The Resultant Forces Act
to the Dam
Lateral Force Component on only
One Side of Dam Ap
(a) The resultant force (FR) due
to water pressure is perpendicular to the dam surface. Here, only concerned with lateral (horizontal) component, so you can use the projected vertical dimension of the dam.
The lateral force (FH) due to fluid pressure (hydrostatic conditions) acting on the projected area Ap can be determined for a unit width as
Fh = γ hc Ap = γ (h/2) Ap
where h is the total water depth.
To find the maximum h, set Fh to the allowable force,
Fm, to give
Fh = Fm
(62.4 lb/ft3) (h/2) [h (1 ft)] = 78,000 lb
h = [78,000 / 31.2]0.5 = 50 ft
Lateral Force Components
Acting on the Dam
(b) If the water depth downstream is 10 ft, then a resultant force will act on that side of the dam, and will counteract the upstream force.
Fh1 - Fh2 = Fm
γ hc1 Ap1- γ hc2 Ap2 = 78,000 lb
62.4 (h/2) [(h)(1)] - 62.4 (5) [(10)(1)] = 78,000
31.2 h2 = 81,120
h = 51.0 ft
Since the 10 ft of water downstream may not exist in times of draught, the dam should be limited to 50 ft of water.