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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 EPFL August (1.5days), EVENT II: Z_SOIL.PC 3D course, at EPFL room CO121, 14:00EPFL 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 11 11 +( 11 / x 1) dx 1 12 +( 12 / x 2) dx 2 12 f1f1 direction 1: ( 11 / x 1) dx 1 dx 2 +( 12 / x 2) dx 1 dx 2 + f 1 dx 1 dx 2 =0 L(u)= ij / x j + f i =0, differential equation(sum on j) x1x1 x2x2 dx 1 Domain Ω, with boundary conditions: -imposed displacements -surface loads and body forces: -gravity(usually) equilibrium

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

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

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WHY elasto-PLASTICITY? 1.non coaxiality of stress and strain increments elastic plastic 2.unloading E E sand

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

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E yy E ep H’ softening hardening CONSTITUTIVE MODEL: ELASTIC- PLASTIC With hardening(or softening) 1- dimensional

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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 d FnFn dndn F n+1 6.Out of balance after 2 iterations Tol.? 2. F 3.linearized problem it.1 1.Converged sol. at t n (Fn,dn) N(d),unknown 4.out of balance force after 1 iteration 5.linearized problem it.2 d n+1 1 F(x,t) d

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TOLERANCESITERATIVE 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) u (u imposed) domain = + u

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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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start by defining the geometry GEOMETRY & BOUNDARY CONDITIONS

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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 2D

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

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ALTERNATIVE SAFETY FACTOR DEFINITIONS SF1:SF1= = m + s SF2:C’=C/SF2tan ’= tan /SF2 SF3: C’=C/SF3

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

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RUN

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Displacement intensities VISUALIZATION OF INSTABILITY

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

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REMARK 1.It is often safer to use driven displacements to avoid taking a numerical instability for a true failure, then: F=u o (x)*LF(t)

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LOAD FUNCTIONS

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ALGORITHM: DRIVEN LOAD DRIVER =single phase axisymmetric analysis)

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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 I 1= tr = kk =3 = 11 + 22 + 33 ;1 st stress invariant J 2 =0.5 tr s** 2=0.5 s ij s ji; 2 nd invariant of deviatoric stress tensor J 3 =(1/3) s ij s jk s ki; 3 rd 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 associated with D-P in deviatoric plane associated with D-P in deviatoric plane M-C(M-W) 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 beforeat failure scales are different!

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REMARK 1.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 F >>d = DIVERGENCE NON CONVERGENCE F >cst.>TOL.

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t LF P=10 kN F(x,t)=P(x)*LF(t) last converged step F ult.=P*LF(t=20)=10*1.5=15 kN COMPUTATION OF ULTIMATE LOAD

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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: 1.Can be solved as single phase 2.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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1.GLOBAL LEVEL 2. LOCAL (MATERIAL 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 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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