Instruct Nirmaana 24-Smart and Lean Construction Through Technology.pdf
Geotechnical Engineering.ppt
1. MB March 10, 2002
University of Arizona
Department of Civil Engineering & Engineering Mechanics
CE544 – Special Topics in Geotechnical Engineering
FIELD INSTRUMENTATION AND MONITORING
Instructor: Muniram Budhu
Date: Apeil 14, 2003
2. MB March 10, 2002
Model
Two requirements for a good model
1: It must accurately describe a large class of
observations with few arbitrarily elements.
2: It must make definite predictions about the
results of future observations.
(extracted from Stephen Hawking ‘Á brief history of time,’ Bantam Books)
3. MB March 10, 2002
Types of Models (CE544)
ELASTIC
LINEAR
NON-LINEAR
PLASTICITY
NON-LINEAR
CAM-CLAY FAMILY
Elasto-plastic
4. MB March 10, 2002
NON-LINEARITY
GEOMETRIC – change of shape, size,
etc.
MATERIAL – change of properties
CAUSES: stress state. History of loading,
change in stiffness, physical conditions, in
situ stress, water content, voids ratio
5. MB March 10, 2002
ELASTIC MODELS
LINEAR – magnitude of response
proportional to excitation
Non-LINEAR – magnitude of response not
proportional to excitation
stress
strain
Linear
Non-linear
6. MB March 10, 2002
CONSTITUTIVE LAWS
SET OF EQUATIONS THAT RELATE
STRESSES TO STRAINS.
F(stress, stress rate, strain, strain rate) = 0
Homogeneity of time.
7. MB March 10, 2002
CONSTITUTIVE EQUATIONS
ij ijkl kl
C
Stiffness matrix
ij ijkl kl
D
Compliance matrix
8. MB March 10, 2002
ELASTIC MODELS
Elastic materials: State of stress is a
function of the current state of
deformation; no history effects
Cauchy – stress is a function of strain
(infinitesimal strain, first order)
Green – based on strain energy function
(Hyper-elastic)
9. MB March 10, 2002
Hooke’s law – simple case
Simple one dimensional case:
E = Young’s modulus (Elastic modulus)
E
11. MB March 10, 2002
Shear stresses and strains
zx
zx zx
2 1
E G
E
G
2 1
G is the shear modulus. Only G or E and u are
required to solve linear elastic problems
12. MB March 10, 2002
Typical values of E and G
Soil Type Description E*
(MPa) G (MPa)
Clay Soft
Medium
Stiff
1 to 15
15 to 30
30 to 100
0.4 to 5
5 to 11
11 to 38
Sand Loose
Medium
Dense
10 to 20
20 to 40
40 to 80
4 to 8
8 to 16
16 to 32
*These are average secant elastic moduli for drained condition
16. MB March 10, 2002
Hooke’s law using stress
invariants
p
K
1
e
p
p is mean stress, K is bulk modulus; the prime denotes effective
)
2
1
(
3
E
p
K e
p
q
G
3
1
e
q
v
1
2
E
G
G
SPECIAL CASE : 1/2; K 0
17. MB March 10, 2002
Constitutive elastic model – stress
invariants
e
q
e
p
G
3
0
0
K
q
p
Decoupling - Mean effective stress causes
volumetric strain; deviatoric stress (shear
stress) causes deviatoric strain
18. MB March 10, 2002
Lame’s constant
G(E 3G)
K 2G /3
3G E
3KE
G
9K E
19. MB March 10, 2002
Poisson’s ratio
2
1
'
K
3
E
also
1
G
2
E
Then
1
1
G
2
2
1
K
3
K
6
G
2
G
2
K
3
20. MB March 10, 2002
Green’s elastic model
The work done by external forces in altering
the configuration of a body from its natural
state is equal to the sum of the kinetic
energy and the strain energy
i i ij i i
v s
ij i i ij i ,j
s v
i ,j ij ij
w Fu dv n u ds
using Gauss's divergence theorem
n u ds ( u) dv
( u) w
Strain tensor - symmetrical
Strain tensor – skew symmetrical
21. MB March 10, 2002
ANISOTROPIC ELASTICITY
Anisotropic materials have different elastic
parameters in different directions.
Structural anisotropy or transverse anisotropy –
manner in which soil is deposited.
Stress induced anisotropy – differences in normal
stresses in different directions.
22. MB March 10, 2002
Transverse anisotropy
- most prevalent in soils
rz
z r
z z
rr
r r
zr
z r
2
1
E E
1
E E
23. MB March 10, 2002
ELASTICITY AND PLASTICITY
Theory of elasticity: uniqueness – behavior
of the material expressed by a set of
equations
Theory of plasticity: discontinuity in stress-
strain relationship (involves discontinuities
and inequalities); deals with initial stress
problems, state of structure at collapse, at
post-yield.
24. MB March 10, 2002
THEORY OF PLASTICITY
TO ADEQUATELY DESCRIBE THE
PLASTIC DEFORMATION OF SOILS
TO USE RELATIONSHIPS DEVELOPED
TO PREDICT FAILURE LOADS AND
SETTLEMENT.
26. MB March 10, 2002
FULL PLASTIC STATE
(COLLAPSE)
Guess a plastic collapse mechanism
For small deformation of this mechanism,
integrate the work consumed in plastic
deformation over the whole body
Equate this to the work supplied to find the
collapse load
(ref: Calladine, C. R. “Engineering plasticity”, Pergamon Press, London)
27. MB March 10, 2002
PLASTICITY THEOREMS
LOWER BOUND –IF ANY STRESS DISTRIBUTION
THROUGOUT THE STRUCTURE CAN BE FOUND WHICH IS
EVERYWHERE IN EQUILIBRIUM INTERNALLY AND BALANCES
CERTAIN EXTERNAL LOADS AND AT THE SAME TIME DOES NOT
VIOLATE THE YIELD CONDITION, THESE LOADS WILL BE CARRIED
SAFELY BY THE STRUCTURE.
UPPER BOUND –IF AN ESTIMATE OF THE PLASTIC
COLLAPSE LOAD OF A BODY IS MADE BY EQUATING INTERNAL RATE
OF DISSIPATION OF ENERGY TO THE RATE AT WHICH EXTERNAL
FORCES DO WORK IN ANY POSTULATED MECHANISM OF
DEFORMATION OF THE BODY, THE ESTIMATE WILL BE EITHER HIGH,
OR CORRECT.