CIVL2611 Chap.3 Forces on Submerged Surfaces
Forces on Submerged Surfaces
Hydrostatic pressure creates a distributed normal load over a submerged surface. This chapter replaces that field with an equivalent resultant, finds its line of action through the centre of pressure, and uses the pressure prism as a geometric cross-check. A reliable solution separates area geometry, force magnitude and the later moment balance.
For a plane area in constant-density liquid, the resultant magnitude equals pressure at the area centroid multiplied by area. The line of action lies deeper than the centroid because the lower region carries greater pressure, and its location depends on the centroidal area second moment about an axis parallel to the free surface.
The distinction between vertical depth, along-surface coordinate and perpendicular moment arm becomes essential on inclined gates. A pressure prism offers an independent construction: prism volume gives force and prism centroid fixes its action. Triangular and trapezoidal pressure diagrams can be split into simple parts and recombined through moments.
Original hatch and panel examples show every intermediate geometry value and check that the final point lies within the wetted surface. Cable or hinge reactions are introduced only after the hydrostatic load is complete. The chapter also explains why placing the same plate deeper moves the resultant relatively closer to its centroid as pressure becomes more uniform across the surface.
What this chapter covers
- 01
Distributed hydrostatic pressure
- 02
Resultant force on plane areas
- 03
Centroid depth
- 04
Centroidal second moment of area
- 05
Centre of pressure
- 06
Pressure prism volume and centroid
Worked example · free
Vertical rectangular hatch
- +1Find area 2.40 square metres and centroid depth 2.00 metres.
- +2Compute resultant 47.1 kilonewtons from centroid pressure times area.
- +1Find the horizontal centroidal second moment as 0.800 metres to the fourth power.
- +2Calculate centre-of-pressure depth as 2.167 metres.
Key terms
- Hydrostatic resultant
- The single force equivalent to the distributed pressure acting normally over a submerged surface.
- Centre of pressure
- The location at which the hydrostatic resultant has the same moment as the pressure distribution.
- Pressure prism
- A geometric solid whose base is wetted area and whose height represents pressure.
Forces on Submerged Surfaces FAQ
Why does centroid pressure give the resultant magnitude?
For a plane surface in a constant-density static liquid, pressure varies linearly with depth. Integrating that linear field over area produces pressure at the area centroid multiplied by the total area.
Why is the centre of pressure below the centroid?
Deeper portions experience larger pressure and therefore contribute more strongly to the moment. The equivalent resultant shifts toward that larger loading, placing its line of action below the geometric centroid for ordinary vertical areas.
Which second moment belongs in the formula?
Use the area second moment about the centroidal axis lying in the surface and parallel to the free surface. Draw that axis first; do not substitute a polar or mass moment of inertia.
Exam move
Practise the same sequence on rectangles, triangles and inclined areas: mark the free surface, find area and centroid, calculate force, choose the correct centroidal axis, locate the centre of pressure and only then take moments about a hinge or support. Begin with a pressure sketch beside the physical surface. Its intensity should increase with vertical depth, regardless of the surface orientation.
On a plane area, separate three distances on the page: vertical centroid depth for pressure, along-surface location when geometry is measured on an incline, and perpendicular lever arm for moments. Give them different symbols so one number is not reused automatically. Create a geometry sheet containing areas, centroids and centroidal second moments for rectangles and triangles, with the relevant axis drawn on every shape.
Test the formula by checking length-to-the-fourth units and whether moving area farther from the axis increases the value. For a rectangle, say aloud which dimension is cubed and why that direction matches the pressure gradient. Solve a vertical-gate problem by both the centre-of-pressure formula and the pressure-prism method.
Split a trapezoidal pressure diagram into a rectangle and triangle, calculate both forces and combine their moments. Agreement between methods is stronger evidence than repeating the same algebra. Then add a hinge and cable only after the hydrostatic force is complete. Take moments about the hinge to remove unknown reactions, and draw the cable line so its moment arm is geometrically correct.
In every answer, check that the centre of pressure lies inside the wetted surface and below the centroid under an ordinary depth-increasing distribution. Move the same plate deeper and predict that the relative offset should shrink as loading becomes more uniform. Time one complete solution with a labelled sketch, symbolic expressions, units and a final location statement.
Review mistakes by category: centroid depth, area, axis, second moment, pressure reference or lever arm. That classification points directly to the step that needs another example. Redraw the final pressure distribution beside the answer and check that its centroid agrees qualitatively with the reported force location. Label every geometric distance clearly.