Factor of Safety Against Heave Failure in a Braced Vertical Excavation:

1. Introduction:
In a braced vertical excavation, it is crucial to determine the factor of safety against heave failure at its base. Heave failure occurs when the lateral pressure exerted by the soil exceeds the resisting capacity of the bracing system, leading to upward movement or heave of the excavation walls. In this scenario, we will calculate the factor of safety against heave failure using relevant parameters.

2. Given Parameters:
- Height of excavation (h) = 5 meters
- Width of excavation (b) = 2.5 meters
- Cohesion of cohesive soil (c) = 20 kN/m²
- Bulk unit weight of soil (γ) = 20 kN/m³

3. Analysis:
To determine the factor of safety against heave failure, we need to calculate the resisting force and the lateral pressure exerted by the soil. The factor of safety is then obtained by dividing the resisting force by the lateral pressure.

4. Lateral Pressure:
The lateral pressure (P) exerted by the soil can be calculated using the Rankine's theory for cohesive soils:

P = 0.5 * γ * h² + c * h

Substituting the given values:
P = 0.5 * 20 * (5)² + 20 * 5
P = 250 kN/m²

5. Resisting Force:
The resisting force (R) can be calculated by multiplying the width of the excavation with the passive earth pressure coefficient (Kp) and the lateral pressure.

R = Kp * P * b

The passive earth pressure coefficient for a braced excavation can be assumed to be approximately 2/3.

R = (2/3) * 250 * 2.5
R = 416.67 kN

6. Factor of Safety:
The factor of safety (FS) against heave failure is obtained by dividing the resisting force by the lateral pressure.

FS = R / P
FS = 416.67 / 250
FS = 1.67

7. Interpretation:
The factor of safety against heave failure at the base of the braced vertical excavation is calculated to be approximately 1.67. This indicates that the resisting force provided by the bracing system is 1.67 times higher than the lateral pressure exerted by the soil. Therefore, the excavation is considered stable against heave failure as long as the factor of safety remains above 1.0.

Conclusion:
By analyzing the given parameters and applying relevant calculations, we have determined the factor of safety against heave failure in a braced vertical excavation. The factor of safety provides a measure of stability and is crucial in ensuring the safety and integrity of the excavation structure.