SOIL MECHANICS, ROCK MECHANICS AND UNDERGROUND STRUCTURES

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SOIL MECHANICS, ROCK MECHANICS AND UNDERGROUND
STRUCTURES ANALYSIS ON MICROCOMPUTERS USING
PLASTICITY THEORY AN INTRODUCTION TO Z_SOIL.PC
2D/3D OUTLINE Short 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, 0900          August  28-29
2008   (1.5days),  EVENT II    Z_SOIL.PC 3D
course , at EPFL room CO121, 1400    
    participants need to bring their own
computer min 1GB RAM
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• LECTURE 1
• - Problem statement
• Stability analysis
• Load carrying capacity
• Initial state analysis

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Starting with an ENGINEERING DRAFT
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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
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DISCRETIZATION IS NEEDED FOR NUMERICAL SOLUTION
e.g. by finite elements

Equilibrium on (dx ? dy)
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EQUILIBRIUM STATEMENT, 1-PHASE
Domain O, with boundary conditions -imposed
displacements -surface loads and body
forces -gravity(usually)
equilibrium
?11
?11(??11/?x1)dx1
x2
x1
dx1
direction 1 (??11/?x1)dx1dx2(??12 /?x2) dx1dx2
f1dx1dx20
L(u) ??ij/?xj fi0, differential equation(sum
on j)
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SOLID(1-phase) BOUNDARY CONDITIONS
2.natural on ?, 0 by default
sliding
fixed
1.essential on d,
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FORMAL DIFFERENTIAL PROBLEM STATEMENT
Deformation(1-phase)
(equilibrium)
(displ.boundary cond.)
(traction bound. cond.)
Incremental elasto-plastic constitutive equation
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WHY elasto-PLASTICITY?

• non coaxiality of stress
• and strain increments

2.unloading
?
elastic
E
plastic
E
?
sand
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CONSTITUTIVE MODEL ELASTIC-PERFECTLY
PLASTIC 1- dimensional
Remark this problem is non-linear
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CONSTITUTIVE MODEL ELASTIC- PLASTIC With
hardening(or softening) 1-
dimensional
hardening
Eep
H
softening
?
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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.)
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SURFACE FOUNDATION FROM LOCAL TO GLOBAL
NONLINEAR RESPONSE
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REMARK The problems we tackle in geomechanics are
always nonlinear, they require linearization,
iterations, and convergence checks
F
4.out of balance force after 1 iteration
N(d),unknown
Fn1
6.Out of balance after 2 iterations ltgtTol.?
2.?F
Fn
5.linearized problem it.2
3.linearized problem it.1
1.Converged sol. at tn?(Fn,dn)
d
d
dn
1
dn1
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TOLERANCES ITERATIVE ALGORITHMS
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INITIAL STATE, STABILITY AND ULTIMATE LOAD
ANALYSIS IN SINGLE PHASE MEDIA
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BOUNDARY CONDITIONS (cut.inp) Single phase problem
?? (? imposed, 0 by default)
?domain ?
? ?? ?u
?u (u imposed)
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WE MUST DEFINE -GEOMETRY BOUNDARY
CONDITIONS -MATERIALS -LOADS -ALGORITHM
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a tutorial is available
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GEOMETRY BOUNDARY CONDITIONS
start by defining the geometry
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Geometry with box-shaped boundary conditions
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MATERIAL WEIGHT MOHR-COULOMB
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GRAVITY LOAD
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ALGORITHM STABILITY DRIVER
2D
Single phase
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Assume
STABILITY ALGORITHM
with
?s
then
Algorithm -set C C/SF tan ?(tan
?)/SF -increase SF till instability
occurs
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ALTERNATIVE SAFETY FACTOR DEFINITIONS
SF1 SF1? ??m??s
SF2 CC/SF2 tan? tan?/SF2
SF3 CC/SF3
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ALGORITHM STABILITY DRIVER
ALTERNATIVE SAFETY FACTOR DEFINITIONS
2D
Single phase
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RUN
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VISUALIZATION OF INSTABILITY
Displacement intensities
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LAST CONVERGED vs DIVERGED STEP
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LOCALISATION 2
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VALIDATION Slope stability
1984
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ELIMINATION OF LOCAL INSTABILITY 1
SF1.4-
Material 2, stability disabled
SF1.4
Slope_Stab_loc_Terrasse.inp
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INITIAL STATE, STABILITY AND ULTIMATE LOAD
ANALYSIS(foota.inp) IN SINGLE PHASE MEDIA
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WE MUST DEFINE -GEOMETRY BOUNDARY
CONDITIONS (-as before) -MATERIALS( -as
before) -LOADS and load function -ALGORITHM
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DRIVEN LOAD ON A SURFACE FOUNDATION
Po(x)
FPo(x)LF(t)
LF
t
foota.inp
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• REMARK
• It is often safer to use driven displacements to
  avoid
• taking a numerical instability for a true
  failure, then

Fuo(x)LF(t)
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LOAD FUNCTIONS
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ALGORITHM DRIVEN LOAD DRIVER
axisymmetric analysis)
single phase
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D-P material
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DRUCKER-PRAGER MISES CRITERIA
DRUCKER-PRAGER
VON MISES
Identification with Mohr-Coulomb requires size
adjustment
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3D YIELD CRITERIA ARE EXPRESSED IN TERMS
OF STRESS INVARIANTS I1tr ? ?kk 3 ?
?11?22?33 1st stress invariant J20.5 tr
s20.5 sij sji 2nd invariant of deviatoric
stress tensor J3(1/3) sij sjk ski 3rd
invariant of deviatoric stress tensor
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SIZE ADJUSTMENTS D-P vs M-C
3-dimensional,external apices
3-dimensional,internal apices
Plane strain failure with (default)
Axisymmetry intermediate adj. (default)
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PLASTIC FLOW
M-C(M-W)
associated with D-P in deviatoric plane
associated with D-P in deviatoric plane
dilatant flow in meridional plane
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run footwt.inp
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SEE LOGFILE
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LOG FILE
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SIGNS OF FAILURE Localized displacements
before at failure
scales are different!
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• REMARK
• When using driven loads,there is always a risk of
  taking
• numerical divergence for the ultimate load use
  preferably
• driven displacements

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DIVERGENCE VS NON CONVERGENCE
DIVERGENCE
F
?F gtgtd
d
NON CONVERGENCE
F
?F gtcst.gtTOL.
d
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COMPUTATION OF ULTIMATE LOAD
LF
2
1.5
1
F(x,t)P(x)LF(t)
10
20
30
t
last converged step
Fult.PLF(t20)101.515 kN
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LAST CONVERGED STEP
DIVERGED STEP
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DISPLACEMENT TIME-HISTORY
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VALIDATION OF LOAD BEARING CAPACITY plane
strain after CHEN 1975
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MORE GENERAL CASES Embedded footing with water
table

• Remarks
• Can be solved as single phase
• Watch for local cut instabilities

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VALIDATION OF LOAD BEARING CAPACITY axisymmetry
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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 t0
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PROOF
-
-
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2. LOCAL (MATERIAL LEVEL)
1.GLOBAL LEVEL
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INITIAL STATE CASE

1. Compute initial state
2. Add stories

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ENV.INP DRIVERS SEQUENCE
simulation of increasing number of stories
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INITIAL STATE ANALYSIS env.inp
Initial state stress level Ultimate load
displacements
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REMARKS
1.The initial state driver applies gravity and
loads which are nonzero at time t0,
progressively, to avoid instabilities
2.Failure to converge may occur during initial
state analysis, switching to driven load may help
identifying the problem 3.Nonlinear behavior,
flow, and two-phase behavior are accounted for
in the initial state analysis
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END LECTURE 1