\bar h_{\text{proj}}$	Vertical centroid depth of the vertical projection	m
$A_{\text{proj}}$	Vertical projected area	$m^2$
$\gamma$	Liquid specific weight	$N/m^3$

$Horizontal-Component Line of Action$

$F_H$ acts through the center of pressure of the vertical projected area, not generally through the geometric centroid of the actual curved surface.

$F_V$

The vertical component equals the weight of the real or imaginary fluid volume bounded by the curved surface and the free surface, with its direction determined from the actual pressure-force orientation.

Vertical Component on a Curved Surface

Uses the equivalent real or imaginary fluid volume above the curved boundary.

|F_V|=\gamma V_{\text{fluid}}

Variables

Symbol	Description	Unit
$F_V$	Vertical force component with direction assigned from geometry	N
$\gamma$	Liquid specific weight	$N/m^3$
$V_{\text{fluid}}$	Equivalent real or imaginary fluid volume	$m^3$

$Do Not Assume the Vertical Component Is Always Downward$

The equivalent-volume construction gives the magnitude of the vertical component. Its direction depends on which side of the curved surface is wetted and the orientation of the pressure normals. The line of action passes through the centroid of the corresponding real or imaginary fluid volume.

Resultant of Curved-Surface Components

Combines perpendicular horizontal and vertical hydrostatic components.

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

\alpha=\tan^{-1}\left(\frac{|F_V|}{|F_H|}\right)

Variables

Symbol	Description	Unit
$R$	Resultant hydrostatic force magnitude	N
$\alpha$	Resultant angle measured from the horizontal component	degrees or rad

Curved-surface force components

Horizontal and vertical components combine to the resultant.

$Thin-Wall Pressure Vessel Idealization$

A cylindrical shell is treated as thin-walled when wall thickness is small compared with diameter so membrane stress can be approximated as uniform through the thickness and bending through the wall is neglected.

Hoop Tension per Unit Cylinder Length

Balances local internal gage pressure on a longitudinally split thin cylinder.

T=\frac{pD}{2}

Variables

Symbol	Description	Unit
$T$	Circumferential membrane tension per unit axial length	N/m
$p$	Local internal gage pressure relative to the external pressure	Pa
$D$	Cylinder inside diameter	m

Thin-Cylinder Hoop Stress

Converts membrane tension to average circumferential stress.

\sigma_h=\frac{pD}{2t}

Variables

Symbol	Description	Unit
$\sigma_h$	Average hoop stress	Pa
$p$	Local internal gage pressure relative to external pressure	Pa
$D$	Cylinder inside diameter	m
$t$	Wall thickness	m

$Hydrostatic Pressure May Vary along a Vertical Tank Wall$

For a tall liquid-filled cylindrical tank, $p$ increases with depth. The thin-wall equation therefore gives a local hoop stress that is greatest where the local pressure is greatest. Do not substitute a single pressure for the whole height unless that pressure model is justified.

$Gravity-Structure Stability Check$

A gravity structure resists applied hydrostatic actions primarily through its weight and foundation reactions. Introductory checks compare overturning moments, sliding actions, uplift, and base bearing response using static equilibrium.

Generic Overturning Safety Ratio

Compares stabilizing and overturning moments about a selected pivot such as the downstream toe.

FS_{\text{OT}}=\frac{\sum M_{\text{resisting}}}{\sum M_{\text{overturning}}}

Variables

Symbol	Description	Unit
$FS_{\text{OT}}$	Overturning safety ratio	dimensionless
$M_{\text{resisting}}$	Stabilizing moment contribution	N·m
$M_{\text{overturning}}$	Overturning moment contribution	N·m

Generic Sliding Safety Ratio

Compares modeled horizontal resistance with driving horizontal action.

FS_{\text{SL}}=\frac{R_H}{D_H}

Variables

Symbol	Description	Unit
$FS_{\text{SL}}$	Sliding safety ratio	dimensionless
$R_H$	Modeled horizontal resistance, such as base friction plus other justified resistance	N
$D_H$	Driving horizontal action	N

$Uplift and Bearing in Gravity Structures$

Uplift reduces the net downward foundation force and therefore can reduce frictional sliding resistance and stabilizing moment. Base-bearing checks use the resultant vertical force and its eccentricity to determine contact-pressure distribution. The appropriate uplift model, allowable bearing pressure, load combinations, drainage assumptions, and required safety factors depend on the governing design basis and project conditions.

$Do Not Treat a Single Safety Factor as Universal$

This lesson introduces equilibrium methods, not a universal dam-design acceptance criterion. Required factors of safety and allowable stresses vary with design code, load case, drainage assumptions, foundation conditions, and project authority. Use specified criteria when a design problem provides them.

Key Takeaways

For a plane surface under a depth-linear net pressure distribution, resultant magnitude is pressure at the centroid times area.
Center-of-pressure formulas depend on the actual pressure distribution; horizontal planes under uniform pressure have their resultant at the centroid.
Area second moments must use the correct in-plane centroidal axis, and the parallel-axis theorem shifts only between parallel axes.
Pressure diagrams and pressure prisms provide a geometric way to recover both force and line of action.
Hinged gates are solved efficiently by replacing distributed pressure with its resultant and taking moments about the hinge.
Curved-surface force is resolved into a horizontal projection force and a vertical equivalent-fluid-weight component whose direction comes from geometry.
Thin-wall hoop stress uses the local net pressure and is a membrane idealization.
Gravity-structure overturning, sliding, uplift, and bearing checks require explicit load models and project-specific acceptance criteria.

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Learning Objectives

Hydrostatic Resultant

Pressure Loading on a Submerged Surface

Interactive Hydrostatic-Force Exploration

Hydrostatic Force on an Inclined Rectangular Gate

Resultant Force on a Plane Surface

Centroid Depth Is Vertical Depth

Center of Pressure

Center of Pressure along an Inclined Plane

Vertical Depth of Center of Pressure

The Center of Pressure Is Not Always Below the Centroid

Eccentricity, eee

Center-of-Pressure Eccentricity

Area Second Moment, IGI_GIG​

Common Centroidal Area Second Moments

Parallel-Axis Theorem

Pressure Prism

Pressure Diagrams and Pressure Prisms

Interactive 3D Pressure Prism & Gate Kinematics

Hydrostatic Pressure Prisms on Submerged Gates 3D

Gate & Hydrostatic Controls

Hinged-Gate Moment Balance

Static Moment Equilibrium about a Gate Hinge

Why Moments Are Taken about the Hinge

Horizontal Component on a Curved Surface, FHF_HFH​

Horizontal Component on a Curved Surface

Horizontal-Component Line of Action

Vertical Component on a Curved Surface, FVF_VFV​

Vertical Component on a Curved Surface

Do Not Assume the Vertical Component Is Always Downward

Resultant of Curved-Surface Components

Thin-Wall Pressure Vessel Idealization

Hoop Tension per Unit Cylinder Length

Thin-Cylinder Hoop Stress

Hydrostatic Pressure May Vary along a Vertical Tank Wall

Gravity-Structure Stability Check

Generic Overturning Safety Ratio

Generic Sliding Safety Ratio

Uplift and Bearing in Gravity Structures

Do Not Treat a Single Safety Factor as Universal

$Learning Objectives$

$Hydrostatic Resultant$

$Pressure Loading on a Submerged Surface$

$Interactive Hydrostatic-Force Exploration$

$Centroid Depth Is Vertical Depth$

$Center of Pressure$

$The Center of Pressure Is Not Always Below the Centroid$

$e$

$I_G$

$Common Centroidal Area Second Moments$

$Pressure Prism$

$Pressure Diagrams and Pressure Prisms$

$Interactive 3D Pressure Prism & Gate Kinematics$

$Hinged-Gate Moment Balance$

$Why Moments Are Taken about the Hinge$

$F_H$

$Horizontal-Component Line of Action$

$F_V$

$Do Not Assume the Vertical Component Is Always Downward$

$Thin-Wall Pressure Vessel Idealization$

$Hydrostatic Pressure May Vary along a Vertical Tank Wall$

$Gravity-Structure Stability Check$

$Uplift and Bearing in Gravity Structures$

$Do Not Treat a Single Safety Factor as Universal$