Forces on a Partially Submerged Gate

A gate, which is hinged at the bottom, is partially submerged under water, and a cable holds the gate closed. Use the sliders to set the angle of the gate, the weight of the gate and the water height. Use the buttons to change the units from (force) and feet (US customary units) to kN and meters (SI units). Check the "show labels" box to display distances. The Demonstration displays the cable tension needed to support the gate. When the tension is too high (greater than 4.23 or 18.82 kN), the cable breaks.

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This Demonstration determines the cable tension necessary to support a gate submerged under water (see Figure 1). The gate is feet wide at the base, feet long diagonally and measures feet of diagonal length.
The magnitude of the resultant force is found by summing the differential forces over the entire surface:
,
where is the resultant force (N),
is the specific weight of the fluid (),
is the vertical distance from the fluid surface to the centroid center (m),
is the coordinate of the height along the diagonal (see Figure 2 for coordinate system),
is the angle (degrees), and
is area ().
For constant and , the resultant force becomes:
.
The integral is the first moment of the area with respect to the axis, so:
,
where is the coordinate of the centroid of area measured from the axis that passes through 0.
The resultant force can be written as:
,
where is the vertical distance from the fluid surface to the centroid of area.
The coordinate of the resultant force is determined by summing the moments around the axis. Thus, the moment of the resultant force must equal the moment of the distributed pressure forces, or:
.
Therefore, since :
.
The integral in the numerator is the second moment of the area (moment of inertia) , with respect to an axis formed by the intersection of the plane containing the surface and the free surface ( axis):
.
Since , where is the second moment of the area with respect to an axis passing through its centroid and parallel to the axis ():
,
where .
A moment balance determines the tension in the cable that is holding up the gate:
.
Rearranging to solve for tension:
,
where is tension (N), and is the weight of the gate (N).
Figure 1
Figure 2
Reference
[1] B. R. Munson, T. H. Okiishi and W. W. Huebsch, Fundamentals of Fluid Mechanics, 6th ed., Hoboken, NJ: John Wiley & Sons, 2009 pp. 57–60.
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