The Resultant Forces Act
Perpendicular
to the Dam
The
Lateral Force Component on only
One Side of Dam A_{p}


(a) The resultant force (F_{R}) 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 (F_{H}) due to fluid pressure (hydrostatic conditions) acting on the projected area A_{p} can be determined for a unit width as
F_{h} = γ h_{c} A_{p} = γ (h/2) A_{p}
where h is the total water depth.
To find the maximum h, set F_{h} to the allowable force,
F_{m}, to give
F_{h} = F_{m}
(62.4 lb/ft^{3}) (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.
F_{h1}  F_{h2} = F_{m}
γ h_{c}1 A_{p1} γ h_{c2} A_{p2} = 78,000 lb
62.4 (h/2) [(h)(1)]  62.4 (5) [(10)(1)] = 78,000
31.2 h^{2} = 81,120
h = 51.0 ft
(c)
Since the 10 ft of water downstream may not exist in times of draught, the dam should be limited to 50 ft of water. 