Question

A dam of height 12 m is trapezoidal in section with top width of 2 m and bottom width of 8 m. The unit weight of the material of dam and water are 20 kN/m$^3$ and 10 kN/m$^3$ respectively. If the dam is allowed to store water upto a height of 10 m, the horizontal pressure on the dam is

• A. 100 kN
• B. 500 kN
• C. 720 kN
• D. 1000 kN

Hint:

The total horizontal hydrostatic force on any vertical plane surface of height 'h' is always equal to the area of the triangular pressure diagram.
Force = (Area of triangle) = $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times (wh) \times h = \frac{1}{2}wh^2$.
This force always acts at a height of $h/3$ from the base.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the total horizontal hydrostatic force (referred to as horizontal pressure) acting on a dam of unit length.

Step 2: Key Formula or Approach:
The horizontal hydrostatic force on a submerged vertical surface is equal to the area of the pressure diagram. The pressure distribution is triangular, with zero pressure at the water surface and maximum pressure at the base.
The total horizontal force ($F_H$) per unit length of the dam is given by:
\[ F_H = \frac{1}{2} w h^2 \] where:
$w$ = unit weight of water
$h$ = height of the water stored behind the dam

Step 3: Detailed Explanation:
We are given:
- Height of water ($h$) = 10 m
- Unit weight of water ($w$) = 10 kN/m$^3$
The dimensions of the dam (height, top width, bottom width) and the unit weight of the dam material are given but are not needed to calculate the horizontal water pressure. They would be needed to check the stability of the dam.
Substitute the relevant values into the formula:
\[ F_H = \frac{1}{2} \times (10 \text{ kN/m}^3) \times (10 \text{ m})^2 \] \[ F_H = \frac{1}{2} \times 10 \times 100 \] \[ F_H = 5 \times 100 = 500 \text{ kN} \] This force is per meter length of the dam (kN/m), but since the options are in kN, it is implied that the question asks for the force on a 1-meter-wide strip of the dam.

Step 4: Final Answer:
The horizontal pressure on the dam is 500 kN (per meter length).