Slope Stability and Ultimate Limit State

Introduction The aim is to discuss the application of finite element method to slope stability analysis. The topics to be covered include: • Options for generation of pore pressures in PLAXIS, with special consideration of slope stability • Options for generation of in situ stresses in PLAXIS • Analysis of stability of slopes with numerical modellingPLAXIS

Slope Stability and Ultimate

Limit State

Applied TheoryProf Minna Karstunen

The aim is to discuss the application of finite element

method to slope stability analysis.

The topics to be covered include:

• Options for generation of pore pressures in PLAXIS, with special consideration of slope stability

• Options for generation of in situ stresses in PLAXIS

• Analysis of stability of slopes with numerical

modelling/PLAXIS

Water Pressures

Water Pressures (general)

• Water pressures

– External water pressures (loads on

boundaries)

– Internal water pressures (pore pressures)

• Effective stress analysis

– Total stress: From weight and external load – Pore pressure:

excess steady

Water Pressures (general)

• p excess are only generated in clusters for

which the type of material behaviour is

specified as Undrained

• Time-dependent dissipation or generation

in consolidation analysis (controlled by

permeability parameters)

• Input of unit weight of fluid is required

(can model other fluids than water)

Water Pressures (general)

• ‘Steady-state’ pore pressures p steady

– Represent stable hydraulic condition

– Generated in the Water condition mode

• ‘Steady-state’ pore pressures p steady

A Generation from phreatic levels

B Generation by groundwater flow calculation

Water Pressure generation

• General phreatic level

– A series of points where the water pressure is zero can be specified

– Pressures increase linearly with depth

depending on γw

– PLAXIS does not allow negative pore

pressures above the phreatic level (all

assumed zero) – this is conservative!

Water Pressure Generation

• General phreatic level

– Crossings of phreatic level and existing

geometry lines do not introduce additional

geometry points, so there is a danger that the external water pressures are not calculated accurately

Water Pressure Generation

• General phreatic level

– Allows to define pressures throughout the model

• To allow for discontinuous pore pressures can

use Cluster phreatic levels

Water Pressure Generation

• Clusters that should have zero pore

pressure can be defined as dry (Cluster

Water Pressure Generation

• General phreatic level

– To generate pore pressures in soil clusters (also for inactive clusters) If you want to

exclude some cluster, choose Cluster dry

option

– To generate external water loads when

outside the mesh

– To generate boundary conditions for

groundwater flow calculation

Water pressure generation

• General phreatic level

(example)

Water Pressure Generation

• Cluster pore pressure distribution

(example:

General boven, interpolate midden, cluster onder)Layer 1

General boven, interpolate midden, cluster onder)

General phreatic level

Cluster phreatic level

Interpolate from adjacent clusters or lines

Layer 1

Layer 3 Layer 2

Water Pressure Generation

• Generation of pore pressure based on

groundwater flow

– Input on boundary conditions on

groundwater head

1 A prescribed groundwater head

2 Closed flow boundary (can only be placed

exactly over existing geometry lines at the outer boundary)

– Flow problems with a free phreatic level may

involve seepage surface on the downstream boundary

Water Pressure Generation

• A seepage surface occurs when a

phereatic level touches an open

downstream boundary (groundwater head=elevation head)

Water Pressure Generation

• Steady-state groundwater flow calculation

Results:

– Flow field

– Groundwater head distribution

– Pore pressure distribution

– Degree of saturation (around phreatic

surface)

– Total discharge through cross section

Water pressure generation

• Steady-state groundwater flow calculation

(example: screen, drain, well)

Water Pressure Generation

• Steady-state groundwater flow

calculation

Water Pressure Generation

• Inactive cluster not taken into account

in groundwater flow calculations, but the pore pressures are determined

afterwards from the General phereatic

level

• Transient groundwater flow calculation with PlaxFlow can be used as input for deformation analysis

• Interfaces are by default fully

impermeable (so may need to be activated for flow calculations)

de-Initial Stresses

σ

'sin

• Advantage:

– No displacements are generated

• Disadvantage:

– Using very high or very low K values

may violate MC failure condition

– No equilibrium for inclined surface, but if not major, could apply plastic nil-step as first calculation phase and reset

displacement to zero

' sin 1

' sin 1

' sin 1

' sin

1

0

φ

φ φ

K

so

• Choose Plastic calculation, Total multipliers

• Set weight multiplier ΣMweight = 1

– Phase 2

• Select Reset displacements to zero to

discard all displacements from raising the gravity

Gravity Loading

Notes

– Undrained material

• Select Ignore undrained behaviour in Phase

1 to prevent the generation of unrealistic excess pore pressures

– K0 procedure has been used first

• In the Initial phase redo the K0 procedure, but with ΣMweight = 0; this will reset all initial stresses to zero.

Gravity loading

Cases where gravity loading should be used instead of K0-procedure:

Slope Stability

• Slope stability problems involve evaluation

of safety ratio (EC7) or factor of safety

mobilized

=

τ

Slope Stability

• Options for slope stability analyses include:

– Limit analysis

– Limit equilibrium analysis

– Discontinuity layout optimisation (NEW METHOD)

– Finite element modelling

Slope Stability

• Limit Analysis

– Lower Bound Theorem: if a statically

admissible stress distribution can be found, uncontained plastic flow will not occur at a lower load Considers only equilibrium and yield and not kinematics.

Slope Stability

• Limit Analysis

– Upper bound theorem: if a kinematically

admissible velocity field can be found,

uncontained plastic flow must have occurred previously Considers only velocity modes

and energy dissipations The stresses do not need to be in equilibrium.

collapse upper bound

Slope Stability

• Limit Analysis

– Applicability for soils controversial, as the

assumption of associated flow overestimates dilatancy in frictional soils

Slope Stability

• Limit Equilibrium Method

– Assume a shape for the failure surface

– Only equilibrium (force and/or momemtum) of the rigid mass enclosed by the failure surface

is considered

– Static alone considered (no kinematics)

– Same factor of safety/partial factors applied along the sliding surface

– More equations than unknowns, so statically indeterminate problems

Slope Stability

• Discontinuity Layout Optimization (DLO)

– Automatically identifies the critical

configuration of sliding soil blocks at failure

– No need to independently consider different failure modes e.g the sliding or bearing

collapse of a gravity wall All possible modes, (anticipated or not anticipated by the

engineer) are simultaneously considered

http://www.limitstate.com/geo/technology

Numerical Analyses of Slope

Stability

• Safety ratio/factor of safety determined

without assuming a slip surface a priori

• Rigorious solution, but still numerical

approximation

• Although advanced soil models may be chosen for deformation analyses, stability analyses performed assuming a Mohr

Coulomb

Numerical Analyses of Slope

Stability

• According to EC7, apply partial factors to strength (φ’ and c’ or c u) and action (loads)

as relevant to a particular design case

• If numerical analysis converges, the slope fulfils the conditions for the ultimate limit

state

• Simulations can be severely

mesh-dependent

3-knotiges Dreieck mit 1 Remeshing

Coarse mesh, 4-noded elements

Factor of safety : no failure

Coarse mesh, 8-noded elements

Fine mesh, 4-noded elements Factor of safety : 1.90

Fine Mesh, 8-noded elements

Coarse mesh, 3-noded elements

Coarse mesh, 8-noded elements

Numerical Analyses of Slope

Stability

• EC7 approach tells you that you are safe, but not how far from the failure you are

• In order to check that, it is possible to do a

phi-c reduction analysis in PLAXIS

• For that, it is best to use characteristic factored) values for strength and actions

(un-Numerical Analyses of Slope

Stability

• In phi-c reduction analysis soil parameters

are automatically progressively reduced and the analysis is repeated, until a clear failure mechanism is identified

Phi-c reduction in PLAXIS

During the analysis the values of c’ and

φ σ

under working conditions

During the analysis the values of c’ and

tan φ’ are incrementally reduced

r r

c

c Msf

FOS

Numerical Analyses of Slope

Stability

• Failure in FEM corresponds to a case

when no equilibrium is found within a

given number of iterations

• Inspect the output to make sure that a

physical failure mechanism has been

found (if not modify the control parameters

in PLAXIS)

Numerical Analyses of Slope

phi-c reduction

shear strength

τf

Example of phi-c reduction

=

′ 1 kN / m , 30

c 2

3 m

Example of phi-c reduction

reduced

c

c′ /

39 1 c

c F

max red

Example of phi-c reduction

Introduction to Consolidation and Phi-c Reduction Workshop

Dr Minna KarstunenUniversity of Strathclyde

• To use PLAXIS to study the consolidation beneath an embankment

• Two different types of analysis:

– Consolidation analysis to model the

dissipation of excess porewater pressures

– Phi-c reduction process to assess the factor of safety at different stages

Problem Peat

Clay

2m

Peat Clay Dense sand

3m 3m

B

A

Problem

Embankment Peat Clay

Dense sand

Clay Dense sand

4m B

B

A

1D Consolidation Theory

0

=

i u

pressures in equilibrium over a large area at t=0

(c) For t > 0 consolidation occurs.

(d) As time increases, pore-pressures

return to equilibrium values

Initial ground surface

0

=

u

1D Consolidation Theory (cont.)

Excess pore-pressure, , varies with time and position according to :

u

c t

1D Consolidation Theory (cont.)

The solution to the consolidation equation is expressed in terms of degree of

consolidation :

t z

where is the excess pore-pressure

within the layer and is the initial

( t z u

i

u

1D Consolidation Theory (cont.)

0 0.2 0.4 0.6 0.8

1D Consolidation Theory (cont.)

• Time factor and d defined as

2 1

( − ν ′ +ν ′

/MN m

10 75

.

6 )

1 (

) 1

)(

2 1

day /

m 10

482

1 day

/ m

=

w v

v

m

k c

k

γ

Example (continues)

At 100 days after construction:

658

0

5 1

100 10

482

1

2

2 2

v

658

0

= From the chart (Fig 3) U is estimated as 0.74, giving: v

kPa 3

8 )

74 0 1 (

=

u

Note: solution depends

on e.g mesh coarseness and step size

Changes of FOS during construction

• When an embankment is built on soft soil,

• These excess pore-pressures reduce the

effective stresses in the soil and this, in turn,

effective stresses in the soil and this, in turn,

reduces the factor-of-safety of the

Changes of FOS during

1.0

(a) Embankment costruction

(b) Variation of factor-of-safety with time

Tutorial Example

• Construction in two stages: each 5 days

• Approximate values of FOS when height is 2 m and 4 m

1.6

1.7

F.O.S =1.1 after embankment raised to 2m

1 1.1 1.2 1.3 1.4 1.5

F.O.S =1.05 after embankment raised to 4m

Long term F.O.S = 1.39

Tutorial 3

Construction of a road

embankment

• Problem is symmetric (model only half?)

• Embankment made of loose sandy soil

• Subsoil 3m of peat and 3 m of clay

• WT at GL

• Dense sand layer not included in the model

Plane strain model

Should generally be > 2L L

Use 15-noded elements

γunsat

γsat

MC Undrained

15

18

MC Undrained

8

11

MC Drained

16

20

-

- kN/m3

kN/m3phreatic level

1000 0.33 2.0

24 0.0

2⋅10-31⋅10-3

350 0.35 5.0

20 0.0

1.0 1.0

3000 0.3 1.0

30 0.0

m/day m/day kN/m2

- kN/m2

°

°

• Create three material sets

• Assign data to appropriate clusters

• Create mesh: medium coarseness

Initial conditions

• Unit weight of water 10 kN/m3

• Water pressures hydrostatic: phreatic level

through points (0.0;6.0) and 40.0;6.0)

• Boundary conditions during consolidation

• Boundary conditions during consolidation (see instructions):

– Left boundary closed due to symmetry

– Right boundary also closed

– Bottom boundary open (sand underneath)

• Generate water pressures

Initial conditions (continues)

• Generate water pressures

• Switch & deactivate the embankment

• Generate initial stress: K0-procedure

– Default values of K =1-sin φ ’ (Jaky’s formula)

– Default values of K0=1-sin φ ’ (Jaky’s formula) adopted

• Input is now complete!

Phases:

– Consolidation 200 days

– Consolidation until excess pore pressures have dissipated (minimum pore pressure)

Need 4 calculation phases in total (see instructions) Select points for curves: A toe of embankment, B at centre of clay layer (to plot excess pore pressures)

Displacement increments after undrained construction of embankment

Excess pore pressures after undrained construction of embankment

Excess pore pressure contours after consolidation to

P excess < 1.0 kN/m2

Output: Curves Program

-10

0

Excess PP [kN/m2]

0 200 400 600 800 -40

Safety Analysis

• Important to consider not just the final

stability, but also the stability during

construction

• Global factor of safety immediately after 1st

• Global factor of safety immediately after 1stand 2nd stages of construction and long

term using Phi-c –reduction calculations (see instructions)

Safety Analysis: Output

Shadings of the total displacement increments indicating the likely long-term failure mechanism

Magnitude of strains not important!

Safety Analysis: Output

1.3

1.4

Sum-Msf

Sum-Msf

long term safety factor 1.38

safety factor (1 st construction stage) 1.13

safety factor (2 nd construction stage) 1.06

Updated Mesh Analysis (optional)

• The embankment settles over 0.5m within

2 years

• Part of the fill will settle to be below the

phreatic level and is therefore subject to buoyancy forces

• This effect can be taken into account by using Updated mesh and Updated water

Updated Mesh Analysis: Output

Settlements of the embankment at centreline, updated mesh

calculation compared with regular calculation

Updated Mesh Analysis: Output

Displacement [m]

0.3 0.4 0.5

Settlements of the toe of the embankment, updated mesh

calculation compared with regular calculation

Time [day]

0 200 400 600 800 1000

0 0.1 0.2