Centre of Pressure

Force, $F$ given by Eq. 2.33 is the resultant force acting on the plane due to the liquid and acts at what is called the Center of Pressure (CP). It does not act at the centroid of the plane as it may seem. Let the coordinates of CP be $(x_p,y_p)$. Noting that the moment of the resultant force is equal to the moment of the distributed force about the same axis, we have

$$x_p~F~=~\int_A~x~p~dA, ~~~~~~~y_p~F~=~\int_A~y~p~dA$$ (2.34)

Before substituting for $F$ in the above equation we note that the atmospheric pressure $p_a$ acting at the free surface also acts everywhere within the fluid and also on both sides of the plane. As such it does not contribute to the net force upon the plane. So we drop term $p_a$ from the equation for $F$. Eq.2.34 becomes

$$y_p~\gamma~\sin \theta ~y_c~A~=~\int_A y \gamma h dA~=~\gamma \sin \theta \int_A y^2~dA$$ (2.35)

The term $\int_A y^2~dA$ is the well-known second moment of area about the x-axis denoted by $I_{xx}$ leading to

$$y_p~=~{{I_{xx}} \over {{A~y_c}}}$$ (2.36)

$I_{xx}$ is related to that about the x-axis passing through the centroid of the area, $I_{xc}$ through the Parallel Axes Theorem given by

$$I_{xx}~=~I_{xc}~+~A~y_c ^2$$ (2.37)

Consequently, we have

$$y_p~=~y_c~+~{I_{xc} \over {{A~y_c}}}$$ (2.38)

Similarly, taking moments about the y-axis, we obtain,

$$x_p~F~=~\int_A \gamma \sin \theta x~y~dA$$

$$\noalign{\texttt{leading to}} ~~~~~~ x_p~=~{I_{xy} \over {A~y_c}}$$ (2.39)

$I_{xy}$ is the product of inertia with respect to x and y axes. Again on the application of the parallel axis theorem we have

$$x_p~=~{I_{xyc} \over {y_c A}}+x_c$$ (2.40)

where $I_{xyc}$ is the product of inertia about the axes passing through the centroid. The coordinates of the Centre of Pressure are thus given by $(x_p,y_p)$ (Eqns. 2.40 and 2.38).

The resulting force upon the immersed surface is therefore given by

$$F~=\gamma h_c A$$ (2.41)

The centre of pressure is given by

$$x_p=x_c+{I_{xyc} \over {y_c A}}~;~ y_p=y_c~+~{I_{xc} \over {{A~y_c}}}$$ (2.42)

 

Expressions for the moments$I_{xy}$, $I_{xyc}$ etc for some of the common shapes are given in the next section.

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