Geotechnical Analysis: Limit equilibrium

Question 1

What theoretical conditions are not satisfied by the Limit Equilibrium method?

Answer 1

Compatibility and Constitutive Behaviour

Question 2

What is the Tresca failure criterion equation in terms of principal stresses?

Answer 2

τf = Su
or
Su = (σ1 - σ3)/2

Question 3

What is the Mohr-Coulomb failure criterion formula?

Answer 3

τf = c’ + σ’n * tan(φ’)
Where:

τf = shear strength
c’ = effective cohesion
σ’n = effective normal stress
φ’ = effective angle of internal friction

Question 4

In the vertical cut problem, what is the unknown we are trying to determine?

Answer 4

The maximum safe excavation depth (H)

Question 5

What geometric assumption is made in modelling the vertical cut problem?

Answer 5

Plane strain geometry is assumed

Question 6

What is the first assumption in Limit Equilibrium analysis?

Answer 6

An arbitrary failure surface is assumed for shape and location

Question 7

According to assumption 2, what is assumed about soil on either side of the failure surface?

Answer 7

Soil is assumed to be rigid and failing everywhere along the surface

Question 8

Is the assumption of rigid failure blocks realistic in actual slope failure?

Answer 8

No – real failing soil deforms and is not perfectly rigid

Question 9

To which part of the soil mass is equilibrium applied in Limit Equilibrium?

Answer 9

Only to the failing block

Question 10

What are the two types of forces acting on the failing block?

Answer 10

• External (e.g. self-weight W)
• Internal (only along the failure surface: normal stress σ’ and shear stress τ)

Question 11

What equilibrium equations are used in Limit Equilibrium analysis?

Answer 11

• Two force equilibrium equations (horizontal and vertical)
• One moment equilibrium equation

Question 12

What three properties must be known to fully define a force?

Answer 12

Magnitude, direction, and line of action

Question 14

Which failure criterion is used in the vertical cut analysis?

Answer 14

Mohr-Coulomb

Question 15

What are the force equilibrium equations in the vertical cut problem?

Answer 15

∑F⊥ = ∫σ’ dl = W * sin(β)
∑F∥ = ∫τ dl = W * cos(β)

Question 16

What is l equal to in the vertical cut problem?

Answer 16

l = H / cos(β)

Question 17

What is W equal to in the vertical cut problem?

Answer 17

W = 0.5 * γ * H² * tan(β)

Question 18

How is the shear force along the failure surface expressed?

Answer 18

∫τ dl = ∫(c’ + σ’ * tan(φ’)) dl

Question 19

What is the expression for safe excavation depth H in terms of soil parameters and β?

Answer 19

H = (2 * c’ * cos(φ’)) / (γ * cos(β) * sin(β + φ’))

Question 20

What is the critical angle β where the minimum H occurs?

Answer 20

β = (π / 4) - (φ’ / 2)

Question 21

What is the Limit Equilibrium solution for H using the critical β?

Answer 21

H = (4 * c’) / (γ * tan(π/4 + φ’/2))

Question 22

What does moment equilibrium allow us to determine in the vertical cut problem?

Answer 22

The line of action of the resultant force N (normal to the failure plane)

Question 23

List the 6 main assumptions of Limit Equilibrium analysis.

Answer 23

1. Arbitrary failure surface assumed
2. Failure everywhere along surface; blocks are rigid
3. Equilibrium applied only to failing block
4. Forces include external (e.g. W) and internal (on failure surface only)
5. Two force and one moment equilibrium equations used
6. Each force defined by magnitude, direction, and line of action