Meriam, Kraige & Bolton — Distributed Forces: Centroids and Centers of Gravity: Problem 5_175

⚡ Mecademy AIENG정역학 · ch5  Problem Statement The upstream side of an arched dam has the form of a vertical cylindrical surface of 500-ft radius and subtends an angle of 60°. If the fresh water is 100 ft deep, determine the total force R exerted by the water on the dam face. Problem 5/175 (a) Determination of total resultant force R 1. Formula: The total resultant hydrostatic force on a vertical submerged curved surface is equivalent to the force acting on the vertical projection of that surface. For a cylindrical surface, the pressure acts radially, and by symmetry, the vertical and transverse components cancel out. The resultant force is horizontal and acts along the axis of symmetry. It is calculated as: where: = specific weight of the fluid (for fresh water, ) = depth to the centroid of the projected vertical area = area of the vertical projection of the submerged surface 2. Substitution: Radius, Depth of water, Total subtended angle, Chord length (width of the projected area), Projected vertical area, Depth to the centroid of the rectangular projected area, Specific weight of water, R R R=γA h ˉ p γγ≈ 62.4 lb/ft 3 h ˉ A p r=500 ft h=100 ft θ=60 ∘ w=2rsin(θ/2)= 2(500)sin(30)= ∘ 500 ft A = p w⋅h=500⋅100= 50,000 ft 2 =h ˉ = 2 h = 2 100 50 ft γ= 62.4 lb/ft 3 Substituting these into the formula: 3. Calculation: Intermediate average pressure: Total force: Final numerical value: 4. Result: ● Final Conclusion: The total resultant force exerted by the water on the dam face is . T