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Resultant force on curved surface ABCDE (S = 1.25)

Short answer

With γ = 12.26 kN/m³, net horizontal force from the 4 m wall is 981 kN; vertical force is 2732.5 kN. Resultant 2903 kN at 70.3° below horizontal.

Find the resultant pressure force on curved surface ABCDE due to liquid with specific gravity S = 1.25, take length of the curved surface (normal to the paper) as 10 m. [Figure: free surface at A, vertical wall A-B 4 m, then curved portions with 2 m radii marked, vertical dimensions 4 m, 2 m, 2 m, 2 m, curve ends on the ground at E and C, D lowest point]

Resultant force on a curved surface = FH2+FV2\sqrt{F_H^2 + F_V^2}, where FHF_H is the force on the vertical projection and FVF_V is the weight of the liquid vertically above the surface up to the free surface.

Assumed geometry (the figure is not fully clear): A is at the free surface; AB is a vertical wall of 4 m; the surface then dips as a semicircular trough of radius 2 m from B to E through the lowest point D, so D is 6 m below the free surface. S=1.25S = 1.25, L=10L = 10 m, γ=1.25×9.81=12.26\gamma = 1.25\times 9.81 = 12.26 kN/m³.

 A |~~~~~~~~~~~~~~~~~~~~~~~~ free surface
   | 4 m
 B |                     E
    \         R = 2 m   /
      '--.   C   .--'   
           ' D '          D lowest (6 m)

Horizontal component

On the trough, the forces on the left and right quarters are equal and opposite horizontally, so they cancel. Only the wall AB (depth 4 m) gives a net horizontal force:

FH=γh22L=12.26×422×10=981.0 kNF_H = \gamma\frac{h^2}{2}L = 12.26\times\frac{4^2}{2}\times 10 = 981.0\ \text{kN}

Vertical component

Volume above the trough = rectangle of 4 m depth over the 4 m width + half-disc of radius 2 m:

A=4×4+π×222=16+6.283=22.28 m2FV=γAL=12.26×22.28×10=2732.5 kN (downward)\begin{aligned} A &= 4\times 4 + \frac{\pi\times 2^2}{2} = 16 + 6.283 = 22.28\ \text{m}^2\\ F_V &= \gamma A L = 12.26\times 22.28\times 10 = 2732.5\ \text{kN (downward)} \end{aligned}

Resultant

F=981.02+2732.52=2903.2 kNθ=tan⁡−12732.5981.0=70.3∘ below the horizontal\begin{aligned} F &= \sqrt{981.0^2 + 2732.5^2} = 2903.2\ \text{kN}\\ \theta &= \tan^{-1}\frac{2732.5}{981.0} = 70.3^\circ \text{ below the horizontal} \end{aligned}

Answer: F≈2903F \approx 2903 kN, inclined at 70.3∘70.3^\circ to the horizontal; FH=981F_H = 981 kN, FV=2732.5F_V = 2732.5 kN.

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