# Depth of Centre of Pressure For Different Vertical Plane Surfaces4 min read

## Hydrostatic Force:

Fluid statics or hydrostatics is the branch of fluid mechanics in which the stresses produced in a fluid system are determined when it is at rest or in a stationary state.

Force exerted by the fluid on the surface with which it is in contact is called Total Hydrostatic Force.

Since for a fluid at rest, shear force does not exist, the total hydrostatic force acts on the surface in the normal direction.

## Centre of Pressure:

The point of application of total hydrostatic force on the surface is known as Centre of Pressure.

In the case of plane surfaces, the total hydrostatic force can be easily calculated because the direction of
force at all points is the same and can be added algebraically.

But in the case of a curved surface, the directions of all forces are different as they are perpendicular to the surface. So, for this case, all forces are resolved into two coordinate systems, and then they have been added algebraically.

Using the principles of hydrostatics, we can compute forces on submerged objects, develop instruments for measuring pressure.

Fluid statics is used in the design of many engineering systems such as dams and liquid storage tanks.

## Depth of Centre of Pressure For Vertical Plane Surface

In the case of vertical plane surfaces, pressure intensity increases with depth. Thus, the centroid of pressure intensity will always be below the centroid of the surface area.

The deeper the surface is submerged, the smaller is the gap between the centre of pressure and the centroid of the plane surface. This is so because as the pressure becomes greater with increasing depth, its variation over a given area becomes smaller in proportion, thereby making the distribution of pressure more uniform.

### 1. Rectangle

• Centre of Gravity (C.G.) or centroid of rectangular section is:

Centroid on X-axis = b / 2

Centroid on Y-axis = h / 2

• The Centre of pressure (C.P.) for rectangular section is:

$\huge C.P.=\frac{2h}{3}$

• The Moment of inertia (IG) of a rectangle with respect to an axis passing through its centroid, is given by

$\huge I_G=\frac{bh^3}{12}$

• The Moment of inertia (IG) of a rectangle with respect to an axis passing through its base, is given by

$\huge I_O_O=\frac{bh^3}{3}$

### 2. Trapezium

• Centre of Gravity (C.G.) or centroid of trapezium section is:

$\huge C.G.=\ \left(\frac{a+2b}{a+b}\right)\frac{h}{3}$

• The Centre of pressure (C.P.) for trapezium section is:

$\huge C.P.=\ \left(\frac{a+3b}{a+2b}\right)\frac{h}{2}$

• The Moment of inertia (IG) of a trapezium with respect to an axis passing through its centroid, is given by

$\huge I_G=\left(\frac{a^2+b^2+4ab}{36\left(a+b\right)}\right)h^3$

• The Moment of inertia (IG) of a trapezium with respect to an axis passing through its base, is given by

$\huge I_{oo}=\left(\frac{a+3b}{12}\right)h^3$

### 3. Triangle

(a)

• Centre of Gravity (C.G.) or centroid of triangle section is:

$\huge =\frac{2h}{3}$

• The Centre of pressure (C.P.) for triangle section is:

$\huge C.P.=\frac{3h}{4}$

• The Moment of inertia (IG) of a triangle with respect to an axis passing through its centroid, is given by

$\huge I_G=\frac{bh^3}{36}$

• The Moment of inertia (IG) of a triangle with respect to an axis passing through its base, is given by

$\huge I_O_O=\frac{bh^3}{4}$

(b)

• Centre of Gravity (C.G.) or centroid of triangle section is:

=h/3

• The Centre of pressure (C.P.) for triangle section is:

$\huge C.P.=\frac{h}{2}$

• The Moment of inertia (IG) of a triangle with respect to an axis passing through its centroid, is given by

$\huge I_G=\frac{bh^3}{36}$

• The Moment of inertia (IG) of a triangle with respect to an axis passing through its base, is given by

$\huge I_O_O=\frac{bh^3}{12}$

### 4. Circle

• Centre of Gravity (C.G.) or centroid of circle section is:

= D/2

• The Centre of pressure (C.P.) for circle section is:

$\huge C.P.=\frac{5D}{8}$

• The Moment of inertia (IG) of a circle with respect to an axis passing through its centroid, is given by

$\huge I_G=\frac{\pi D^4}{64}$

• The Moment of inertia (IG) of a circle with respect to an axis passing through its base, is given by

$\huge I_O_O=\frac{5\pi D^4}{64}$

### 5. Semi Circle

• Centre of Gravity (C.G.) or centroid of semi-circle section is:

= 2D/3Π

• The Centre of pressure (C.P.) for semi-circle section is:

$\huge C.P.=\ \frac{3\pi D}{32}$

• The Moment of inertia (IG) of a semi-circle with respect to an axis passing through its centroid, is given by

$\huge I_G=\ \left(\frac{\pi}{8}-\frac{8}{9\pi}\right)\frac{D^4}{16}$

• The Moment of inertia (IG) of a semi-circle with respect to an axis passing through its base, is given by

$\huge I_{OO}=\frac{\pi D^4}{128}$

### 6. Parabola

(a)

• Centre of Gravity (C.G.) or centroid of parabola section is:

=3h/5

• The Centre of pressure (C.P.) for parabola section is:

$\huge C.P.=\frac{5h}{7}$

(b)

• Centre of Gravity (C.G.) or centroid of parabola section is:

= 2h/5

• The Centre of pressure (C.P.) for parabola section is:
$\huge C.P.=\frac{4h}{7}$