8/31/2024
CE- 203 FLUID MECHANICS
Lecture 13
Forces on Submerged Surfaces
Sreeja Pekkat
Associate Professor
Dept. of Civil Engineering
IIT Guwahati
Hydrostatic Thrusts on Submerged Curved Surfaces
 On a curved surface, the direction of normal changes from point to point
 Hence the pressure forces on individual elemental surfaces differ in their
directions
 Therefore, a scalar summation of them cannot be made
 Instead, the resultant thrusts in certain directions are to be determined
 And these forces may then be combined vertically
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Hydrostatic Thrusts on Submerged Curved Surfaces
 A rectangular Cartesian coordinate system
o
x
is introduced whose x-y plane coincide
with the free surface of the liquid
 The
z-axis
is
directed downward
y
below the x-y plane
z
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Hydrostatic Thrusts on Submerged Curved Surfaces
o
x
Consider an arbitrary
submerged
curved
surface
A
zc
yc
y
zp
C
C
yp
P
P
Fx
z
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Hydrostatic Thrusts on Submerged Curved Surfaces
o
 Consider an elemental area dA at a
x
depth z from the surface of the
liquid
 the force acts in the y
A
zc
direction normal to
yc
the area dA
z
zp
C
P
C
yp
Fx
P
y
dA
z
dF
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Hydrostatic Thrusts on Submerged Curved Surfaces
o
x
 The hydrostatic force
on the elemental area
dA is
A
dF   gzdA
zc
yc
y
z
zp
P
C
C
yp
Fx
P
y
dA
z
dF
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Hydrostatic Thrusts on Submerged Curved Surfaces
o
x
 The components of the force
dF in x, y and z direction can
dAz
be taken
A
zc
yc
y
z
zp
P
C
C
yp
Fx
P
y
dA
dAx
z
dFx
dF
dFy
dFz
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Hydrostatic Thrusts on Submerged Curved Surfaces
 The components of the force dF in x, y and z direction are
dFx  ldF  l  gzdA 

dFy  mdF  m  gzdA  A 

dFz  ndF  n  gzdA 
Where
l, m, n – direction cosines of the
normal to dA
 The components of the surface element A projected on the yz, xz and xy plane are
respectively
dAx  ldA 

dAy  mdA  B 

dAz  ndA 
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Hydrostatic Thrusts on Submerged Curved Surfaces
 dFx   gzdAx
 Put (B) in (A)
dFy   gzdAy
dFz   gzdAz
 The components of the total hydrostatic force along the coordinate axes are
Fx    gzdAx 


Fy    gzdAy  (C )

Fz    gzdAz 

Where
Fx   gzc Ax
Fy   gzc Ay
zc - z-coordinate of the centroid of the area Ax and Ay (the projected areas of the curved surface on yz
and xz planes
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Hydrostatic Thrusts on Submerged Curved Surfaces
 Consider the vertical component of hydrostatic component of hydrostatic force on the
curved surface
Fz    gzdAz
  gV
 where
V - volume of the body of the liquid within the region extending vertically above the submerged
surface to the free surface of the liquid
∴ The vertical component of the hydrostatic force on a submerged curved surface is equal to the
weight of the liquid volume vertically above the solid surface to the free surface of the liquid in that
volume
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Centre of Pressure of Submerged Curved Surfaces
 Equating the moment of the resultant forces about the x-axis to the summation of the
moments of the component forces, we have
z p Fx   zdFx
z p   gzdAx   z  gzdAx
z p  zdAx   z 2 dAx
 z dA
z 
 zdA
2
x
p
x
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Centre of Pressure of Submerged Curved Surfaces
 The ordinate of the centre of area of the plane surface Ax is defined as
z dA

z 
 zdA
2
x
zc 
1
zdAx
Ax 
 zp 
 Ax zc   zdAx
p
x
1
z 2 dAx

Ax Z c
zp 
I yy
Ax Z c
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Centre of Pressure of Submerged Curved Surfaces
 If zp and yp are taken to be the coordinates of the point of action of Fx on the projected area Ax
on the yz plane
I yy
1
2
z
dA

x
Ax Z c 
Ax Z c
I yz
1
yp 
yzdA

x
Ax Z c 
Ax Z c
 zp 
Where
 Iyy - moment of inertial of area area Ax about the y-axis
 Iyz - product of inertia of Ax with respect to the axes y and z
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Centre of Pressure of Submerged Curved Surfaces
 In a similar fashion z’p and x’p, the coordinates of the point of action of force Fy on area Ay can be
written as
I 
1
z 2 dAy  xx 

Ay Z c
Ay Z c 
D
I xz 
1
xp 
xzdAy 
Ay Z c 
Ay Z c 
z p 
Where
 Ixx is the moment of inertia of the area Ax about x-axis
 Ixz is the product of inertia of Ay about axes x and z
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Hydrostatic Thrusts on Submerged Curved Surfaces
 From the above equations (C to D) for a curved surface, the component of hydrostatic
force in a horizontal direction is equal to the hydrostatic force on the projected plane
surface perpendicular to that direction and acts through the centre of pressure of the
projected area.
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