Hydrostatic Force on a submerged surface

The other important utility of the hydrostatic equation is in the determination of force acting upon submerged bodies. Among the innumerable applications of this are the force calculation in storage tanks, ships, dams etc.

Figure 2.13 : Force upon a submerged object

 

First consider a planar arbitrary shape submerged in a liquid as shown in the figure. The plane makes an angle $\theta$ with the liquid surface, which is a free surface. The depth of water over the plane varies linearly. This configuration is efficiently handled by prescribing a coordinate frame such that the y-axis is aligned with the submerged plane. Consider an infinitesimally small area $dA=dx.dy$ at a (x,y). Let this small area be located at a depth $h$ from the free surface. From Eq.2.18 we know that

$$p = p_a + \gamma h$$ (2.30)

where $p_a$ is the pressure acting on the free surface. The hydrostatic force on the plane is given by,

$$F~=~\int p~dA~ =~\int_A~(p_a+\gamma~h)dA$$

$$=~p_a~A~+~\gamma~\int_A~h~dA~$$

$$=~p_a~A~+~\int_A \gamma~y~\sin \theta dA$$

$$\texttt{i.e.,} F~ =~p_a~A~+~\gamma~\sin \theta \int_A y~dA$$ (2.31)

The integral, $\int_A y dA$ is the first moment of surface area about x axis. If $y_c$ is the y-coordinate of the centroid of the area we have,

$$\int_A~y~dA~=~y_c~A$$ (2.32)

Consequently, Eq. 2.31 is rewritten as

$$F~=~p_a~A~+~\gamma~\sin \theta~y_c~A~=~(p_a~+~\gamma h_c)A~=~p_c~A$$ (2.33)

where $h_c~=~y_c~\sin \theta$ and $p_c$ is the pressure acting at the centroid.

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney