So we note from this that the resultant force is the average pressure

times the area.

And that simple formula works also for

a wall which is partially submerged, like this.

So if I'm just interested in this portion of a gate or a plug, for example.

Which is below the water surface then again, the resultant force is the pressure

at the centroid of that gate multiplied by area which is equal to

gamma h1 + h2 over 2, where h1 is the depth of the top of that gate.

Below the water surface, and h2 is the depth of the bottom of the gate.

The pressure distribution is shown over on the right here, and

that is equal to p1 + p2 over 2,

where p1 is the pressure at the top of the gate, p2 is the pressure at the bottom.

The area is h2- h1 times b, and the same formula applies.

But note that that only applies for a gate of constant width,

in other words a rectangular wall.

Because in that case the force in both cases doesn't act at the average depth.

It also applies to an inclined plane, the same formulas.

And these are the corresponding pages or reference from the Reference Handbook,

which just give the same formulas that I just derived.

Now let's do some numerical examples.

The first one, we have a water tank here which is connected to a round pipe.

So this is a round pipe here, 6 feet in diameter.

And it's sealed off by a circular plug.