Download presentation

Presentation is loading. Please wait.

Published byLaney Lassiter Modified over 2 years ago

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 CONVERGENCE**

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

53
**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

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

Similar presentations

OK

Summer School for Integrated Computational Materials Education 2015 Computational Mechanics: Basic Concepts and Finite Element Method Katsuyo Thornton.

Summer School for Integrated Computational Materials Education 2015 Computational Mechanics: Basic Concepts and Finite Element Method Katsuyo Thornton.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on happy new year 2013 Ppt on australian continental drift Ieee papers ppt on sustainable buildings How to run ppt on ipad 2 Ppt on carbon and its compounds class Ppt on origin of life Ppt on bill gates as a leader Ppt on condition monitoring maintenance Ppt on mapping space around us Download ppt on facebook advantages and disadvantages