Resultant force on a submerged curved surface

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Presentation transcript:

Resultant force on a submerged curved surface It is most straightforward to calculate the horizontal and vertical components and combine these to obtain the resultant force and its direction. In the diagram below the liquid is resting on top of a curved base.

Horizontal forces Considering the horizontal forces, none can act on CB as there are no shear forces in a static fluid so the forces would act on the faces AC and AB as shown below.

The resultant horizontal force of a fluid above a curved surface is: RH = Resultant force on the projection of the curved surface onto a vertical plane. It is calculated as before. Vertical forces The diagram below shows the vertical forces which act on the element of fluid above the curved surface.

The resultant vertical force of a fluid above a curved surface is: RV = Weight of fluid directly above the curved surface. and it will act vertically downward through the centre of gravity of the mass of fluid.

Resultant force The overall resultant force is found by combining the vertical and horizontal components vectorialy, The angle the resultant force makes to the horizontal is

The figure below shows a situation where there is a curved surface which is experiencing fluid pressure from below.

Horizontal force The resultant horizontal force, RH acts as shown in the diagram. The resultant horizontal force of a fluid below a curved surface is:

RH = Resultant force on the projection of the curved surface on a vertical plane. That is it cab be calculated as before. Vertical force The vertical force are acting are as shown on the figure below. If the curved surface were removed and the area it were replaced by the fluid. The resultant vertical force of a fluid below a curved surface is: Rv =Weight of the imaginary volume of fluid vertically above the curved surface.

Resultant force The overall resultant force is found by combining the vertical and horizontal components vectorialy, The angle the resultant force makes to the horizontal is

Example: Determine the total hydrostatic pressure and center of pressure on a 3 m long and 2 m height quadrant gate. RH 1.33 RV 0.85

Solution: The horizontal component is equal to the hydrostatic pressure on vertical projection of the gate RH = γ ŷ A = 9810 x (0.5 x 2) x (2x3) = 58860 N The pressure center of the horizontal component is at (2/3) of the height i.e. 1.33 m The vertical component equal the weight over the gate RV = γ V = 9810 (0.25 x π 22) 3 = 92460 N The pressure center of the vertical component is at (4r/3π) = 0.85 m The resultant and the direction