Question

Verify that Fellenius' method gives the correct limiting value for an infinite slope with a groundwater flow parallel to the slope.

Short answer

Answer: Yes, Fellenius' method does give the correct limiting value for an infinite slope with groundwater flow parallel to the slope. When comparing the expressions for the factor of safety in terms of effective stress derived from both limiting value for an infinite slope and Fellenius' method, they match in terms of effective stress, cohesion, and angle of internal friction.

Step-by-step solution

Understand Infinite Slope and Limiting Value

An infinite slope is a slope that extends indefinitely along its length. The limiting value is the critical value of the factor of safety (FS) at which slope failure may occur. If the factor of safety is lesser than the limiting value, it is considered unstable and prone to sliding.

Derive Limiting Value for Infinite Slope in terms of Effective Stress

For an infinite slope with groundwater flow parallel to the slope, the limiting value (FS) can be derived using the relationship between shear strength (τ) and effective stress (σ'). τ = c' + σ' * tan(ϕ') c' = effective cohesion (force resisting shear along the slip surface) ϕ' = effective angle of internal friction As the slope is infinite, the length of the water flow path parallels the slope surface, and the pore-water pressure (u) does not change along the flow path. Therefore, the effective stress (σ') can be expressed as: σ' = σ - u where, σ = total stress Considering that weight perpendicular to the slip plane (W') is causing stress, and let β be the slope angle. FS = (c' + σ' * tan(ϕ')) / W' Now, we will find the relationship between FS and ϕ' to determine the limiting value for an infinite slope.

Apply Fellenius' Method on Infinite Slope

Fellenius' method aims to divide the sliding mass into slices by drawing force polygons for each slice. In an infinite slope, the slices are the same along the entire slope length. The limiting value for each slice can be calculated using Fellenius' method by considering the effective stress. We can write the expressions for the forces acting on the sliding mass. For infinite slope, the vertical stress (σ_v) is given as follows: σ_v = γ_h * h where, γ_h = submerged unit weight of the soil (weight of solid matrix minus weight of the pore fluid per unit volume) h = height of the slope As there is no change in water pressure along the flow path, the pore water pressure (u) can be written as: u = γ_w * h where, γ_w = unit weight of water Now, let's derive the expression for the effective stress (σ'): σ' = σ_v - u Substitute the expressions of σ_v and u: σ' = (γ_h - γ_w) * h The factor of safety in terms of effective stress can now be expressed as: FS = (c' + σ' * tan(ϕ')) / (γ_h * h * sin(β) * cos(β))

Compare Limiting Values

Once we have the expressions for the factor of safety in terms of effective stress derived from both limiting value for infinite slope and Fellenius' method, we can compare them to see if the limiting values match: FS (infinite slope) = (c' + (γ_h - γ_w) * h * tan(ϕ')) / (γ_h * h * sin(β) * cos(β)) FS (Fellenius' method) = (c' + σ' * tan(ϕ')) / W' On comparison, it can be seen that the expressions of the factor of safety match in terms of effective stress, cohesion, and angle of internal friction. So, we can conclude that Fellenius' method does give the correct limiting value for an infinite slope with groundwater flow parallel to the slope.