Problem Statement
An inclined circular gate with water on one side is shown in the figure. Determine the total resultant force acting on the gate and the location of the center of pressure.
Solution
1. Calculate Area of the Gate
The area of the gate is:
\( A = \frac{\pi \cdot 1^2}{4} = 0.785 \, \text{m}^2 \)
2. Location of Center of Gravity (CG)
The CG is located at:
\( y_{\text{CG}} = 1.8 + \frac{1.0 \cdot \sin 60}{2} = 2.23 \, \text{m} \)
3. Resultant Force on the Gate
The resultant force is:
\( F = \gamma \cdot A \cdot y_{\text{CG}} \)
\( F = 9810 \times 0.785 \times 2.23 = 17173 \, \text{N} = 17.173 \, \text{kN} \)
4. Location of Center of Pressure (CP)
Using the moment of inertia about the CG:
\( I_G = \frac{\pi}{64} \cdot 1^4 = 0.049 \, \text{m}^4 \)
The location of the CP is:
\( y_p = y_{\text{CG}} + \frac{I_G \cdot \sin^2 \theta}{A \cdot y_{\text{CG}}} \)
\( y_p = 2.23 + \frac{0.049 \cdot \sin^2 60}{0.785 \cdot 2.23} = 2.25 \, \text{m} \)
Results:
- Resultant Force: \( F = 17.173 \, \text{kN} \)
- Center of Pressure: \( y_p = 2.25 \, \text{m} \)
Explanation
- Area Calculation: The area of the circular gate is derived from its diameter, divided by four for a quarter circle.
- Center of Gravity (CG): The CG represents the average depth of the quarter-circle gate, considering its inclined orientation.
- Resultant Force: The total hydrostatic force on the gate is calculated based on the area and the depth of the CG.
- Center of Pressure (CP): The CP is slightly below the CG due to the pressure distribution increasing with depth.
Physical Meaning
This problem illustrates the hydrostatic pressure distribution on a curved surface. The calculations are essential for designing gates and structures that can withstand fluid forces, ensuring their stability and integrity.
