FEM applied to 3D stability analyses

R.Castellanza: Influence of material model on the settlement resultsProf. Ing. Riccardo Castellanza – FEM Course for PhD students – October 2020

Prof. Riccardo Castellanza

Associate Professor Geotechnical Engineering

Department of Earth and Envinromental Sciences

Università degli Studi di Milano Bicocca

Geotechnical modelling (3D FEM based) for slope stability and underground geostructures - Lection 1

FEM applied to 3D stability analyses

35th PhD Cycle

http://www.geo.unimib.it/index.php


1.Geotechnical modelling

2.FEM approach

3.Strength Reduction Method

4.Landslides case studies

5.Geo-structures case studies

6.2D demonstrative case

Outline


“Reason is like an eye staring at reality, greedily taking it in, recording its connections andimplications, penetrating reality, moving from one thing to another yet conserving all ofthem in memory, trying to embrace everything. A human being faces reality using reason.Reason is what makes us human. Luigi Giussani (1997) , “The Religious sense”, McGill-Queen’s University Press )

Geotechnical Modelling: the best starting points

“Modelling forms an implicit part of all engineering design but many engineers engage inmodelling without consciously considering the nature, validity and consequences of thesupporting assumptions. Many engineers make use of numerical modelling but may not havestopped to think about the approximations and assumpions that are implicit in thatmodelling – still less about the nature of the constitutive models that may have beeninvoked. ” David Muir Wood (2004), “Geotechnical Modelling”, Taylor&Francis

Aristotle said “ί έ ᾶ ᾕ ή ᾕ ίή”: any prediction is basedeither on a rational calculation or on intuitive perception. Although the latter has been fora long time the starting-point of any construction and still plays a relevant role in design, itis the former that allows the definition of the structure’s dimensions and safetyassessment. In fact, it allows rational prediction of the structure’s behavior in the differentconstruction phases and during its life. Roberto Nova (2012)”Soil mechanics”, Wiley- ISTE


Geotechnical Modelling: main steps

Real case: description of phenomena

Model:

prediction of the behaviour

Work:

Making the job

Engineering Geology Geotechnical Engineering


Geotechnical Modelling: main

Starting from focusing and observing the physical processes of the problem the following steps will be considered (based on my experience):Step 1: framing and defining the preliminary working hypothesisStep 2: topographic and hydro/geological surveys necessary to understand the boundary value problem (BVP). Definition of two-dimensional or three-dimensional geometric features to be included in the model,Step 3: performing the in situ and laboratory experimental campaign for the characterization of geomaterials based on the phenomenological models that will be used,Step 4: choice of constitutive and phenomenological models to be used, estimation of parameters and state variables on the basis of the experimental evidences, validation of the chosen theoretical frameworkStep 5: construction of the FEM model, execution of the numerical analyses in stages of the boundary value problem,Step 6: critical assessment of the results achieved by comparing them with simple analytical and/or empirical models.

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Design and construction of the geotechnical project (e.g remediational measures)



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1) Geometry & experiments 2) theoretical framework

0ij

w iz

j i

h

x x

+ + =

1

2

h k

hk

k h

U U

x x

= − +

ij p

ij hk, ,ijhk

hkij

ij

Ct t

=

2

2

v

i

hk

tx

− =

Equilibrium solid scheleton

Compatibility

Constitutive model of soil/rock

Continuity equations fluid/soil

+ Boundary condition + Initial condition

FEM discretization + integration

new tunnel

3) numerical predictions



0ij

w iz

j i

h

x x

+ + =

1

2

h k

hk

k h

U U

x x

= − +

2

2

v

i

hk

tx

− =

Equilibrium solid scheleton

Compatibility

Continuity equations fluid/soil

+ Boundary condition + Initial condition

new tunnel

This is the hearth of the geotechnical modeling

This is the body of the geotechnical modeling

ij p

ij hk, ,ijhk

hkij

ij

Ct t

=

Constitutive model of soil/rock

FEM discretization + integration

1) Geometry & experiments 2) theoretical framework 3) numerical predictions




2.FEM approach





Outline


Numerical Solution to Boundary Value Problems

Prof. Ing. Riccardo Castellanza

The field equations which govern a typical soil mechanics problem –

expressing in mathematical terms the balance of mass for the pore fluid

(continuity equation), the balance of momentum for the solid skeleton

(equilibrium equations) and the strain-displacement compatibility

conditions, as well as the constitutive equations for the solid skeleton and

the pore fluid – were defined as a set of partial differential equations.

0ij

w iz

j i

h

x x

+ + =

1

2

h k

hk

k h

U U

x x

= − +

ijhk

hkijCt t

=

2

2

v

i

hk

tx

− =

i ij

j

hV k

x

= −

ij ij iju −

with:

ij p

ij hk, , =

hkij hkij

ij

C C

with:

where


14•Example: An acid solution, seeping through the pores, dissolves a part of the solid skeleton (e.g. bonds) and produce a gas (CO2).

Fully non linear

coupled system of

PDEs

•A complete theoretical study required a multiphase continuum approach with

Basic assumptions for modelling geotechnical environmental BVP

E1 Solid : Momentum Equilibrium Equations

E4 Fluid : Balance of Mass for water

E2 Solid : Kinematic Equations

E3 Solid : Constitutive equations

E5 Contaminant : Balance of mass for contaminant

species (diffusion, advection ad reactions )

+ Boundary and initial conditions

•In our approach we consider 3 different fields: 1) displacements, 2) pore pressure, 3) concentrations of chem. species

3 phases (solid,liquid,gas) and different species.

E6 Gas : Balance of Mass for gas

R.Castellanza: Influence of material model on the settlement resultsProf. Ing. Riccardo Castellanza – FEM Course for PhD students – October 2020Prof. Ing. Riccardo Castellanza

Thus, in order to determine the quantities relevant to the design process

from an engineering viewpoint (i.e. displacement components at specific

points of the soil mass or the structures, earth pressures on retaining

structures, pore water pressure distributions, collapse load of foundations,

etc.), the set of governing partial differential equations is required to be

integrated, together with the chosen constitutive equations and the

appropriate initial and boundary conditions for the specific problem at hand

However, in almost all problems of practical interest, the integration of the

governing system of partial differential equations defined can only be

performed by means of approximate numerical methods.

Finite Element MethodFinite Difference Method

In the continuum approach…

the main methods are…

Numerical Solution to Boundary Value Problems


16


The finite difference method essentially consists of the discretization of the governing partial

differential equations. According to this method, the derivative operator (i.e. the limit of the

difference quotient) is replaced by the difference quotient itself. The denser the chosen

discretization, the closer the results obtained are to the “exact” solution of the problem. ,

When the PDE is a second order elliptic partial differential equation, the finite difference

method proves to be very efficient .

Finite Difference Method

=Ah b

where A is a n x n matrix and h and b are two n-

component vectors, will be obtained.

Overall an algebraic system of n equations with

n unknowns of the kind:


17


When hydraulic and static problems are coupled, the governing field

equations are no longer linear; the order of differentiation of the unknown

functions with respect to time increases and the equation governing the

evolution in space and time of the pore water pressure is parabolic. In this

case, the efficiency and accuracy of the finite difference method are

significantly reduced. For this reason, this approach has been practically

abandoned since the development of the finite element method (FEM).

Finite Element Method


18


On the other hand, the FEM starts discretizing the continuous medium, and

substituting the unknown functions with some suitably selected approximating

functions with local support, characterized by a limited number of unknown

scalar coefficients representing the values of the unknown functions at some

specific points (e.g. the displacement nodal degrees of freedom).

Subsequently, these approximating functions are introduced into the field

equations, which are recast in integral form over the entire domain, and the

constitutive equations are enforced. The algebraic equations governing the

discretized problem then arise “naturally” as a consequence of the initial

discretization.

Finite Element Method



2.FEM approach





Outline


Strength Reduction Method

• Limit Equilibrium methods (2D and 3D):• Several methods available:

• Bishop’s modified method, • Spencer’s method, • Janbu’s generalized slice procedure …

• Soil mass is divided in slices and assumptions are made to satisfy equilibrium➢ The most likely failure mechanism is determined

• Finite element method (2D and 3D):• Equilibrium is obtained by continuum mechanics• No restriction regarding the geometries or the heterogeneities

(reinforcement) that can be considered

• No arbitrary assumption required for the shape of the failure surface• Various elasto-plastic soil models can be considered➢ Both safety factor and failure behaviour are determined

Approaches to slope stability assessment


Differences between the two approaches

Finite Elements Limit equilibrium

Equilibrium Satisfied on the continuum Satisfied only for slices

Stresses Computed on the continuum Computed approximately on certain surfaces

Deformation Computed on the continuum Not considered

FailureYield condition checked in

every point of the continuum

Failure allowed only

on certain pre-defined

surfaces no check on yield

condition elsewhere

KinematicsThe failure mechanism

satisfies kinematic constraints

Kinematics are not

considered –

failure mechanisms

may not be feasible



• Increase of the applied load:• All material properties are kept constant (possibly weighted by a safety

factor)

• The load (weight of superstructure, groundwater level…) is increased until loss of stability

➢ The failure mechanism and the ultimate load are determined

• Reduction of the strength characteristics of the soil (c-f):• Self-weight and additional loads are applied at their nominal value

• The strength characteristics of the soil are reduced until loss of stability

➢ The failure mechanism and the safety factor on material propertiesare determined.

– The uncertainty is usually larger on material properties than on applied loads.

➢ The strength reduction method is preferred

Alternative finite element approaches



Factor of Safety



How is instability (failure) determined in FE analysis?

• Non-convergence of a non-linear static analysis is a suitable indicator for instability (Griffiths and Lane, 1999)

• No convergence = No stress distribution can be found that is simultaneously able to satisfy both the plastic failure criterion and the global equilibrium

• Slope failure and numerical non-convergence occur simultaneously, and are accompanied by a drastic increase in the nodal displacements

• Criteria for non-convergence:• The Energy, Displacement or Force norms are monitored during

the incremental-iterative solution procedure.• A step is non-converged when the maximum number of iterations

(default value = 50) is reached without convergence of the norm.

➢ Accuracy in the determination of the stability limit is depending on the non-linear analysis controls :➢ Size of the safety factor steps and maximum number of steps➢ Maximum number of iterations➢ Type and value of the target convergence norm



Strength reduction procedure



• Example 1:

• Parameters :

Benchmark Examples

1.2H 2H

H

Model mesh & dimensions

2 320 , 10 kN m , 20 kN m

10 m

0.05

c

H

c H

f

= = =

=

=



• Result contour plots:

Displacements on deformed shape

Maximum shear strain

Benchmark Examples



• Factor of Safety results : Comparison with equilibrium methods

Mogenstern & Price : 1.449

Janbu : 1.361

Bishop : 1.452 Spencer : 1.449

1.421.671.461.901.421.471.391.51

Q8Q4T6T3Q8Q4T6T3

GTSPhase2Software

Element type

FS

Benchmark Examples



• Example 2:

• Parameters :

Model mesh & dimensions

2H 2H

H

2H

H

1uc

2uc

2 3

1

1 2 1

0 , 50 kN m , 20 kN m

10 m

0.25, 0.6

u u

u u u

c

H

c H c c

f

= = =

=

= =

Benchmark Examples

Griffiths, D.V. and Lane, P.A., (1999) "Slope stability analysis by finite elements", Geotechnique, 49, 3, 387-403,



• Result contour plots:

Displacements on deformed shape


Benchmark Examples



Bishop : 1.505 Spencer : 1.505

Janbu : 1.393 Mogenstern & Price : 1.505

• Factor of Safety results : Comparison with equilibrium methods

1.391.351.35

Q8Q4T6T3Q8Q4T6T3

GTSPhase2Software

Element type

FS

Benchmark Examples



• Example 3:

– Parameters :

3D Examples

Model geometry Model mesh

Parameters



• Result contour plot:


3D Examples

•Safety Factor 1.19 (see work tree)



1uc

2uc

SRM for Mohr Coulomb failure criteria

1 0tantan

SRFf

−

= f

ff

0

SRFf

C

cc =

[kPa]

[kPa]

ff0

f

0c

fc B

BA

A

' tan c= + f

Local level (in the Gauss point of the element)

Global level - BVP



ff 0

f

0c

fc

1 0tantan

SRFf

−

= f

ffLet’s define and

0

SRFf

C

cc =

if SRF SRFcf

Generalized SRM for Mohr-Coulomb failure criteria

Loss of convergence



ff 0

f

0c

fc

1 0tantan

SRFf

−

= f


0

SRFf

C

cc =

if SRF < SRFcf


e.g. bonded geomaterials - rock



ff 0

f

0c

fc

1 0tantan

SRFf

−

= f


0

SRFf

C

cc =

if SRF >SRFcf




ff

0f

0 0fc c= =

1 0tantan

SRFf

−

= f


0

SRFf

C

cc =

if SRF SRFcf

Purely Granular soil


c



0 0f= =f f

1 0tantan

SRFf

−

= f


0

SRFf

C

cc =

if SRF SRFcf

Purely Cohesive Soil/Rock

0c

fc




Generalized SRM for other failure criteria (e.g. Hoek&Brown)

0.0

200.0

400.0

600.0

800.0

1000.0

-500 0 500 1000 1500 2000 2500 3000

[K

Pa]

[KPa]

Mohr-Coulomb failure loci

Hoek-Brown failure loci



Generalized SRM for other failure criteria (e.g. Hoek&Brown)

0.0

200.0

400.0

600.0

800.0

1000.0

-500 0 500 1000 1500 2000 2500 3000

[K

Pa]

[KPa]

Mohr-Coulomb failure loci

Hoek-Brown failure loci



Generalized SRM for other failure criteria (e.g. Nova model)

p’, p*

q

pt

ps

pm

f0: fresh rock

fw

: partially weathered

fu: uncemented soil

fwf

u

f0A

Bps

pt

pm

pc

pto

0.1.0

xd

pm.t.s

unbonded soilFresh rock

When the shrinking yield surface reaches point A, the stress-strain state move to Bfor respecting consistency.

EXAMPLE: Weathering (or SRM) in oedometric condition (initial stress A)

CHEMO-MECHANICAL

COUPLING

•Nova R., Castellanza R., Tamagnini C. (2003), A constitutive model for bonded geomaterials subject to mechanical and/or chemicaldegradation, Int. J. Num. Anal. Meth. Geomech.; 27(9), 705–732.




2.FEM approach





Outline


Landslides case studies

Starting point: include or not the failure surface(s)

A) Failure surface(s) known and prescribed B) Failure surface

undefined

C) Failure surfacepartially known


Case study #1: 3D geometry from DTM



Case study #1: 3D geometry from DTM



Case study #1: creation of the solid model of the chosen area



Case study #1: creation of the solid model of the chosen area

deposit Adeposit B



Surf. 1A : deep surface between the intact rock and the deposit. It is obtained from the interpolation of 3D points (e.g. coring and geophysical measurements)

Case study #1: definition of deposit A surfaces



Surf. 2A : intermediate surface obtained from the interpolation of 3D points (e.g. coring and geophysical measurements)





Surf. 2A : shallow surface obtained from the interpolation of 3D points (e.g. coring and geophysical measurements)



Case study #1: subdivision of the solid (deposit A)















Case study #1: definition of deposit B surfaces

Surf. 1B : deep surface between the intact rock and the deposit. It is obtained from the interpolation of 3D points (e.g. coring and geophysical measurements)




Surf. 2B : intermediate surface. It is obtained from the interpolation of 3D points(e.g. coring and geophysical measurements)




Surf. 2B : intermediate surface. It is obtained from the interpolation of 3D points(e.g. coring and geophysical measurements)




Surf. 3B : shallow surface. It is obtained from the interpolation of known points (e.g. coring and geoelectic)



Case study #1: subdivision of the solid (deposit B)













Check on 2D sections



Case study #1: definition of the deposits



Case study #1: mesh generation (dep. A)



Deposit A: 3D analysis, c-f reduction - Fs > 2.5

Case study #1: SRM analysis and results (plastic strains)



Deposit A: 3D analysis, c-f reduction - Fs > 2.5




Case study #1: mesh generation (deposit B)



Deposit B: 3D analysis, c-f reduction - Fs > 3.1




Deposit B: 3D analysis, c-f reduction - Fs > 3.1




Sez. A

Sez. B

Sez. C

Sez. A Sez. B Sez. C

Case study #1: 2D detail analysis (dep. A sections)



Sez. A Sez. B Sez. C

Case study #1: extraction of 2D surf. and rotation on x-y plane (sect. A)



Case study #1: extraction of 2D surf. and rotation on x-y plane (sect. A)



Case study #1: material properties assignment



Case study #1: mesh of section A



Plastic strains

Case study #1: SRM analysis and results of sect. A; Fs=1.68



Displacements

Case study #1: SRM analysis and results of sect. A; Fs=1.68



Sez. D

Sez. E

Sez. F

Sez. G

Case study #1: extraction of 2D surf. and rotation on x-y plane (sect. E)



Case study #1: mesh of section E



Case study #1: mesh of section E



83

Dott. Ing. Riccardo Castellanza

Plastic strains

Case study #1: SRM analysis and results of sect. E; Fs=1.78



84

Dott. Ing. Riccardo Castellanza

Case study #1: SRM analysis and results of sect. E; Fs=1.78

Displacements (deformed)



3D SRM Fs 2.5 (A) and 3.1(B)


2D SRM Fs beetwen 1.6 and 1.9



Case study #2: Vajont back-analysesCase study #2: Vajont landslide (1960-1963)



Case study #2: Vajont landslide (1960-1963)9 october 1963, about 300ml m3 , about 2000 deaths



Case study #2: Vajont landslide (1960-1963)

before after

9 ottobre h 22.39’46’’







after

9 ottobre h 22.39’46’’

5

62

63

(a

(b

Modified by R. Selli, L. Trevisan, (1964)

Water level October 1963


lake level Groundwater level

5

62

63

(a

(b

Modified by R. Selli, L. Trevisan, (1964)



lake level Groundwater level





after

9 ottobre h 22.39’46’’



Progressive Shear strengthreduction until loss of

convergence

0

50

100

150

200

250

300

1214161820222426

coh

esi

on

[kP

a]

friction angle [deg]

Failure → progressive weakening

Ave. Material propertiesleading to a generalized

instability

3D FEM: ave material properties

3D effect of lake impounding on slope stability and activityThin shear zone along the 3D reconstruction of the failure zone→ Bistacchi et al.

Case study #2: Vajont back-analyses



3D Mesh

3D FEM: ave material properties Case study #2: Vajont back-analyses

prescribed failure surface




# 1/1: Lake level 700 m a.s.l.

(c=300 kPa, f=26°)

Displacement

Plastic strain

Displacement

Plastic strains



(c=200 kPa, f=22°)

Displacement

Plastic strain



Displacement

Plastic strains





(c=100 kPa, f=18°)

Displacement

Plastic strain

Displacement

Plastic strains





(c=50 kPa, f=14°)

Displacement

Plastic strain

Displacement

Plastic strains





(c=25 kPa, f=13°)

Displacement

Plastic strain

Loss of convergence

Displacement

Plastic strains





Landslide case studies

Case study #3: 3D rockslide



Case study #3: rockslide (in collaboration with Studio Associato Cancelli)

G. B. Crosta, C. di Prisco, G. Frigerio, P. Frattini, R. Castellanza, F. Agliardi (2013) Chasing a completeunderstanding of the triggering mechanisms of a large rapidly evolving rockslide, Landslides,



Case study #3: 3D rockslide



Case study #3: 3D slope (in collaboration with Studio Associato Cancelli)



Case study #3: 3D rockslide (in collaboration with Studio Associato Cancelli)

Real Failure Surface

MC: parameters:Friction angle : 27°

Cohesion c’ = 23 kPa




Real Failure Surface




G. B. Crosta, C. di Prisco, G. Frigerio, P. Frattini, R. Castellanza, F. Agliardi (2013) Chasing a completeunderstanding of the triggering mechanisms of a large rapidly evolving rockslide, Landslides,




Giovanni B. Crosta, Paolo Frattini, Riccardo Castellanza, Gabriele Frigerio, Claudio di Prisco, Giorgio Volpi, Mattia De Caro, Paolo Cancelli, AndreaTamburini, Walter Alberto, and Davide Bertolo, Investigation, Monitoring and Modelling of a Rapidly Evolving Rockslide: The Mt. de la Saxe CaseStudy, 349 Engineering Geology for Society and Territory - Volume 2. pp 349-354



1 0tantan

SRFf

−

= f

ff

0

SRFf

C

cc =

An higher reduction in cohesion than in the friction angle is more suitable for simulating the rock weathering

[kPa]

[kPa]

ff0

f

0c

fc B

BA

A

' tan c= + fSRM for Mohr Coulomb failure criteria




ff

0f

0c

fc

SRF SRFc f

cohesion

[kPa]

friction angle [°]

SRFc

coesione Cs

(kPa) SRFФ' Cs (°)

1.00 350 1.00 33.0

1.40 250 1.02 32.5

1.80 194 1.03 32.0

2.20 159 1.05 31.5

2.60 135 1.06 31.0

2.75 127 1.08 30.5

2.90 121 1.10 30.0

3.05 115 1.12 29.5

3.20 109 1.14 29.0

3.35 104 1.16 28.5

3.50 100 1.18 28.0

1 0tantan

SRFf

−

= f

ff0

SRFf

C

cc =

An higher reduction in cohesion than in the friction angle is more suitable for simulating the rock weathering.




SRFc

coesione Cs

(kPa) SRFФ' Cs (°)

1.00 350 1.00 33.0

1.40 250 1.02 32.5

1.80 194 1.03 32.0

2.20 159 1.05 31.5

2.60 135 1.06 31.0

2.75 127 1.08 30.5

2.90 121 1.10 30.0

3.05 115 1.12 29.5

3.20 109 1.14 29.0

3.35 104 1.16 28.5

3.50 100 1.18 28.0

1 0tantan

SRFf

−

= f

ff0

SRFf

C

cc =

An higher reduction in cohesion than in the friction angle is more suitable for simulating the rock weathering.

SRFC

Norm

aliz

ed

cohesive

and

fric

tionalst

rength

[%]

0

ff

f

0

c

c




Giovanni B. Crosta, Paolo Frattini, Riccardo Castellanza, Gabriele Frigerio, Claudio di Prisco, Giorgio Volpi, Mattia De Caro, Paolo Cancelli, AndreaTamburini, Walter Alberto, and Davide Bertolo, Investigation, Monitoring and Modelling of a Rapidly Evolving Rockslide: The Mt. de la Saxe CaseStudy, 349 Engineering Geology for Society and Territory - Volume 2. pp 349-354

Documents

FEM applied to 3D stability analyses