# 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 (1.5days), EVENT I:    Z_SOIL.PC 2D course , at EPFL room CO121, 09:00        August    (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
Domain Ω, with boundary conditions: -imposed displacements -surface loads and body forces: -gravity(usually) 12 +(12 /x2)dx2 12 f1 equilibrium 11 11+(11/x1)dx1 x2 x1 dx1 direction 1: (11/x1)dx1dx2+(12 /x2) dx1dx2+ f1dx1dx2=0 L(u)= ij/xj + fi=0, differential equation(sum on j)

SOLID(1-phase) BOUNDARY CONDITIONS
2.natural: on , 0 by default sliding fixed 1.essential: on d,

FORMAL DIFFERENTIAL PROBLEM STATEMENT
Deformation(1-phase): (equilibrium) (displ.boundary cond.) (traction bound. cond.) Incremental elasto-plastic constitutive equation:

WHY elasto-PLASTICITY?
non coaxiality of stress and strain increments 2.unloading elastic E plastic E sand

CONSTITUTIVE MODEL: ELASTIC-PERFECTLY PLASTIC
1- dimensional E y Remark: this problem is non-linear

CONSTITUTIVE MODEL: ELASTIC- PLASTIC
With hardening(or softening) dimensional hardening E y Eep H’ softening

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

The problems we tackle in geomechanics are always nonlinear,
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 Fn+1 6.Out of balance after 2 iterations <=>Tol.? 2.F Fn F(x,t) 5.linearized problem it.2 3.linearized problem it.1 1.Converged sol. at tn(Fn,dn) d d dn 1 dn+1

TOLERANCES ITERATIVE ALGORITHMS

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

 ( imposed, 0 by default)
BOUNDARY CONDITIONS (cut.inp) Single phase problem  ( imposed, 0 by default) domain   = +u u (u imposed)

WE MUST DEFINE: -GEOMETRY & BOUNDARY CONDITIONS -MATERIALS -LOADS -ALGORITHM

a tutorial is available

GEOMETRY & BOUNDARY CONDITIONS
start by defining the geometry

Geometry with box-shaped
boundary conditions

MATERIAL & WEIGHT: MOHR-COULOMB

ALGORITHM: STABILITY DRIVER
Single phase

Assume STABILITY ALGORITHM with s then Algorithm: -set C’= C/SF tan ’=(tan )/SF -increase SF till instability occurs

ALTERNATIVE SAFETY FACTOR DEFINITIONS
SF1: SF1= =m+s SF2: C’=C/SF2 tan’= tan/SF2 SF3: C’=C/SF3

ALGORITHM: STABILITY DRIVER
ALTERNATIVE SAFETY FACTOR DEFINITIONS 2D Single phase

RUN

VISUALIZATION OF INSTABILITY
Displacement intensities

LAST CONVERGED vs DIVERGED STEP

LOCALISATION 1 Transition from distributed to localized strain

LOCALISATION 2

VALIDATION Slope stability 1984

ELIMINATION OF LOCAL INSTABILITY 1 SF=1.4- Material 2, stability
disabled SF=1.4+ 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 FOUNDATION Po(x) F(x,t)
F=Po(x)*LF(t) LF t foota.inp

REMARK It is often safer to use driven displacements to avoid taking a numerical instability for a true failure, then: F=uo(x)*LF(t)

axisymmetric analysis) =single phase

D-P material

DRUCKER-PRAGER & MISES CRITERIA
VON MISES Identification with Mohr-Coulomb requires size adjustment

STRESS INVARIANTS 3D YIELD CRITERIA ARE EXPRESSED IN TERMS OF
I1=tr = kk =3 = 11+22+33 ; 1st stress invariant J2=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 ADJUSTMENTS D-P vs M-C 3-dimensional,external apices 3-dimensional,internal apices Plane strain failure with (default) Axisymmetry intermediate adj. (default)

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

run footwt.inp

SEE LOGFILE

LOG FILE

SIGNS OF FAILURE: Localized displacements
before at failure scales are different!

REMARK When using driven loads,there is always a risk of taking numerical divergence for the ultimate load: use preferably driven displacements

DIVERGENCE VS NON CONVERGENCE
F F >>d = d NON CONVERGENCE F F >cst.>TOL. d

LF P=10 kN 2 1.5 1 F(x,t)=P(x)*LF(t) 10 20 30 t last converged step Fult.=P*LF(t=20)=10*1.5=15 kN

LAST CONVERGED STEP DIVERGED STEP

DISPLACEMENT TIME-HISTORY

plane strain after CHEN 1975

MORE GENERAL CASES: Embedded footing with water table Remarks: Can be solved as single phase Watch for local “cut” instabilities

VALIDATION OF LOAD BEARING CAPACITY axisymmetry

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 - -

2. LOCAL (MATERIAL LEVEL)
1.GLOBAL LEVEL

INITIAL STATE CASE Compute initial state Add stories

ENV.INP DRIVERS SEQUENCE simulation of increasing number of stories

INITIAL STATE ANALYSIS
env.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 problem 3.Nonlinear behavior, flow, and two-phase behavior are accounted for in the initial state analysis

END LECTURE 1