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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

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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 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)

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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 E y Remark: this problem is non-linear

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**CONSTITUTIVE MODEL: ELASTIC- PLASTIC**

With hardening(or softening) dimensional hardening E y 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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**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

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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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** ( imposed, 0 by default)**

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**

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: C’=C/SF2 tan’= tan/SF2 SF3: C’=C/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 1 Transition from distributed to localized strain

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LOCALISATION 2

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VALIDATION Slope stability 1984

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**ELIMINATION OF LOCAL INSTABILITY 1 SF=1.4- Material 2, stability**

disabled SF=1.4+ Slope_Stab_loc_Terrasse.inp

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**ULTIMATE LOAD ANALYSIS(foota.inp)**

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) F(x,t)**

F=Po(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: F=uo(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**

VON MISES Identification with Mohr-Coulomb requires size adjustment

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**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

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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**

F F >>d = d NON CONVERGENCE F F >cst.>TOL. d

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**COMPUTATION OF ULTIMATE LOAD**

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

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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 t=0

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PROOF - -

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**2. LOCAL (MATERIAL LEVEL)**

1.GLOBAL LEVEL

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INITIAL STATE CASE Compute initial state 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 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

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END LECTURE 1

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