Problem Statement
A dam has a parabolic profile as shown in the figure. Compute the horizontal and vertical components of the force on the dam due to the water. The width of the dam is 15 m. The area of the parabolic section is given as:
\( A = \frac{2}{3} \cdot b \cdot d \)
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
- \( A = b \cdot h = 15 \cdot 6.9 = 103.5 \, \text{m}^2 \)
- \( \bar{y} = \frac{h}{2} = \frac{6.9}{2} = 3.45 \, \text{m} \)
Substituting the values:
\( F_H = 9810 \cdot 103.5 \cdot 3.45 = 3502906 \, \text{N} = 3502.906 \, \text{kN (right)} \)
2. Vertical Force \( (F_V) \)
The vertical force is the weight of the water volume above the parabolic section \( AB \):
\( F_V = \gamma \cdot \text{Volume above AB} \)
The volume of the parabolic section is:
\( \text{Volume} = \frac{2}{3} \cdot b \cdot d \cdot w \)
Substituting values:
- \( b = 3 \, \text{m}, d = 6.9 \, \text{m}, w = 15 \, \text{m} \)
\( F_V = 9810 \cdot \frac{2}{3} \cdot 3 \cdot 6.9 \cdot 15 \)
\( = 2030670 \, \text{N} = 2030.67 \, \text{kN (down)} \)
Result:
- Horizontal Force: \( F_H = 3502.906 \, \text{kN (right)} \)
- Vertical Force: \( F_V = 2030.67 \, \text{kN (down)} \)
Explanation
- Horizontal Force: This is due to the hydrostatic pressure acting horizontally on the dam. The force is calculated using the average depth of the water (center of pressure) and the area of the dam’s face in contact with the water.
- Vertical Force: This represents the weight of the water above the dam’s parabolic section. The volume of water is determined using the given dimensions and the parabolic area formula \( \frac{2}{3} \cdot b \cdot d \cdot w \).
- Equations Used: Hydrostatic equations consider the pressure distribution linearly increasing with depth, and the vertical force is derived from the volume’s weight.
Physical Meaning
This problem illustrates the two main components of hydrostatic forces acting on a dam:
- The horizontal force demonstrates how water pressure increases linearly with depth, exerting a force that could potentially topple the dam if not balanced by structural design.
- The vertical force highlights the weight of the water directly above the dam’s profile. It must be considered to ensure the foundation can support this load without excessive settlement or failure.
Understanding these forces is crucial for designing safe and stable dams that can resist hydrostatic pressures while minimizing risks of structural failure.
