Model boundariesModel boundariesGeneral considerations

– Type of analysis: Deformation, stability, dynamics, flow, ….– Type of behaviour: Drained or undrained.– Is the situation (fully) symmetric? Can we model only half the problem?– Boundaries should not influence results.– Changes in stress and strain at boundaries should be low (except for symmetry g ( p y y

boundaries).– What is the consequence of taking boundaries closer or further away?

Model boundariesModel boundariesStability analysis:

– Mechanism must fit in model– Only plastic deformation is relevanty p– Undrained failure mechanisms are generally smaller than drained ones– Stress state may not be disturbed by boundaries (arching!)

M d l ll b ll th f d f ti l i– Model can generally be smaller than for deformation analysis

Model boundariesModel boundariesDeformation analysis:

– Deformations may still occur at a large distance from the action, especially for undrained analysis (preservation of volume!).

– Both elastic and plastic displacements are of influence.– Model should generally be larger than for stability analysis

drained undrained

Model boundariesModel boundariesDynamic analysis:

– Vibrations may occur at very large distance from the action.– Even very small displacements (vibrations) are of influence.– Even if measures are taken to avoid spurious reflections at boundaries, it is 

better to take boundaries far away (considering wave speed and duration of analysis).

– Model should generally be larger than for deformation analysis

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Model boundariesModel boundariesStability analysisStability analysis

DrainedDrained deformation analysis

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Undrained deformation analysis

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D i l iDynamic analysis

Model boundaries – Shallow foundationsModel boundaries  Shallow foundations

wa a

a

initial stress distribution

after loading

(0.1 to 0.2) acceptedw

limitdepth

Suggestions: Stability analysis: a 2wDrained deformation: a 3wUndrained deformation: a 4w

Model boundaries – Shallow foundationsModel boundaries  Shallow foundations

Take account of the following:

– For deformation analysis:• When using Mohr‐Coulomb, use different layers with increasing stiffness; 

bottom layer with height w should have large small‐strain stiffness.• When using Hardening‐Soil, use bottom layer with height w with large small‐

strain stiffness for Eurref.B l i HS ll d l• Best results using HSsmall model.

– For horizontal loading components: Increased width in loading direction.

Model boundaries – EmbankmentsModel boundaries – Embankmentswa a

h

a

Similarity with shallow footingsSimilarity with shallow footings

Suggestions: Stability analysis: a 2wDrained deformation: a 3wa ed de o a o a 3Undrained deformation: a 4w

Model boundaries – EmbankmentsModel boundaries – EmbankmentsTake account of the following:

– Embankments are considered to follow similar rules as shallow foundations with the same base width w

– For stability analysis, a can be much smaller if mechanism is purely in embankment itself

Model boundaries – ExcavationsModel boundaries – Excavations

Suggestion for deformation analysis (based on Dutch situations): (K.J. Bakker, PAO course Damwanden)

Model boundaries – ExcavationsModel boundaries – ExcavationsSettlement behind a sheetpile wall

0.2

-0.10

0.1

0 0.5 1 1.5 2 2.5

ent d

v/dh

-0.4-0.3-0.2

disp

lace

me

-0.7-0.6-0.5

rela

tive

d ClaySand low stiffSand medium stiffSand Higher friction

Motivation: Settlement trough behind the wall from FE analyses

-0.8relative distance x/h

Sand Higher friction

Motivation: Settlement trough behind the wall from FE analyses (K.J. Bakker, PAO course Damwanden)

See also Peck, 1969

Model boundaries – ExcavationsModel boundaries – Excavationsa w a

dl

a

d

General Stability analysis: a l and a 2dsuggestions: Structural forces analysis: a 2d (Vermeer & Wehnert, 2005)

Deformation analysis: a = 2d to 3d or 2w to 3w (max. of d and w) (Meiβner (2002)(Meiβner (2002)

Deformation analysis: a 2d ; bottom layer d with small-strain stiffness for Eur

ref (Vermeer & Wehnert, 2005)

Model boundaries – ExcavationsModel boundaries – ExcavationsTake account of the following:

1. Suggested model depth requires that large small‐strain stiffness is used below the excavation.

2. When using Hardening‐Soil, use bottom layer with height d with large small‐strain stiffness for Eurref.

3. Using a ‘normal’ stiffness for Eurref will result in unrealistic heave of excavation urbottom (and wall); a lower model depth should then be  considered (but 2 is preferred).

4. Ignoring small‐strain stiffness will generally result in a too wide settlement trough behind the wall, regardless the model width.

5. For  a < 3d significant settlements may be expected at upper model corners. This is even more pronounced for undrained behaviour.

Model boundaries – ExcavationsModel boundaries – ExcavationsConsidering the wall:

1. Unrealistic heave of excavation bottom gives unrealistic heave of wall > use large stiffness below excavation

2. For  a < 2d vertical model boundaries influence wall displacements3. Model depth and width seem to have little influence on the wall forces 

(bending moments)(Vermeer & Wehnert, 2005)

Model boundaries – TunnelsModel boundaries – TunnelsD ww ww

D D

w

D

a a

D

TBM orNATM excavationNATM excavation

Suggestions: Face stability: a ½D ; w 2D (Ruse, 2003)Structural forces, probably w 2DDrained deformation: w 3DUndrained deformation: w 4D

Model boundaries – TunnelsModel boundaries – TunnelsTake account of the following:

1. Large unloading and small‐strain stiffness below the tunnel2. Suggested model depth requires that large small‐strain stiffness is used below 

the tunnel (MC different sub layers HS large E ref)the tunnel (MC: different sub‐layers; HS: large Eurref).3. Ignoring small‐strain stiffness will result in unrealistic heave of tunnel;  a lower 

model depth should then be considered (but 2 is preferred).4 Ignoring small‐strain stiffness will generally result in a too wide settlement4. Ignoring small‐strain stiffness will generally result in a too wide settlement 

trough above the tunnel, regardless the model width.5. For  w < 3D significant settlements may be expected at the upper model 

corners. This is even more pronounced for undrained behaviour.6. For deep tunnels the overburden may be modelled as load, provided that at 

least a height  w above the tunnel is included in the model.

ConclusionsConclusionsConclusions:• Model size and boundaries depend, a.o., on type of analysis and type of 

behaviour (stability analysis, drained deformation undrained deformation, dynamic analysis)., y y )

• Small‐strain stiffness and relatively large models are needed to accurately predict deformations.