A gravity dam can resist a maximum lateral force of F_m = 78, 000 lb. (a) What is the maximum height of the dam if there is no downstream (right side of dam)? (b) What if the water depth downstream is 10 ft? (c) Would it be a good idea to design the dam for the condition of part (b)?

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/ft³) (h/2) [h (1 ft)] = 78,000 lb
     h = [78,000 / 31.2]^0.5 = 50 ft

(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_c1 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 h² = 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.