ME 310 Chap.3 Hydrostatic Forces on Plane and Curved Surfaces
Hydrostatic Forces on Plane and Curved Surfaces
From pressure to force
Once pressure is known everywhere in a liquid, the design question becomes the force it exerts on a wall, gate or hatch, and where that force acts.
The lectures on flat surfaces treat this with the geometric properties of the surface: its area, the location of its centroid and its second moment of area.
Plane surfaces
For a flat surface of any shape, integrating the linearly varying pressure gives a resultant equal to the pressure at the centroid times the area.
The line of action lies below the centroid, because deeper parts are pushed harder, and the offset along the surface equals the centroidal second moment of area divided by the product of centroid distance and area. That offset shrinks as the surface goes deeper.
Distances measured along an inclined surface and vertical depths are different quantities, and mixing them is the most frequent slip in gate problems.
Moments on gates
Gate problems finish with a moment balance about a hinge. Homework in this unit asks how the moment needed to hold a submerged circular gate changes as the gate is lowered.
Working it out with equations rather than intuition shows that the moment of the pressure force about the centroidal axis is independent of depth, even though the resultant force grows with depth. Results like this reward writing the general expression before substituting numbers.
Curved surfaces
On a curved surface the pressure direction changes from point to point.
Instead of integrating directly, isolate the block of fluid next to the surface. The horizontal force component equals the force on the vertical projection of the surface; the vertical component equals the weight of fluid, real or imaginary, directly above it. A homework problem on a submerged road tunnel uses this idea: the two sides of a symmetric roof cancel horizontally, leaving only a vertical load.
For a circular arc, the resultant passes through the centre of curvature.
Checklist for any surface-force problem
Fix the free surface and the reference for pressure, find the centroid depth and the wetted area, compute the resultant, locate it with the second moment of area, and only then take moments about the hinge or support that the question names.
What this chapter covers
- 01
Resultant force equals centroid pressure times area
- 02
Centre of pressure below the centroid
- 03
Second moments of area for common shapes
- 04
Hinge moments and latch forces on gates
- 05
Depth-independent moment about the centroid
- 06
Curved surfaces by horizontal projection and fluid weight
Worked example · free
Force on an inclined rectangular gate
- 2Centroid depth: h_c = 1.0 + 1.0 sin 60° = 1.866 m; area 3.0 m².
- 2Resultant: F = γ h_c A = 9810 × 1.866 × 3.0 = 54.9 kN.
- 2Incline distance to centroid: y_c = h_c / sin 60° = 2.155 m; I_xc = 1.5 × 2.0³ / 12 = 1.0 m⁴.
- 1Centre of pressure: y_R = 2.155 + 1.0 / (2.155 × 3.0) = 2.309 m along the incline.
Key terms
- Centre of Pressure
- The centre of pressure is the point on a submerged surface through which the resultant hydrostatic force acts.
- Second Moment of Area
- The second moment of area is a geometric property describing how an area is distributed about an axis; it sets the depth of the centre of pressure.
- Vertical Projection
- The vertical projection of a curved surface is its shadow on a vertical plane, used to find the horizontal hydrostatic force component.
Hydrostatic Forces on Plane and Curved Surfaces FAQ
Why is the centre of pressure always below the centroid?
Pressure grows with depth, so the lower part of a submerged surface carries more force than the upper part. The resulting force distribution is weighted toward the bottom, which pulls the line of action beneath the centroid.
How do I find the vertical force on a curved surface when the liquid is underneath it?
Use the weight of the imaginary liquid that would fill the region between the surface and the free-surface level. The magnitude is the same as if that liquid were present, but the force points upward instead of downward.
Does a deeper gate always need a larger holding moment?
Not about every axis. The resultant force grows with depth, but its moment about the horizontal axis through the centroid equals specific weight times the sine of the incline times the centroidal second moment, which does not depend on depth.
Which second moment of area do I use for a gate?
The one about the horizontal axis through the centroid of the wetted area, parallel to the free surface. For a rectangle of width b and height h it is b times h cubed over twelve; for a circle of radius R it is pi R to the fourth over four.
Exam move
Draw the pressure distribution on every gate before computing, and mark the hinge; the lever arm is where most errors hide.
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