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SOIL MECHANICS, ROCK MECHANICS AND UNDERGROUND STRUCTURES ANALYSIS ON MICROCOMPUTERS USING PLASTICITY THEORY: AN INTROD

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),

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SOIL MECHANICS, ROCK MECHANICS AND UNDERGROUND STRUCTURES ANALYSIS ON MICROCOMPUTERS USING PLASTICITY THEORY: AN INTROD

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  1. 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

  2. LECTURE 1 • - Problem statement • Stability analysis • Load carrying capacity • Initial state analysis

  3. Starting with an ENGINEERING DRAFT

  4. 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

  5. DISCRETIZATION IS NEEDED FOR NUMERICAL SOLUTION e.g. by finite elements Equilibrium on (dx ● dy)

  6. 12 +(12 /x2)dx2 f1  12 EQUILIBRIUM STATEMENT, 1-PHASE Domain Ω, with boundary conditions: -imposed displacements -surface loads and body forces: -gravity(usually) 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)

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

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

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

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

  11. y E CONSTITUTIVE MODEL: ELASTIC- PLASTIC With hardening(or softening) 1- dimensional hardening Eep H’ softening 

  12. 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.)

  13. SURFACE FOUNDATION: FROM LOCAL TO GLOBAL NONLINEAR RESPONSE

  14. F(x,t) 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 5.linearized problem it.2 3.linearized problem it.1 1.Converged sol. at tn(Fn,dn) d d dn 1 dn+1

  15. TOLERANCES ITERATIVE ALGORITHMS

  16. INITIAL STATE, STABILITYAND ULTIMATE LOAD ANALYSIS IN SINGLE PHASE MEDIA

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

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

  19. a tutorial is available

  20. GEOMETRY & BOUNDARY CONDITIONS start by defining the geometry

  21. Geometry with box-shaped boundary conditions

  22. MATERIAL & WEIGHT: MOHR-COULOMB

  23. GRAVITY LOAD

  24. ALGORITHM: STABILITY DRIVER 2D Single phase

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

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

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

  28. RUN

  29. VISUALIZATION OF INSTABILITY Displacement intensities

  30. LAST CONVERGED vs DIVERGED STEP

  31. LOCALISATION 1 Transition from distributed to localized strain

  32. LOCALISATION 2

  33. VALIDATION Slope stability 1984

  34. ELIMINATION OF LOCAL INSTABILITY 1 SF=1.4- Material 2, stability disabled SF=1.4+ Slope_Stab_loc_Terrasse.inp

  35. INITIAL STATE, STABILITY AND ULTIMATE LOAD ANALYSIS(foota.inp) IN SINGLE PHASE MEDIA

  36. WE MUST DEFINE: -GEOMETRY & BOUNDARY CONDITIONS (+-as before) -MATERIALS( +-as before) -LOADS and load function -ALGORITHM

  37. F(x,t) DRIVEN LOAD ON A SURFACE FOUNDATION Po(x) F=Po(x)*LF(t) LF t foota.inp

  38. 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)

  39. LOAD FUNCTIONS

  40. ALGORITHM: DRIVEN LOAD DRIVER axisymmetric analysis) =single phase

  41. D-P material

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

  43. 3D YIELD CRITERIA ARE EXPRESSED IN TERMS OF STRESS INVARIANTS 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

  44. SIZE ADJUSTMENTS D-P vs M-C 3-dimensional,external apices 3-dimensional,internal apices Plane strain failure with (default) Axisymmetry intermediate adj. (default)

  45. 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

  46. run footwt.inp

  47. SEE LOGFILE

  48. LOGFILE

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

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