A plain concrete dam will be constructed in a gallery of 4 m x 3 m cross-section against a water head of 90 m. If the shear strength of coal is 500 kg/cm2, the actual thickness of... A plain concrete dam will be constructed in a gallery of 4 m x 3 m cross-section against a water head of 90 m. If the shear strength of coal is 500 kg/cm2, the actual thickness of the water dam should be:
Understand the Problem
The question asks about determining the required thickness of a plain concrete dam built in a gallery, considering the water head and shear strength of the concrete. This involves applying principles of structural engineering and material mechanics to ensure the dam's stability against water pressure.
Answer
The required thickness is approximately $0.6 m$.
Answer for screen readers
The required thickness of the plain concrete dam is approximately $0.6 m$.
Steps to Solve
- Calculate the water pressure at the base of the dam.
The water pressure $P$ at the base of the dam is given by: $P = \gamma_w * h$ where $\gamma_w$ is the specific weight of water (approximately $9.81 kN/m^3$) and $h$ is the water head ($6 m$). $P = 9.81 \frac{kN}{m^3} * 6 m = 58.86 \frac{kN}{m^2}$
- Calculate the horizontal force due to water pressure.
The total horizontal force $F_h$ acting on the dam per unit width is given by: $F_h = \frac{1}{2} * P * h = \frac{1}{2} * \gamma_w * h^2$ $F_h = \frac{1}{2} * 58.86 \frac{kN}{m^2} * 6 m = \frac{1}{2} * 9.81 \frac{kN}{m^3} * (6 m)^2 = 176.58 \frac{kN}{m}$
- Calculate the required thickness of the dam based on shear strength.
The shear force $F_s$ that the dam can withstand is given by: $F_s = \tau * A = \tau * t * 1$ (per unit width), where $\tau$ is the shear strength of the concrete and $t$ is the thickness.
To ensure stability, the shear force must be greater than or equal to the horizontal force: $F_s \ge F_h$ $\tau * t \ge F_h$ $300 \frac{kN}{m^2} * t \ge 176.58 \frac{kN}{m}$ $t \ge \frac{176.58 \frac{kN}{m}}{300 \frac{kN}{m^2}}$ $t \ge 0.5886 m$
- Round up to a practical thickness.
Since the thickness should be a practical value, round it to the nearest tenth of a meter. $t \approx 0.6 m$
The required thickness of the plain concrete dam is approximately $0.6 m$.
More Information
The calculation assumes a simplified scenario. In real-world dam design, factors such as uplift pressure, safety factors, dynamic loads, and material properties variations are considered.
Tips
A common mistake is forgetting to convert units to be consistent (e.g., using meters for length and $kN/m^3$ for specific weight). Another mistake is not applying a proper safety factor.
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