3.2.1. Numerical example for Design of Gravity Dam

 Let's go through a numerical example for the design of a gravity dam, using a simplified real-life scenario. This example will cover key aspects such as stability against overturning, sliding, and checking the bearing capacity of the foundation.

Numerical Example: Design of a Gravity Dam

Scenario:

Let's design a gravity dam with the following specifications:

• Height of Dam (H): 50 meters
• Base Width (B): 45 meters
• Top Width (T): 5 meters
• Density of Concrete (γc): 24 kN/m³
• Density of Water (γw): 9.81 kN/m³
• Coefficient of Friction (μ): 0.75
• Allowable Bearing Capacity of Foundation: 1500 kN/m²

Objective: Check for stability against overturning, sliding, and ensure that the foundation pressure does not exceed the allowable bearing capacity.

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1. Geometry of the Dam

The dam has a trapezoidal cross-section with the following dimensions:

• Upstream face: Vertical
• Downstream face: Sloped (linearly tapered)
• Height (H): 50 m
• Top Width (T): 5 m
• Base Width (B): 45 m

The slope of the downstream face ensures that the dam's center of gravity is closer to the upstream face, which helps in resisting overturning.

2. Forces Acting on the Dam

A. Hydrostatic Pressure (Water Pressure)

Formula:

$$P = \frac{1}{2} \gamma_w H^2$$

• Water Pressure at the Base:

  $$P = \frac{1}{2} \times 9.81 \times 50^2$$$$= \frac{1}{2} \times 9.81 \times 2500$$ $$P = 12262.5 \ \text{kN/m} \text{ (total force per meter width of dam)}$$
• Point of Action: Hydrostatic pressure acts at a height of $\frac{H}{3}$ from the base, which is $\frac{50}{3} \approx 16.67$ m.

B. Self-Weight of the Dam

Area of Cross-section:

• The dam has a trapezoidal shape. The area ($A$) can be calculated as:

  $$A = \frac{1}{2} \times (B + T) \times H$$$$A = \frac{1}{2} \times (45 + 5) \times 50$$ $$A = \frac{1}{2} \times 50 \times 50$$ $$A = 1250 \ \text{m}^2$$

• Self-Weight ($W$):

  $$W = A \times \gamma_c$$ $$W = 1250 \times 24$$ $$W=30000\text{ kN/m }$$

• Point of Action: Acts at the centroid of the trapezoid, approximately $\frac{2H}{3}$ from the base (from the upstream face), which is $\frac{2 \times 50}{3} \approx 33.33$ m.

3. Stability Analysis

A. Check for Overturning

• Overturning Moment (M$_\text{overturning}$):

  $$M_{\text{overturning}} = P \times \text{Distance to Toe}$$ $$M_{\text{overturning}} = 12262.5 \times (45 - 16.67)$$$$M_{\text{overturning}} = 12262.5 \times 28.33$$ $$M_{\text{overturning}} = 347187.375 \ \text{kNm/m}$$

• Resisting Moment (M$_\text{resisting}$):

  $$M_{\text{resisting}} = W \times \text{Distance to Toe}$$ $$M_{\text{resisting}} = 30000 \times (45/2)$$ $$M_{\text{resisting}} = 30000 \times 22.5$$ $$M_{\text{resisting}} = 675000 \ \text{kNm/m}$$

• Factor of Safety (FOS) against Overturning:

  $$\text{FOS} = \frac{M_{\text{resisting}}}{M_{\text{overturning}}}$$ $$\text{FOS} = \frac{675000}{347187.375}$$ $$\text{FOS} \approx 1.94$$

  Interpretation: The FOS against overturning is 1.94, which is greater than the required minimum of 1.5, indicating that the dam is safe against overturning.

B. Check for Sliding

• Resisting Force (F$_\text{resisting}$):

  $$F_{\text{resisting}} = \mu \times W$$ $$F_{\text{resisting}} = 0.75 \times 30000$$ $$F_{\text{resisting}} = 22500 \ \text{kN/m}$$

• Driving Force (F$_\text{driving}$): This is the horizontal component of the water pressure.

  $$F_{\text{driving}} = P = 12262.5 \ \text{kN/m}$$

• Factor of Safety (FOS) against Sliding:

  $$\text{FOS} = \frac{F_{\text{resisting}}}{F_{\text{driving}}}$$$$\text{FOS} = \frac{22500}{12262.5}$$$$\text{FOS} \approx 1.84$$

  Interpretation: The FOS against sliding is 1.84, which is greater than the required minimum of 1.5, indicating that the dam is safe against sliding.

C. Check for Bearing Capacity

• Maximum Foundation Pressure (σ$_\text{max}$): $$\sigma_{\text{max}} = \frac{W}{\text{Base Width}} + \frac{6 \times M_{\text{overturning}}}{B^2}$$ $$\sigma_{\text{max}} = \frac{30000}{45} + \frac{6 \times 347187.375}{45^2}$$ $$\sigma_{\text{max}} = 666.67 + \frac{2083124.25}{2025}$$ $$\sigma_{\text{max}} = 666.67 + 1028.96$$ $$\sigma_{\text{max}} = 1695.63 \ \text{kN/m}^2$$

Interpretation: The calculated maximum foundation pressure is 1695.63 kN/m², which is slightly above the allowable bearing capacity of 1500 kN/m². This suggests that the foundation may need to be strengthened, or the dam's base could be widened to distribute the load more effectively.

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4. Conclusion

From the above calculations:

1. The dam is stable against overturning with a FOS of 1.94.
2. The dam is stable against sliding with a FOS of 1.84.
3. However, the maximum foundation pressure exceeds the allowable bearing capacity, indicating a need for further design modifications to ensure safety.

This example illustrates the critical steps and considerations in the design of a gravity dam, emphasizing the importance of stability analysis to ensure the safety and functionality of the structure.