Find the horizontal and vertical forces per m of width on the tainter gate shown in the fig.

Forces on Tainter Gate

Problem Statement

Find the horizontal and vertical forces per meter of width on the tainter gate shown in the figure.

The horizontal force is given by:

\( F_H = \gamma A \bar{y} \)

Substitute the values:

\( F_H = 9810 \cdot (7.5 \cdot 1) \cdot \frac{7.5}{2} = 275906 \, \text{N} = 275.906 \, \text{kN (right)} \)

The horizontal force acts at a distance of:

\( \frac{7.5 \cdot 2}{3} = 5 \, \text{m} \)

from the water surface.

The vertical force is the weight of the imaginary volume of water vertically above ABCA:

\( F_V = \gamma \left[ \text{Volume}_{\text{sector AOBC}} – \text{Volume}_{\text{triangle AOB}} \right] \)

Substitute the values:

\( F_V = 9810 \cdot \left[ \frac{60}{360} \cdot \pi \cdot (7.5)^2 \cdot 1 – 0.5 \cdot 7.5 \cdot (7.5 \cdot \cos 30) \cdot 1 \right] \)

Calculation:

\( F_V = 9810 \cdot (29.45 – 5.09) = 49986 \, \text{N} = 49.986 \, \text{kN (up)} \)

The vertical force acts through the centroid of the segment ABCA.

Result:

Horizontal Force: The horizontal force is caused by the hydrostatic pressure acting on the gate’s vertical projection. It depends on the width, depth, and location of the centroid of the submerged surface.
Vertical Force: The vertical force arises due to the imaginary weight of water above the curved and flat surfaces of the gate. This is calculated using the volume of the water segment above the gate.

This problem highlights the interaction between hydrostatic forces and the geometry of curved structures like tainter gates:

Horizontal Force: Ensures the gate can resist water pressure and maintain structural stability.
Vertical Force: Represents the lifting effect of water on the curved gate, balancing the weight of the gate or other forces acting downward.
Applications: These calculations are critical in designing gates, spillways, and other water-retaining structures to ensure safety and efficiency.