Geotechnical Analysis: Stress fields Flashcards
What theoretical requirements does the Stress Field method satisfy and not satisfy?
Satisfies: Equilibrium only.
Does not satisfy: Compatibility, Constitutive Behaviour, Boundary Conditions
In Limit Equilibrium, where is equilibrium applied?
Only to the failing soil block between the failure surface and boundary.
What are the main assumptions of the Stress Field method?
- Soil is at failure everywhere
- Combine failure criterion with equilibrium equations
What are the stress components for a soil element in plane strain?
σx, σy, τxy, γ
What are the plane strain equilibrium equations?
∂σx/∂x + ∂τxy/∂y = 0 (1)
∂σy/∂y + ∂τxy/∂x + γ = 0 (2)
What do σ₁′ and σ₃′ represent in a Mohr circle?
Major and minor principal effective stresses.
What is the Mohr-Coulomb failure criterion in component stresses?
(σx′ - σy′)/2 = τxy
τ = c’ + ((σx′ + σy′)/2) * sin(φ′)
What is the final combined failure criterion used in Stress Field method?
[(σx′ - σy′)/2]² + τxy² = [c′ + ((σx′ + σy′)/2) * sin(φ′)]²
How many unknowns and equations are present in the Stress Field method?
3 unknowns: σx, σy, τxy; 3 equations (2 equilibrium + 1 failure criterion)
What type of equations arise in Stress Field analysis?
Two hyperbolic partial differential equations (PDEs)
What are the characteristic direction equations?
dy/dx = tan(θ - π/4 - φ′/2)
dy/dx = tan(θ + π/4 + φ′/2)
Do Stress Field solutions apply to the whole continuum?
No, only to a partial zone of interest.
What do characteristic lines represent in Stress Field analysis?
Directions along which stresses vary.
Why is the solution near a retaining wall called a “partial stress field”?
Because it does not extend through the full soil mass.
What does σT represent in tunnel analysis using stress fields?
The tunnel support pressure
What does the characteristic mesh represent in tunnel analysis?
The network of stress propagation paths from the tunnel wall.
What is the initial stress state in Rankine theory?
σx′ = K₀ * σy′
When does the active stress state develop?
When lateral extension reduces σx′ until failure occurs.
What is the Rankine active earth pressure expression?
σx′ = Ka * σy′
When does the passive stress state develop?
When lateral compression increases σx′ until failure occurs.
What is the Rankine passive earth pressure expression?
σx′ = Kp * σy′
In Rankine theory, what is always true about τxy?
τxy = 0
Therefore, σx′ and σy′ are principal stresses.
What is the equation for σxa′ in the Rankine active case?
σxa′ = σy′ - 2 * a * A
What is the final expression for active pressure from Mohr circle?
σxa′ = σy′ - 2 * c′ * cot(φ′)
