Problem Statement
The cylinder in the figure is 1.5 m long, and its radius is 1.25 m. Compute the horizontal and vertical components of the pressure force on the cylinder.
Solution
1. Horizontal Force \( (F_H) \)
The horizontal force is given by:
\( F_H = \gamma \cdot A \cdot \bar{y} \)
Here:
- \( \gamma = 9810 \, \text{N/m}^3 \): Specific weight of water
- \( A = 2.13 \cdot 1.5 = 3.195 \, \text{m}^2 \): Area of the vertical projection
- \( \bar{y} = \frac{2.13}{2} = 1.065 \, \text{m} \): Depth of the centroid
Substituting the values:
\( F_H = 9810 \cdot 3.195 \cdot 1.065 = 33380 \, \text{N} = 33.38 \, \text{kN (right)} \)
2. Vertical Force \( (F_V) \)
The vertical force is the weight of the water volume above the cylinder:
\( F_V = \gamma \cdot \Bigg( \text{Volume}_1 + \text{Volume}_2 + \text{Volume}_3 \)
\( + \text{Volume}_4 \Bigg) \)
Here:
- \( \text{Volume}_1 = \frac{1}{2} \cdot \pi \cdot (1.25)^2 \cdot 1.5 \)
- \( \text{Volume}_2 = 0.88 \cdot 1.25 \cdot 1.5 \)
- \( \text{Volume}_3 = 0.5 \cdot 0.88 \cdot 0.88 \cdot 1.5 \)
- \( \text{Volume}_4 = \frac{1}{8} \cdot \pi \cdot (1.25)^2 \cdot 1.5 \)
Substituting values across two lines for better visibility:
\( F_V = 9810 \cdot \Bigg[ \frac{1}{2} \cdot \pi \cdot (1.25)^2 \cdot 1.5 + 0.88 \cdot 1.25 \cdot 1.5 \)
\( + 0.5 \cdot 0.88 \cdot 0.88 \cdot 1.5 + \frac{1}{8} \cdot \pi \cdot (1.25)^2 \cdot 1.5 \Bigg] \)
Simplifying:
\( F_V = 67029 \, \text{N} = 67.03 \, \text{kN (up)} \)
Result:
- Horizontal Force: \( F_H = 33.38 \, \text{kN (right)} \)
- Vertical Force: \( F_V = 67.03 \, \text{kN (up)} \)
Explanation
- Horizontal Force: The horizontal force is calculated as the product of the water pressure, the area of the vertical projection, and the depth of the centroid. It represents the net force pushing the cylinder horizontally.
- Vertical Force: The vertical force is due to the weight of the water above the cylinder. This force includes contributions from different parts of the volume: semicircular, rectangular, triangular, and a small curved section.
- Volumes: Each volume contribution is computed separately, considering the geometry of the cylinder and the water distribution above it.
Physical Meaning
This problem highlights the pressure forces acting on a submerged cylindrical surface:
- Horizontal Force: Represents the force exerted due to the water’s hydrostatic pressure on the vertical projection of the cylinder.
- Vertical Force: Accounts for the weight of water directly above the cylinder, impacting its stability and structural design.
- Applications: Such calculations are crucial in designing submerged structures, pipelines, and containers to ensure they can withstand hydrostatic forces effectively.
