SOIL MECHANICS, ROCK MECHANICS AND UNDERGROUND STRUCTURESANALYSIS ON MICROCOMPUTERS USING PLASTICITY THEORY:

AN INTRODUCTION TO Z_SOIL.PC 2D/3D

OUTLINEShort courses taught by  A. Truty, K.Podles, Th. Zimmermann & coworkers

in Lausanne, Switzerland

  August 27-28 2008 (1.5days), EVENT I:    Z_SOIL.PC 2D course , at EPFL room CO121, 09:00

       

August  28-29 2008   (1.5days),  EVENT II:    Z_SOIL.PC 3D course , at EPFL room CO121, 14:00        participants need to bring their own computer: min 1GB RAM

LECTURE 1

- Problem statement- Stability analysis- Load carrying capacity- Initial state analysis

Starting with an ENGINEERING DRAFT

PROBLEM COMPONENTS

- EQUILIBRIUM OF 2-PHASE PARTIALLY SATURATED MEDIUM

- NON TRIVIAL INITIAL STATE- NONLINEAR MATERIAL BEHAVIOR(elasticity is not applic.)- POSSIBLY GEOMETRICALLY NONLINEAR BEHAVIOR- TIME DEPENDENT -GEOMETRY

-LOADS -BOUNDARY CONDITIONS

DISCRETIZATION IS NEEDED FOR NUMERICAL SOLUTION

e.g. by finite elements

Equilibrium on (dx ● dy)

EQUILIBRIUM STATEMENT, 1-PHASE

11 11+(11/x1)dx1

12 +(12 /x2)dx2

12

f1

direction 1:

(11/x1)dx1dx2+(12 /x2) dx1dx2+ f1dx1dx2=0

L(u)= ij/xj + fi=0, differential equation(sum on j)

x1

x2

dx1

Domain Ω, with boundary conditions: -imposed displacements

-surface loadsand body forces: -gravity(usually)

equilibrium

SOLID(1-phase) BOUNDARY CONDITIONS

2.natural: on ,0 by default

1.essential: on d,

fixed

sliding

u

on

uon

FORMAL DIFFERENTIAL PROBLEM STATEMENT

, 0ij j if on xTime

Deformation(1-phase):

k uu on Γ xT

i tt on Γ xT

;ij small displacements assumed

ep

i, j j,i

Δσ = D Δε

Δε 0.5( u + u ) u

Incremental elasto-plastic constitutive equation:

(equilibrium)

(displ.boundary cond.)

(traction bound. cond.)

WHY elasto-PLASTICITY?

1. non coaxiality of stress and strain increments

elastic

plastic

2.unloading

E

E

sand

E

y

CONSTITUTIVE MODEL: ELASTIC-PERFECTLY PLASTIC 1- dimensional

Remark: this problem is non-linear

epE

E

y

EepH’

softening

hardening

CONSTITUTIVE MODEL: ELASTIC- PLASTIC

With hardening(or softening) 1- dimensional

:alternatively p pΔσ = E (Δε - Δε ) or Δσ = H'Δε

ep

let ande pepΔε = Δε + Δε Δσ = E Δε

NB:-softening will engender mesh dependence of the solution -some sort of regularization is needed in order to recover mesh objectivity -a charateristic length will be requested from the user when a plastic model with softening is used (M-W e.g.)

SURFACE FOUNDATION:FROM LOCAL TO GLOBAL NONLINEAR RESPONSE

REMARKThe problems we tackle in geomechanics are always nonlinear, they require linearization, iterations, and convergence checks

F

d

Fn

dn

Fn+1 6.Out of balance after 2 iterations<=>Tol.?

2.F

3.linearized problem it.1

1.Converged sol. at tn(Fn,dn)

N(d),unknown4.out of balance force after 1 iteration

5.linearized problem it.2

dn+11

F(x,t)

d

TOLERANCES ITERATIVE ALGORITHMS

INITIAL STATE, STABILITY AND ULTIMATE LOAD ANALYSIS IN SINGLE PHASE MEDIA

BOUNDARY CONDITIONS (cut.inp)Single phase problem

( imposed, 0 by default)

u (u imposed)

domain = +u

WE MUST DEFINE:

-GEOMETRY & BOUNDARY CONDITIONS-MATERIALS-LOADS-ALGORITHM

a tutorial is available

start by defining the geometry

GEOMETRY & BOUNDARY CONDITIONS

Geometry with box-shaped boundary conditions

MATERIAL & WEIGHT: MOHR-COULOMB

GRAVITY LOAD

ALGORITHM: STABILITY DRIVER

Single phase

2D

s

STABILITY ALGORITHM

sd

sdSF

s

sy

with

tanny C then

tan

( / ) (tan / )y n

sn

s s

C d s

d s C SF SF d sSF

Algorithm: -set C’= C/SF tan ’=(tan )/SF

-increase SF till instability occurs

Assume

ALTERNATIVE SAFETY FACTOR DEFINITIONS

SF1: SF1= =m+s

SF2: C’=C/SF2 tan’= tan/SF2

SF3: C’=C/SF3

ALGORITHM: STABILITY DRIVER

Single phase

2D

ALTERNATIVE SAFETY FACTOR DEFINITIONS

RUN

Displacement intensities

VISUALIZATION OF INSTABILITY

LAST CONVERGED vs DIVERGED STEP

LOCALISATION 1Transition from distributed to localized strain

LOCALISATION 2

VALIDATIONSlope stability

1984

SF=1.4+

SF=1.4-

ELIMINATION OF LOCAL INSTABILITY 1

Material 2, stabilitydisabled

Slope_Stab_loc_Terrasse.inp

INITIAL STATE, STABILITY AND

ULTIMATE LOAD ANALYSIS(foota.inp) IN SINGLE PHASE MEDIA

WE MUST DEFINE:

-GEOMETRY & BOUNDARY CONDITIONS(+-as before)

-MATERIALS( +-as before)-LOADS and load function-ALGORITHM

DRIVEN LOAD ON A SURFACE FOUNDATIONF(x,t)

F=Po(x)*LF(t)

Po(x)

LF

t

foota.inp

REMARK

1. It is often safer to use driven displacements to avoid taking a numerical instability for a true failure, then:

F=uo(x)*LF(t)

LOAD FUNCTIONS

ALGORITHM: DRIVEN LOAD DRIVER

=single phase

axisymmetric analysis)

D-P material

DRUCKER-PRAGER & MISES CRITERIA

ijijij SJar

kJaIF

)2/(1 2

21

DRUCKER-PRAGER

VON MISES

ijij

VM

SJr

kJF

)2/(1 2

2

Identification with Mohr-Coulombrequires size adjustment

3D YIELD CRITERIA ARE EXPRESSED IN TERMS OFSTRESS INVARIANTS

I1=tr = kk =3 = 11+22+33 ; 1st stress invariantJ2=0.5 tr s**2=0.5 sij sji ; 2nd invariant of deviatoric stress tensor

J3=(1/3) sij sjk ski ; 3rd invariant of deviatoric stress tensor

SIZE ADJUSTMENTSD-P vs M-C

))sin3(3/()cos6));sin3(3/(sin2 Cka

3-dimensional,external apices

3-dimensional,internal apices

))sin3(3/()cos6));sin3(3/(sin2 Cka

Plane strain failure with (default)

)cos;3/sin Cka

0

Axisymmetry intermediate adj. (default)

)sin9/(cos36);sin9/(sin32 22 Cka

PLASTIC FLOW

associated with D-P in deviatoric plane

associated with D-P in deviatoric plane

M-C(M-W)

dilatant flow in meridional plane

run footwt.inp

SEE LOGFILE

LOG FILE

SIGNS OF FAILURE: Localized displacements

before at failure

scales are different!

REMARK

1. When using driven loads,there is always a risk of takingnumerical divergence for the ultimate load: use preferablydriven displacements

DIVERGENCE VS NON CONVERGENCE

F

F

d

d

F >>d =

DIVERGENCE

NON CONVERGENCE

F >cst.>TOL.

t

LF2

1

10 20 30

1.5

P=10 kN

F(x,t)=P(x)*LF(t)

last converged step

Fult.=P*LF(t=20)=10*1.5=15 kN

COMPUTATION OF ULTIMATE LOAD

LAST CONVERGED STEP

DIVERGED STEP

DISPLACEMENT TIME-HISTORY

VALIDATION OF LOAD BEARING CAPACITYplane strain

after CHEN 1975

MORE GENERAL CASES:Embedded footing with water table

Remarks:1. Can be solved as single phase2. Watch for local “cut” instabilities

VALIDATION OF LOAD BEARING CAPACITYaxisymmetry

INITIAL STATE ANALYSIS (env.inp)

Superposition of gravity+o(gravity)+preexisting loads*

yields: (gravity)+ (prexist. loads)and NO DEFORMATION

*/ the ones with non-zero value at time t=0

PROOF

--

1.GLOBAL LEVEL

2. LOCAL (MATERIAL LEVEL)

INITIAL STATE CASE

1. Compute initial state2. Add stories

ENV.INP DRIVERS SEQUENCE

simulation of increasing number of stories

INITIAL STATE ANALYSISenv.inp

Initial state stress level

Ultimate load displacements

REMARKS

1.The initial state driver applies gravity and loads which are nonzero at time t=0, progressively, to avoid instabilities

2.Failure to converge may occur during initial state analysis,switching to driven load may help identifying the problem3.Nonlinear behavior, flow, and two-phase behavior are accounted for in the initial state analysis

END LECTURE 1