FLUID MECHANICS  THEORY



In this section, the discussion is focused on
how to determine hydrostatic forces exerted on a plane surface submerged
in a static fluid. Determining this hydrostatic force is particularly crucial
in the design of engineering structures, such as dams, storage tanks and
hydraulic systems. The concept of using a pressure prism to obtain the
hydrostatic force will also be introduced. 





Hydrostatic Force on a Plane Surface

Hydrostatic Force on an
Inclined Arbitrary Plane Surface


Consider an inclined plane submerged in
a static fluid as shown in the figure. The resultant force F_{R} is
acting perpendicular to the plane since no shear force is present when
the fluid is at
rest. F_{R }has a line of action that passes through the point
(x_{cp} , y_{cp}), which is called the center of pressure.
Note that the pressure acting perpendicular to the plane surface is
also referred to as the normal stress.
Now take a small differential element dA at a depth of h. The differential
force dF acting on dA is given by
dF = ρgh dA
The magnitude of the resultant force can be obtained by integrating the
differential force over the whole area






The integral represents
the first moment of the area about the x axis, which is equal
to
where y_{c }is the y coordinate of the centroid of the plane
surface.
From trigonometry,
h_{c} = y_{c} sinθ
where h_{c }is the vertical distance from the fluid surface to the centroid of the plane surface. The resultant force is simplified to




Parallel Axis Theorem


The center of pressure, x_{cp} and y_{cp} can be obtained by summing moments about the y
and x axis, respectively. First, by equating the sum of moments of all
pressure forces about the x axis to the moment of the resultant force:
where is
the second moment of the area or the area moment of inertia (I_{x}) about the x axis. According to the parallel axis theorem, the moment of inertia can also be written as
I_{x } = I_{x}' + Ay_{c}^{2}
where I_{x}' is the second moment of the area with respect to the centroidal axis, which is parallel to the x axis.
Hence, the center of pressure coordinate y_{cp} is given by






Similarly, x_{cp} can be obtained by equating the
sum of moments of all pressure forces about the y axis to the moment
of the resultant force
where is
the product of inertia (I_{xy})
of the area about the x and y axes. Once again according to the parallel
axis theorem, it can also be written as
I_{xy } = I_{xy}' + Ax_{c}y_{c}
where I_{xy}' is the product of inertia of the area with respect
to the centroidal axes.
Hence, the coordinate x_{cp} is given by
The values for the second moment of inertia and product of inertia with respect to the centroidal axes for various common geometries are given in the appendix.
From the formulations of x_{cp} and y_{cp}, it is noted that the center of pressure is always lower in the liquid than the centroid of the plane area. 





Pressure Prism 
Hydrostatic Pressure Force Exerted
on a Vertical Plane Surface
Pressure Prism


An alternate approach of determining the hydrostatic force is by means of a pressure prism. Consider a vertical plane submerged in a static fluid, as shown in the figure. The pressure increases linearly with the depth. One can then easily construct a corresponding threedimensional diagram of the pressure distribution, and such a volume is called a pressure prism. The resultant force is the total volume of the pressure prism, that is
F_{R} = Volume = 1/2 (ρgh) (bh)
The resultant force passes through the centroid of the pressure prism. For this particular example, the centroid of a triangular element is located at a distance of h/3 from its base and lies in the vertical symmetry axis.
As illustrated, this method is particularly convenient when the shape of the pressure prism is a common geometry, in which the volume and centroid can be readily obtained.



