Intro to Soil Mechanics

Transcription

Intro to Soil Mechanics:
the what, why & how
Thursday, June 23, 2011
José E. Andrade, Caltech
The What?
What is Soil
Mechanics?
erdbaumechanik
The application of the laws of
mechanics (physics) to soils as
engineering materials
Karl von Terzaghi is credited as
the father of erdbaumechanik
sands & gravels
clays & silts
The Why?
Sandcastles
what holds them up?
Palacio de Bellas Artes
Mexico, DF
uniform settlement
The leaning tower of
Pisa
differential settlement
!
Teton dam
dam failure
Niigata earthquake
liquefaction
Katrina
New Orleans
levee failure
MER: Big Opportunity
xTerramechanics
MER: Big Opportunity
xTerramechanics
The How?
Topics in classic Soil Mechanics
• Index & gradation
• Soil classification
• Compaction
• Permeability, seepage, and effective stresses
• Consolidation and rate of consolidation
• Strength of soils: sands and clays
Index & gradation
Definition: soil mass is a collection of particles and
voids in between (voids can be filled w/ fluids or air)
solid particle
fluid (water)
Each phase
has volume
and mass
gas (air)
Mechanical behavior governed by phase interaction
Index & gradation
Key volumetric ratios
Vv
e=
Vs
Vv
η=
Vt
void ratio
[0.4,1] sand
[0.3,1.5] clays
porosity
[0,1]
solid
water+air=voids
Key mass ratio
water
content
Mw
w=
<1 for most soils
Ms
>5 for marine, organic
Key link mass & volume
ρ = M/V
moist, solid, water, dry, etc.
Vw
S=
Vv
saturation
[0,1]
ratios used in practice to
characterize soils & properties
Gradation & classification
Grain size is main classification feature
sands &
gravels
• can see grains
• mechanics~texture
• d>0.05 mm
clays &
silts
• cannot see grains
• mechanics~water
• d<0.05 mm
Soils are currently classified using USCS (Casagrande)
Fabric in coarsely-grained soils
“loose packing”, high e
“dense packing”, low e
relative
emax greatest possible, loosest packing
emin lowest possible, densest packing
ID =
emax −e
emax −emin
relative density
strongly affects engineering
behavior of soils
Vv
e=
Vs
Typical problem(s)xb
in Soil Mechanics
• Compact sand fill
• Calculate consolidation of clay
• Calculate rate of consolidation
• Determine strength of sand
• Calculate F.S. on sand (failure?)
• Need: stresses & matl behavior
SAND FILL
PISA CLAY
ROCK (UNDERFORMABLE, IMPERMEABLE)
Modeling tools
Theoretical
framework
• continuum mechanics
• constitutive theory
!s
"
"#
• computational inelasticity
x
X
!f
• nonlinear finite elements
f
"#
x2
x1
Theoretical
framework
balance of mass
ṗ
+∇·v
φ
Kf
∇·σ+γ
= −∇ · q
= 0
balance of momentum
Theoretical
framework
q
σ̇
!
k
darcy
hooke
permeability tensor
controls fluid flow
cep
= k · ∇h
ep
= c : "˙
mechanical stiffness
controls deformation
Theoretical
framework
!$
tr
!n+1
!n+1
Fn+1
Fn
!n
!"
!#
Theoretical
framework
Displacement node
Pressure node
Finite Element Method (FEM)
• Designed to approximately solve PDE’s
• PDE’s model physical phenomena
• Three types of PDE’s:
• Parabolic: fluid flow
%!"!#$
)
&
'(1
'(0
'(0
'(/
'(/
'(.
'(.
'(-
'(-
'(,
'(,
'(+
'(+
'(*
'(*
'()
'&
'
'()
'()
'(*
'(+
'(,
'(-
!!"!#$
• Hyperbolic: wave eqn
• Elliptic: elastostatics
)
'(1
'(.
'(/
'(0
'(1
)
'
FEM recipe
Strong from
Weak form
Galerkin form
Matrix form
Multi-D deformation with FEM
∇ · σ + f = 0 in Ω
u = g on Γg
σ · n = h on Γh
equilibrium
e.g., clamp
e.g., confinement
Γg
Constitutive relation:
Ω
given u
Γh
get σ
e.g., elasticity, plasticity
Modeling Ingredients
1. Set geometry
2.Discretize domaiin
3. Set matl parameters
4. Set B.C.’s
5. Solve
H4.417
B
1. Set geometry
2. Discretize domain
4. Set B.C.’s
5. Solve
1. Set geometry
4. Set B.C.’s
5. Solve
1. Set geometry
!r
4. Set B.C.’s
5. Solve
!a
1. Set geometry
2. Discretize domain !r
4. Set B.C.’s
5. Solve
!a
FEM Program
TIME STEP LOOP
ITERATION LOOP
ASSEMBLE FORCE VECTOR
AND STIFFNESS MATRIX
ELEMENT LOOP: N=1, NUMEL
GAUSS INTEGRATION LOOP: L=1, NINT
CALL MATERIAL SUBROUTINE
CONTINUE
CONTINUE
CONTINUE
T = T + !T
constitutive
model
Material behavior: shear strength
• Void ratio or relative density
• Particle shape & size
• Grain size distribution
Engineers have developed
models to account for most
of these variables
• Particle surface roughness
• Water
• Intermediate principal stress
• Overconsolidation or pre-stress
Elasto-plasticity
framework of choice
A word on current characterization methods
Direct Shear
Pros: cheap, simple, fast,
good for sands
Cons: drained, forced failure,
non-homogeneous
Triaxial
Pros: control drainage & stress
path, principal dir. cnst.,
more homogeneous
Cons: complex
Material models for
sands should capture
• Nonlinearity and irrecoverable
deformations
v
• Pressure dependence
A
• Difference tensile and
compressive strength
• Relative density dependence
!v
p
C
B
• Nonassociative plastic flow
log -p’
Material models for
deformations
q (kPa)
200
!’vc
191 kPa
241
310
380
100
0
100
200
-p’ (kPa)
300
Material models for
deformations
!’$
Von Mises
Mohr Coulomb
!’"
!’#
Loose sand
Dense sand
deformations
"!#$!
a
r" (kPa)
Material models for
1000
800
600
!a
400
!r
200
0
0
2
4
6
8
10
%a (%)
4
3
2
%v (%)
1
0
-1
Dense Sand
Loose Sand
Material models for
deformations
q
Yield Function
Plastic Potential
Flow vector
-p’
Elasto-plasticity in one slide
Hooke’s law σ̇ = c : �˙
e
p
Additive decomposition of strain �˙ = �˙ + �˙
Convex elastic region F (σ, α) = 0
p
g := ∂G/∂σ
Non-associative flow �˙ = λ̇g,
ep
K-T optimality
c
ep
λ̇F = 0
λ̇H = −∂F/∂α · α̇
Elastoplastic constitutive tangent
1 e
e
e
e
=c − c :g⊗f :c ,
χ=H −g :c :f
χ
Examples
Example of elasto-plastic model
!3
"=1
q
"=7/9
N=0
N=0.5
$
#i
v
M
!2
!1
CSL
CSL
(a)
(b)
F = F (σ , πi )
v1
#i
�
Figure 3: Three
~ invariant yield surface on (a) deviatoric plane at diﬀerent values of � and (b)
" at diﬀerent values of N .
meridian plane
vc
v2
p'
with
#
G = G(σ , π̄i )


 M [1 + ln (πi /p� )]
η=
�
�

N/(1−N
)
�
 ln-p’
M/N 1 − (1 − N ) (p /πi )
�
if N = 0
H = H(p , πi , ψ)
�
if N > 0.
(2.4)
-p’
-!i
The function ζ controls the cross-sectional shape of the yield function on a deviatoric plane
model validation:
txccompression
and ps
Figure 2.4:drained
Triaxial
calibrations, (a) and (c), and plane strain compression
predictions, (b) and (d), versus experimental data for loose and dense Brasted Sand
specimens.
undrained txc loose sands
true triaxial b=constant
H − H̄L
Plane-strain liquefaction numerical simulation
H − H̄L
Plane-strain liquefaction numerical simulation
250
200
150
100
50
(a) Pore Pressure (in kPa)
.0025
.0020
.0015
.0010
.0005
.0000
(b) Deviatoric Strain
ELASTIC
600000
400000
200000
H − HL
0
Field scale prediction
Levee failure
(recall Katrina)
References

Intro to Soil Mechanics

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