Problem Statement
Gate AB in the figure is 4.8 m long and 2.4 m wide. Neglecting the weight of the gate, compute the water level \( h \) for which the gate will start to fall.
Solution
1. Area of the Gate
The area of the gate is:
\( A = \frac{2.4 \cdot h}{\sin 60} = 2.77h \, \text{m}^2 \)
2. Location of Center of Gravity (CG)
The depth of the CG from the free surface is:
\( y_{\text{CG}} = \frac{h}{2} \, \text{m} \)
3. Resultant Force on the Gate
The resultant force is:
\( F = \gamma \cdot A \cdot y_{\text{CG}} \)
\( F = 9810 \cdot 2.77h \cdot \frac{h}{2} = 13587h^2 \, \text{N} \)
4. Moment of Inertia about CG
The moment of inertia about the CG is:
\( I_G = \frac{1}{12} \cdot 2.4 \cdot \left(\frac{h}{\sin 60}\right)^3 = 0.308h^3 \, \text{m}^4 \)
5. Vertical Distance of CP from Free Surface
The vertical distance of the center of pressure (CP) is:
\( y_p = y_{\text{CG}} + \frac{I_G \cdot \sin^2 60}{A \cdot y_{\text{CG}}} \)
\( y_p = \frac{h}{2} + \frac{0.308h^3 \cdot \sin^2 60}{2.77h \cdot \frac{h}{2}} = 0.667h \, \text{m} \)
6. Distance of Force from Point B
The distance of the resultant force from point B is:
\( \Delta y = \frac{h – 0.667h}{\sin 60} = 0.384h \, \text{m} \)
7. Solve for Water Level \( h \)
Taking moments about point B:
\( 5000 \cdot 4.8 – 13587h^2 \cdot 0.384h = 0 \)
\( h = 1.66 \, \text{m} \)
Result:
- The water level at which the gate will start to fall is \( h = 1.66 \, \text{m} \).
Explanation
- Area of the Gate: The area is calculated considering the inclined gate surface and its dimensions.
- Center of Gravity: The CG is located at the midpoint of the water height.
- Resultant Force: The hydrostatic force depends on the area and the depth of the CG.
- Moment of Inertia: The moment of inertia is determined for the inclined gate about its CG.
- Center of Pressure: The CP is slightly below the CG due to pressure distribution, and its position is adjusted using the moment of inertia.
- Moment Balance: The critical water height is found by taking moments about the hinge point and solving for \( h \).
Physical Meaning
This problem illustrates the principles of hydrostatic forces acting on inclined gates. It highlights the importance of calculating the critical water level to ensure stability and avoid gate failure under fluid pressure.
