Geotechnical Analysis: Limit equilibrium Flashcards
What theoretical conditions are not satisfied by the Limit Equilibrium method?
Compatibility and Constitutive Behaviour
What is the Tresca failure criterion equation in terms of principal stresses?
τf = Su
or
Su = (σ1 - σ3)/2
What is the Mohr-Coulomb failure criterion formula?
τf = c’ + σ’n * tan(φ’)
Where:
τf = shear strength
c’ = effective cohesion
σ’n = effective normal stress
φ’ = effective angle of internal friction
In the vertical cut problem, what is the unknown we are trying to determine?
The maximum safe excavation depth (H)
What geometric assumption is made in modelling the vertical cut problem?
Plane strain geometry is assumed
What is the first assumption in Limit Equilibrium analysis?
An arbitrary failure surface is assumed for shape and location
According to assumption 2, what is assumed about soil on either side of the failure surface?
Soil is assumed to be rigid and failing everywhere along the surface
Is the assumption of rigid failure blocks realistic in actual slope failure?
No – real failing soil deforms and is not perfectly rigid
To which part of the soil mass is equilibrium applied in Limit Equilibrium?
Only to the failing block
What are the two types of forces acting on the failing block?
- External (e.g. self-weight W)
- Internal (only along the failure surface: normal stress σ’ and shear stress τ)
What equilibrium equations are used in Limit Equilibrium analysis?
- Two force equilibrium equations (horizontal and vertical)
- One moment equilibrium equation
What three properties must be known to fully define a force?
Magnitude, direction, and line of action
Which failure criterion is used in the vertical cut analysis?
Mohr-Coulomb
What are the force equilibrium equations in the vertical cut problem?
∑F⊥ = ∫σ’ dl = W * sin(β)
∑F∥ = ∫τ dl = W * cos(β)
What is l equal to in the vertical cut problem?
l = H / cos(β)
What is W equal to in the vertical cut problem?
W = 0.5 * γ * H² * tan(β)
How is the shear force along the failure surface expressed?
∫τ dl = ∫(c’ + σ’ * tan(φ’)) dl
What is the expression for safe excavation depth H in terms of soil parameters and β?
H = (2 * c’ * cos(φ’)) / (γ * cos(β) * sin(β + φ’))
What is the critical angle β where the minimum H occurs?
β = (π / 4) - (φ’ / 2)
What is the Limit Equilibrium solution for H using the critical β?
H = (4 * c’) / (γ * tan(π/4 + φ’/2))
What does moment equilibrium allow us to determine in the vertical cut problem?
The line of action of the resultant force N (normal to the failure plane)
List the 6 main assumptions of Limit Equilibrium analysis.
- Arbitrary failure surface assumed
- Failure everywhere along surface; blocks are rigid
- Equilibrium applied only to failing block
- Forces include external (e.g. W) and internal (on failure surface only)
- Two force and one moment equilibrium equations used
- Each force defined by magnitude, direction, and line of action
