Coupled Consolidation

Question 1

Coupled theory definition

Answer 1

Combination of seepage equation with equilibrium and constitutive equations

Combines PW flow with stress and strain behaviour and account for time dependency (short term vs long term)

Question 2

Drained vs Undrained

Answer 2

-Drained change in PWP is zero
-Undrained the change in volume is zero not PWP

Combination of these two can be used to solve most problems

Question 3

Undrained assumption of change in stress = change in PWP

Answer 3

Applies when the soil it elastic, perfectly plastic and saturated with water assumes as compressible

Makes the change in effective stress zero

Question 4

Consolidation

Answer 4

-Drained, leads to a change in effective stress and a change in volume with increased bearing capacity which is larger than in the undrained case

Question 5

Formulation of coupled theory

Answer 5

-Assumes nodal PWP are unknown along with displacements
-Equations to be satisfied
* Equilibrium
* Constitutive
* Darcys Law: {v} = -[k]{grad h}

h = Pf/gamma_w . (x.i_gx + y.i_gy +z.i_gz)
-With {i_g} being the unit vector // and opposite to gravity
-For an isotropic material Kxx = Kyy = Kzz while Kxy = Kxz = Kzy = 0

• Continuity: the difference between water flowing out an water flowing in is the same as the change in volumetric strain with respect to time

partial Vi/di - Q = partial epsilon_vol/dt

Question 6

Coupled theory equation

Answer 6

Both equations are solves simultaneously therefore change in dis leads to change in PWP and vice versa
-beta = time stepping factor reflecting the change in PWP
-phi_g = permeability/gamma_w
-{delta Pf}ng = global PWP vector
-[neta g] = [K].{ig}
- Bottom row on RHS is flow
-[Lg] = off diagonal cross-coupling sub matrix]

[Lg] .(change in PWP) =>
*effective stress can change due to change in PWP
*change in effective stress in general produces a force which shoul dbe in equilibrium with other applied forces

[Lg] tansposed . change in displacement =>
*flow due to displacements in the soil structure

phi_g . shange in PWP =>
* flow due to difference in hydraulic head, Darcys law

Question 7

Getting beta

Answer 7

-Curve of a Pf - t plot is unknown but for two known points the area under it needs to be found

A = [beta.({Pf}ng)_2 + (1-beta).({Pf}ng)_1 ].dt

• beat usually >= 0.5

Question 8

Pore pressure DoFs

Answer 8

• Quadratic variation in PP requires every node to have a PWP DoF
• Quadratic disp. leads to linear stress/strains which means sigma’ and PWP are of different orders which isnt a real issue but some ppl dont like it
• Linear PWP for DoFs only on the corners

-Consolidating nodes require PWP DoFs and so does the interface between them and non-consolidating nodes

Question 9

Hydraulic BC

Answer 9

-Defines a prescribed nodal PWP (1) or a prescribed flow(2)
1.Affects {delta Pf}ng
2.Can effect Q in the form of source sinks and infiltration

-Must be defined at the edges of the domain and at the interface between consolidating and non-consolidating
-Can be tied which affects the whole system of equations

Question 10

Prescribed PWP

Answer 10

-change Pf = 0 @ boundary assuming its far enough that its unaffected
-Pf = 0 at the bottom of a pumped excavation

Question 11

Prescribed flows

Answer 11

-zero for an impermeable boundary]
-sources/sinks can be applied at certain nodes
-infiltration specifies flows across a boundary and like a boundary stress can be resolved to nodal flows

Question 12

Precipitation BC

Answer 12

Automatically changes from a no flow to no PWP boundary
- used in tunnelling for example