Centre of Pressure for Vertical and Inclined Plane Surface

Centre of Pressure for Vertical Plane Surface

The point of application of the total pressure on a plane surface is known as centre of pressure. For a plane surface immersed horizontally since the pressure intensity is uniform the total pressure would pass through the centroid of the area i.e., in this case the centroid of the area and the centre of pressure coincide with each other. For a plane surface immersed vertically the centre of pressure does not coincide with the centroid of the area. Since the pressure intensity increases with the increase in the depth of liquid, the centre of pressure for a vertically immersed plane surface lies below the centroid of the surface area. The position of the centre of pressure for a vertically immersed plane surface can be determined as shown below.

Fig. 1 Total Pressure on a Vertical Plane Surface 

As shown in Fig. 1 let ‘ħ’ be the vertical depth of the centre of pressure for the plane surface immersed vertically. Then the moment of the total pressure ‘P’ about axis OO is equal to (P ℏ).

The total pressure on the strip

dP = pdA = wx(bdx)

and its moment about axis OO is

(dP) x = wx² (bdx)

Likewise, by considering a number of small strips and summing the moments of the total pressure on these strips about axis OO, the sum of the moments of the total pressures on all the strips becomes

∫ (dP) x = w ∫ x²(bdx)

By using the “Principle of Moments”, which states that the moment of the resultant of a system of forces about an axis is equal to the sum of the moments of the components about the same axis, the moment of the total pressure about axis OO is

P ℏ = w ∫ x² (bdx)                  (1)

In Eq. (1), ∫ x²(bdx) represents the sum of the second moment of the areas of the strip about axis OO, which is equal to the moment of inertia I0 of the plane surface about axis OO. That is

I₀ = ∫ x² (bdx)                      (2)

Eq. (1) in Eq. (2) and solving for ℏ,

Substituting for the total pressure from Equation P = wAx̅, we obtain

From the “parallel axes theorem” for the moment of inertia, 

I₀ = I_G + A x̅                         (4)

where I_G is the moment of inertia of the area about an axis passing through the centroid of the area and parallel to axis OO.

Introducing Eq. (4) in Eq. (5), it becomes 

Equation (5) gives the position of the centre of pressure on a plane surface immersed vertically in a static mass of liquid. Since for any plane surface the factor (I_G/Ax̅) is always positive, Eq. (5) indicates that ℏ > x̅, i.e., the centre of pressure is always below the centroid of the area. 

Further it is seen that deeper the surface is submerged, i.e., the greater is the value of x̅, the factor (I_G/Ax̅) becomes smaller and the centre of pressure comes closer to the centroid of the plane surface. This is 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. Thus where the variation of pressure is negligible the centre of pressure may be taken as approximately at the centroid. This is justifiable in liquids, only if the depth is very large and the area is small, and in gases because in them the pressure changes very little with depth.

The lateral location of the centre of pressure can also be readily determined by taking moments about any convenient axis in the vertical direction. Thus if ‘OX’ is the reference axis (as shown in Fig.1) in the vertical direction, lying in the same vertical plane in which the plane surface is lying, from which ȳ is the distance of the centre of pressure of the plane surface and ȳ is the distance of the centre of pressure of the small strip on the plane surface, then the distance ȳ may be determined by taking the moments about axis OX. The moment of dP about axis OX is (dP) y =wx (bdx) y and the sum of the moments of the total pressure on all such strips considered on the plane surface is

∫(dP)y = w ∫ xy (bdx) 

which by the principle of moments is equal to the moment of P about axis OX, i.e., (P ȳ).

Thus

The centre of pressure of the plane surface immersed vertically in a static mass of liquid is therefore, at a vertical depth ℏ below the free surface of the liquid and at a distance ȳ from an assumed vertical reference axis OX. If the plane surface has a vertical axis of symmetry passing through its centroid, then this axis may be taken as the reference axis OX, in which case ∫ xy (bdx) = 0, and the centre of pressure lies on the axis of symmetry at a vertical depth ℏ below the free surface of the liquid.

Centre of Pressure for Inclined Plane Surface

Fig. 2 Total pressure on inclined plane surface

As shown in Fig. 2, let ℏ be the vertical depth of the centre of pressure for the inclined plane surface below the free surface of the liquid and its inclined distance from the axis OO be y_p.

Total pressure on the strip shown in Fig. 2 is [w(y sin θ) dA ] and its moment about axis OO is

(dP) y = (w sinθ y²  dA)

By summing the moments of the total pressures on such small strips about axis OO and using the “Principle of Moments”, we get

Py_p = w sin θ ∫ y²  dA                   (6)

Again in Eq. (6), ∫ y²dA represents the sum of the second moments of the areas of the strips about axis OO, which is equal to the moment of inertia I0 of the plane surface about axis OO. That is,

I₀ = ∫ y² dA                       (7)

Introducing Eq. (7) in Eq. (6) and solving for y_p, we obtain

Further from the “parallel axes theorem” for the moments of inertia, 

I₀ = I_G +A ȳ ²                     (9)

where I_G is the moment of inertia of the area about an axis passing through the centroid of the area and parallel to axis OO. Introducing Eq. (9) in Eq. (8) and substituting for the total pressure from P = wA (ȳ sin θ), we get

        (11)

Eq. (10) gives the vertical depth of centre of pressure below free surface of liquid, for an inclined plane surface, wholly immersed in a static mass of liquid. The lateral location of the centre of pressure in this case may also be determined in the same manner as in the case of vertically immersed plane surface, by considering a reference axis OY perpendicular to the axis OO (or OZ) (as shown in Fig. 2) and lying in the same plane in which the inclined plane surface is lying. From the axis OY let z_p be the distance of the centre of pressure of the plane surface and z be the distance of the centre of pressure of the small strip on the plane surface. The moment of the total pressure dP on the small strip about axis OY is (dP) y = wx(dA) z and the sum of the moments of the total pressure on all such strips considered on the plane surface is

∫ (dP) z = w ∫ xz (dA)

which by the principle of moments is equal to the moment of P about axis OY, i.e., (Pz_p). Thus