Problem Statement
The gate AB shown is hinged at A and is in the form of a quarter-circle wall with a radius of 12 m. If the width of the gate is 30 m, calculate the force required \( P \) to hold the gate in position.
Solution
1. Horizontal Force \( F_H \)
The horizontal force is calculated as:
\( F_H = \gamma \cdot A \cdot \bar{y} \)
\( F_H = 9810 \cdot (30 \cdot 12) \cdot \frac{12}{2} = 21189600 \, \text{N} = 21189.6 \, \text{kN} \; (\text{right}) \)
The horizontal force acts at a distance of:
\( \frac{12}{3} = 4 \, \text{m} \)
above the hinge A.
2. Vertical Force \( F_V \)
The vertical force is equal to the weight of the volume of water vertically above AB:
\( F_V = \gamma \cdot \text{Volume}_{AOB} \)
\( F_V = 9810 \cdot \left(\frac{\pi \cdot 12^2}{4} \cdot 30\right) = 33284546 \, \text{N} = 33284.546 \, \text{kN} \; (\text{downward}) \)
The vertical force acts at a distance of:
\( \frac{4r}{3\pi} = \frac{4 \cdot 12}{3 \cdot 3.1416} = 5.1 \, \text{m} \)
from the vertical AO.
3. Moment Balance about A
Taking the moment about A:
\( P \cdot 12 = F_H \cdot 4 + F_V \cdot 5.1 \)
\( P \cdot 12 = 21189.6 \cdot 4 + 33284.546 \cdot 5.1 \)
\( P = \frac{21189.6 \cdot 4 + 33284.546 \cdot 5.1}{12} \)
\( P = 21209 \, \text{kN} \)
Result:
- Force required to hold the gate in position: \( P = 21209 \, \text{kN} \)
Explanation
- Horizontal Force: The horizontal force is determined by considering the hydrostatic pressure acting on the curved surface and its projection.
- Vertical Force: The vertical force corresponds to the weight of water above the gate, calculated from the volume of water.
- Moment Balance: The moments due to the horizontal and vertical forces are balanced by the force \( P \), calculated using the principle of equilibrium.
Physical Meaning
This problem illustrates the forces acting on a curved gate due to hydrostatic pressure and water weight. Engineers use such calculations to ensure the stability of gates and structures in fluid environments, ensuring that the applied forces keep the system in equilibrium.
