A vertical dam has a semicircle gate shown in the figure. Find the hydrostatic force against the gate. The water level is 3 m, the gate width is 4 m, and the overall height is 15 m.

Question

I got 563486.28 when solving - please confirm or check answer and solve step by step, I do not just want an answer!!!

Answer 1

Find Force on Semicircular gate through a Dam wall

Solution:

The force on gate is caused by variable pressure profile.

Therefore, integrate F = ∫ P dA

Consider pressure on bottom of lake (bed) due to a column of water of height h

P = Weight/Cross-section area (CSA) of column

  = mg/CSA = ρVg/CSA, V – Volume of column and ρ density of water

  = ρ x g x h because V = h x CSA. So, this pressure of water occurs on gate wall also.

Further, consider force on tiny segment of gate wall of area dA = r dθ dh

Small force dF = P x dA = ρ x g x h x r dθ dh. Find total force, integrate

      _{3   π}                                                                         

F = ∫ ∫ ρ x g x h x r dθ dh = ½ x ρ x g x h² x r x π

      ^0 0                                                  

    = ½ x ρ x g x 9 x r x π

                                                                                                                 ⁰

We can substitute values density r = 1000 kg/m³, radius r = 2 m, g = 9.81 m/s²

                                                                                                               

F = ½ x 1000 x 9.81 x 9 x 2 x π = 277371 N

For further details please contact me and verify solution. I will send you a pdf with diagram. I hope this helps.

Sincerely,

Shailesh (Sky) Kadakia, Former Professor

University of Pittsburgh, PA

Electrical Engineering Technology Department