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
      
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

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
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)

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

11. CONSTITUTIVE MODEL: ELASTIC- PLASTIC
With hardening(or softening) dimensional
hardening E  y
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. 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

15. TOLERANCES ITERATIVE ALGORITHMS

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

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

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

20. a tutorial is available

21. GEOMETRY & BOUNDARY CONDITIONS
start by defining the geometry

22. Geometry with box-shaped
boundary conditions

23. MATERIAL & WEIGHT: MOHR-COULOMB

24. GRAVITY LOAD

25. ALGORITHM: STABILITY DRIVER
Single phase

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

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

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

29. RUN

30. VISUALIZATION OF INSTABILITY
Displacement intensities

31. LAST CONVERGED vs DIVERGED STEP

32. LOCALISATION 1
Transition from distributed
to localized strain

33. LOCALISATION 2

34. VALIDATION
Slope stability
1984

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

36. ULTIMATE LOAD ANALYSIS(foota.inp)
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 Po(x) F(x,t)
F=Po(x)*LF(t) LF t
foota.inp

39. 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)

40. LOAD FUNCTIONS

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

42. D-P material

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

44. 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

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
M-C(M-W)
associated with D-P
in deviatoric plane
associated with D-P
in deviatoric plane
dilatant flow in meridional plane

47. run footwt.inp

48. SEE LOGFILE

49. LOG FILE

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

51. REMARK
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 CONVERGENCEF
F >>d = d
NON CONVERGENCE F
F >cst.>TOL. d

53. COMPUTATION OF ULTIMATE LOADLF
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

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

62. INITIAL STATE CASE
Compute initial state
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