Problem Statement
Find the net hydrostatic force per unit width on rectangular panel AB in the figure and determine its line of action.
Solution
1. Area of the Panel
The area of the panel is:
\( A = 2 \cdot 1 = 2 \, \text{m}^2 \)
2. Location of Center of Gravity (CG)
The depth of the CG for the water side is:
\( y_{\text{CG}} = 2 + 1 + \frac{2}{2} = 4 \, \text{m} \)
The depth of the CG for the glycerin side is:
\( y_{1, \text{CG}} = 1 + \frac{2}{2} = 2 \, \text{m} \)
3. Resultant Force on the Panel
The hydrostatic force on the water side is:
\( F_{\text{water}} = \gamma \cdot A \cdot y_{\text{CG}} \)
\( F_{\text{water}} = 9.81 \cdot 2 \cdot 4 = 78.49 \, \text{kN} \)
The hydrostatic force on the glycerin side is:
\( F_{\text{glycerin}} = \gamma \cdot A \cdot y_{1, \text{CG}} \)
\( F_{\text{glycerin}} = 12.36 \cdot 2 \cdot 2 = 49.44 \, \text{kN} \)
The net force is:
\( F = F_{\text{water}} – F_{\text{glycerin}} = 78.49 – 49.44 = 29.04 \, \text{kN} \)
4. Moment of Inertia about CG
The moment of inertia about the CG is:
\( I_G = \frac{1}{12} \cdot 1 \cdot 2^3 = 0.666 \, \text{m}^4 \)
5. Line of Action of Forces
The vertical distance of \( F_{\text{water}} \) from the CG is:
\( y_{\text{cp1}} = y_{\text{CG}} + \frac{I_G}{A \cdot y_{\text{CG}}} \)
\( y_{\text{cp1}} = 4 + \frac{0.666}{2 \cdot 4} = 4.083 \, \text{m} \)
The vertical distance of \( F_{\text{glycerin}} \) from the CG is:
\( y_{\text{cp2}} = y_{1, \text{CG}} + \frac{I_G}{A \cdot y_{1, \text{CG}}} \)
\( y_{\text{cp2}} = 2 + \frac{0.666}{2 \cdot 2} = 2.166 \, \text{m} \)
Taking moments about B:
\( 29.04y = 78.49 \cdot (5 – 4.083) – 49.44 \cdot (3 – 2.166) \)
\( y = 0.945 \, \text{m} \)
Results:
- Net Hydrostatic Force: \( F = 29.04 \, \text{kN} \)
- Line of Action: \( y = 0.945 \, \text{m} \)
Explanation
- Area: The area is based on the dimensions of the rectangular panel.
- Forces: Hydrostatic forces are computed separately for water and glycerin, considering their densities and depths.
- Net Force: The difference in forces due to water and glycerin gives the net force.
- Moments: Moments are taken about the CG to determine the line of action of the resultant force.
Physical Meaning
This problem demonstrates the principles of hydrostatic force distribution on submerged surfaces. Understanding the line of action helps ensure structural stability in fluid environments.
