Hydrostatic Forces on Curved Submerged Surfaces x Pz

Hydrostatic Forces on Curved, Submerged Surfaces x Pz P q Px Z q q d. A =d. Acos(q) x d. Az=d. Asin(q) Pressure is always acting perpendicular to the solid surface since there is no shear motion in static condition.

Projected Forces x h d. Az Z Integrated over all elements

Buoyancy Force acting down FD= rg. V 1 from Buoyancy = FU-FD =rg(V 2 -V 1)=rg. V V: volume occupied by the object Force acting up FU = rg. V 2 from

Horizontal Forces x h Projected area Ax d. Az Z Integrated over all elements h d. Ax Equivalent system: A plane surface perpendicular to the free surface Finding Fx is to determine the force acting on a plane submerged surface oriented perpendicular to the surface. Ax is the projection of the curved surface on the yz plane. Similar conclusion can be made to the force in the y direction Fy.

Examples Determine the magnitude of the resultant force acting on the hemispherical surface. x 2 m R=0. 5 m z equals minus

Line of Action Horizontal direction: line of action goes through z’ z’ Vertical direction: the line of action is 3 R/8 away from the center of the hemisphere z. C Projection in x-direction z The resultant moment of both forces with respect to the center of the hemisphere should be zero: Fx(2. 03125 -2)-Fz(0. 1875) =15386(0. 03125)-2564(0. 1875)=0 location of the centroid for a hemisphere is 3 R/8=0. 1875(m) away from the equator plane