SOIL MECHANICS, ROCK MECHANICS AND UNDERGROUND STRUCTURES ANALYSIS ON MICROCOMPUTERS USING...
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Transcript of SOIL MECHANICS, ROCK MECHANICS AND UNDERGROUND STRUCTURES ANALYSIS ON MICROCOMPUTERS USING...
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