When a static mass of fluid comes in contact with a surface, either plane or curved, a force is exerted by the fluid on the surface. This force is known as total pressure. Since for a fluid at rest no tangential force exists, the total pressure acts in the direction normal to the surface. The point of application of total pressure on the surface is known as centre of pressure. As indicated later an engineer is often required to compute the magnitude of total pressure and to locate its point of application in the design of several hydraulic structures.
Total Pressure on a Horizontal Plane Surface
Consider a plane surface immersed in a static mass of liquid of specific weight w, such that it is held in a horizontal position at a depth ‘h’ below the free surface of the liquid, as shown in Fig. 1. Since every point on the surface is at the same depth below the free surface of the liquid, the pressure intensity is constant over the entire plane surface, being equal to p = wh. Thus if ‘A’ is the total area of the surface then the total pressure on the horizontal surface is
P = pA = (wh) A = wAh (1)
The direction of this force is normal to the surface, as such it is acting towards the surface in the vertical downward direction at the centroid of the surface.
Total Pressure on a Vertical Plane Surface
Fig. 2 shows a plane surface of arbitrary shape and total area ‘A’, wholly submerged in a static mass of liquid of specific weight ‘w’. The surface is held in a vertical position, such that the centroid of the surface is at a vertical depth of ‘x’ below the free surface of the liquid. It is required to determine the total pressure exerted by the liquid on one face of the plane surface.
In this case since the depth of liquid varies from point to point on the surface, the pressure intensity is not constant over the entire surface. As such the total pressure on the surface may be determined by dividing the entire surface into a number of small parallel strips and computing the total pressures on each of these strips. A summation of these total pressures on the small strips will give the total pressure on the entire plane surface.
Consider on the plane surface a horizontal strip of thickness ‘dx’ and width ‘b’ lying at a vertical depth ‘x’ below the free surface of the liquid. Since the thickness of the strip is very small, for this strip the pressure intensity may be assumed to be constant equal to p = wx. The area of the strip being dA = (b × dx), the total pressure on the strip becomes
dP = pdA = wx(bdx) (2)
Total pressure on the entire plane surface is
P = ∫ dP = w∫ x(bdx)
But ∫ x (bdx) represents the sum of the first moments of the areas of the strips about an axis OO, (Which is obtained by the intersection of the free surface of the liquid with the vertical plane in which the plane surface is lying) which from the basic principle of mechanics is equal to the product of the area A and the distance x of the centroid of the surface area from the same axis OO. That is
∫ x (bdx) = Ax̄
P = wA x̄ (3)
Equation (3) thus represents a general expression for total pressure exerted by a liquid on a plane surface. Since w x̄ is the intensity of pressure at the centroid of the surface area, it can be stated that the total pressure on a plane surface is equal to the product of the area of the surface and the intensity of pressure at the centroid of the area.
Total pressure on a horizontal plane surface can also be determined by Eq. (3), since in this case
x̄ = h.
Total Pressure on Inclined Plane Surface
Consider a plane surface of arbitrary shape and total area ‘A’, wholly submerged in a static mass of liquid of specific weight ‘w’. The surface is held inclined such that the plane of the surface makes an angle ‘θ’ with the horizontal as shown in Fig. 3. The intersection of this plane with the free surface of the liquid is represented by axis OO, which is normal to the plane of the paper.
Let x̄ be the vertical depth of the centroid of the plane surface below the free surface of the liquid and the inclined distance of the centroid from axis OO measured along the inclined plane is ȳ.
Consider on the plane surface, a small strip of area ‘dA’ lying at a vertical depth of ‘x’ and its distance from axis OO being ‘y’. For this strip the pressure intensity may be assumed to be constant equal to p = wx.
Total pressure on the strip is
dP = wx (dA)
Since x = y sin θ
dP = w (y sin θ) (dA)
By integrating the above expression the total pressure on the entire surface is obtained as
P = (w sin θ) ∫ y (dA)
Again ∫ y dA represents the sum of the first moments of the areas of the strips about axis OO, which is equal to the product of the area A and the inclined distance of the centroid of the surface area from axis OO. That is
∫ y dA = A ȳ
∴ P = wA (ȳ sin θ) (4)
But x̄ = ȳ sin θ
P = wA x̄ (5)
Eq. 5 is same as Eq. 3, thereby indicating that for a plane surface wholly submerged in a static mass of liquid and held either vertical or inclined, the total pressure is equal to the product of the pressure intensity at the centroid of the area and the area of the plane surface.
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