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, 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
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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.
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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
- 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
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
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