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Vertical planes are a special case of submerged surfaces.
For a rectangular plane the second moment of area (Ixx,c) can be calculated using:
substituting this into the centre of pressure equation we get:
When the plane is vertical θ=90o so Sinθ=1 so can be removed from the equation leaving:
The resultant force is found using
where A is the surface area of the submerged surface and Pc is the centroid pressure. This is the force that acts at the point yp.
The pressure at the centroid is the average pressure, and acts at the centre of mass of the object. For a uniform rectangle this would be half way down the plate.
It can be found using:
Where rho is the density of the liquid, g is the acceleration due to gravity (9.81m/s^2) and z is the average height.
Example
A vertical submerged gate of 5m height and 2m wide closes a channel filling with water. It is hinged at B and 500kN is required to open the gate. The water level is 3m above B. Will the gate open, and where will this force act?
First the magnitude of the force can be found using F=APC
A = 5 x 2 = 10m2
Pc = z ⍴ g
z is found by halving the height of the gate and adding that to the height of water above the plate.
= 5.5 x 1000 x 9.81
= 53955 N/m2
using these values
F= 10 x 53955= 539550 N = 540 kN
Now it needs to be found where this force acts using the equation:
Atmospheric pressure can be ignored due to it acting down on the water and on the other side of the wall so cancels out leaving:
This shows that a force of 540kN acts 5.9m deep (2.9 m down the gate) showing the force is great enough to open the gate.
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