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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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Presentation on theme: "SOIL MECHANICS, ROCK MECHANICS AND UNDERGROUND STRUCTURES ANALYSIS ON MICROCOMPUTERS USING PLASTICITY THEORY: AN INTRODUCTION TO Z_SOIL.PC 2D/3D OUTLINE."— Presentation transcript:

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 (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

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 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

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

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

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

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

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

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 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

15 TOLERANCESITERATIVE ALGORITHMS

16

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

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

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

20 a tutorial is available

21 start by defining the geometry GEOMETRY & BOUNDARY CONDITIONS

22 Geometry with box-shaped boundary conditions

23 MATERIAL & WEIGHT: MOHR-COULOMB

24 GRAVITY LOAD

25 ALGORITHM: STABILITY DRIVER Single phase 2D

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

27 ALTERNATIVE SAFETY FACTOR DEFINITIONS SF1:SF1=  =  m  +  s SF2:C’=C/SF2tan  ’= tan  /SF2 SF3: C’=C/SF3

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

29 RUN

30 Displacement intensities VISUALIZATION OF INSTABILITY

31 LAST CONVERGED vs DIVERGED STEP

32

33 LOCALISATION 2

34 VALIDATION Slope stability 1984

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

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

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

38 DRIVEN LOAD ON A SURFACE FOUNDATION F(x,t) F=P o (x)*LF(t) P o (x) LF t foota.inp

39 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)

40 LOAD FUNCTIONS

41 ALGORITHM: DRIVEN LOAD DRIVER =single phase axisymmetric analysis)

42 D-P material

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

44 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

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

46 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

47 run footwt.inp

48 SEE LOGFILE

49 LOG FILE

50 SIGNS OF FAILURE: Localized displacements beforeat failure scales are different!

51 REMARK 1.When using driven loads,there is always a risk of taking numerical divergence for the ultimate load: use preferably driven displacements

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

53 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

54 LAST CONVERGED STEP DIVERGED STEP

55 DISPLACEMENT TIME-HISTORY

56 VALIDATION OF LOAD BEARING CAPACITY plane strain after CHEN 1975

57 MORE GENERAL CASES: Embedded footing with water table Remarks: 1.Can be solved as single phase 2.Watch for local “cut” instabilities

58 VALIDATION OF LOAD BEARING CAPACITY axisymmetry

59 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

60 PROOF - -

61 1.GLOBAL LEVEL 2. LOCAL (MATERIAL LEVEL)

62 INITIAL STATE CASE 1.Compute initial state 2.Add stories

63 ENV.INP DRIVERS SEQUENCE simulation of increasing number of stories

64 INITIAL STATE ANALYSIS env.inp Initial state stress level Ultimate load displacements

65 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

66 END LECTURE 1


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