fundamentals of soil mechanics

1. Shear Strength of Soils
Geol 4111– Fundamentals of soil and rock mechanics
Presented by: Elias A.
1

2. CONCEPT OF EFFECTIVE STRESS
PERMEABILITY
SEEPAGE AND FLOW NETS
SHEAR CHARACTERISTICS OF SOILS
COULOMB’S EQUATION FOR SHEAR STRENGTH
DETERMINATION OF SHEAR STRENGTHS OF SOILS
2

3. When you complete this unit, you should be able to:
 Determine the shear strength of soils.
 Understand the differences between drained and
undrained shear strength.
 Determine the type of shear test that best simulates
field conditions.
 Interpret laboratory test results to obtain shear
strength parameters.
Shear Strength of Soils
3

4. Strength of different
materials
Steel
Tensile
strength
Concrete
Compressive
strength
Soil
Shear
strength
Presence of pore water
Complex
behavior
4

5. 5

6. 6

7. 7

8. 8

9. Embankment
Strip footing
Shear failure of soils
Soils generally fail in shear
Failure surface
Mobilized shear
resistance
At failure, shear stress along the failure surface
(mobilized shear resistance) reaches the shear strength.
9

10. Retaining
wall
Shear failure of soils
Soils generally fail in shear
10

11. Retaining
wall
Shear failure of soils
At failure, shear stress along the failure surface
(mobilized shear resistance) reaches the shear strength.
Failure
surface
Mobilized
shear
resistance
Soils generally fail in shear
11

12. Shear failure
mechanism
failure surface
The soil grains slide
over each other along
the failure surface.
No crushing of
individual grains.
12

13. Shear failure mechanism
At failure, shear stress along the failure surface ()
reaches the shear strength (f).



13

14. Mohr-Coulomb Failure Criterion
(in terms of total stresses)

f is the maximum shear stress the soil can take without
failure, under normal stress of .

c
 f  c  tan

Cohesio
n
Friction
angle
f

14

15. Mohr-Coulomb Failure Criterion
(in terms of effective stresses)
f is the maximum shear stress the soil can take without
failure, under normal effective stress of ’.

’
c’
 f  c' ' tan
'
’
Effective
cohesion Effective
friction angle
f
’
 '
  u
u = pore water
pressure
15

16. Mohr-Coulomb Failure Criterion
’f
f
’
Shear strength consists of two
components: cohesive and frictional.

'
c’
’f tan ’
c’
 f  c''f tan'
frictional
component
16

17. c and  are measures of shear strength.
Higher the values, higher the shear strength.
17

18. Mohr Circle of stress
Soil element
’
1
’
1
’
3
’
3
’


2 2
2
1
1 3
' 3
Cos2

 '
 '
 '
'

 '
'
 1 3
Sin2
Resolving forces in  and  directions,
2
1
2
2
2 
3





 

'


'
'


'

2
'


1 3  

18

19. 1
2
2 
3





 

'

2

'
'


'

2
2
'


1 3  

Soil element
’


Soil element
Soil element
Mohr Circle of stress
’1
’
1
’
3
’
3
’




’
1 3
2
 '
2
 '
 '
1 3
'
3
 '
1
 '

19

20. 1
2
2 
3





 

'

2

'
'


'

2
2
'


1 3  

Soil element
’


Soil element
Soil element
Mohr Circle of stress
’1
’
1
’
3
’
3
’




’
1 3
2
 '
2
 '
 '
1 3
'
3
 '
1
 '

PD = Pole w.r.t. plane

’, 
20

21. Soil elements at different locations
Failure surface
Mohr Circles & Failure Envelope
X X
Y
Y
Y ~ stable
X ~ failure

’
 f  c' ' tan
'
21

22. Mohr Circles & Failure Envelope
Y c
c
Initially, Mohr circle is a point
c
 +

The soil element does not fail if
the Mohr circle is contained
within the envelope
GL

c
22

23. Mohr Circles & Failure Envelope
Y c
c
As loading progresses, Mohr
circle becomes larger…
.. and finally failure occurs
when Mohr circle touches the
envelope
GL

c
23

24. 
’
1 3
2
 '
 '
'
3
 1
 '
PD = Pole w.r.t. plane

f
’,  
Orientation of Failure Plane
’


’1
’1
’3
’3
’


Failure envelope
 –

’
Therefore,
 –    ’ =
  45 + ’/2
24

25. Mohr circles in terms of total & effective stresses
h’
X X
u
u
= +
v
h
 ’
effective stresses
u
v
 ’ h
X
v v’
h
total stresses

 or ’
25

26. Failure envelopes in terms of total & effective
stresses
h’
X X
u
u
= +
h’
effective stresses
u
v
v’ h
X
v v’
h

 or ’
If X is on
failure

total stresses
Failure envelope in
terms of total stresses
’
c’ c
Failure envelope in terms
of effective stresses
26

27. Mohr Coulomb failure criterion with Mohr circle
of stress
’v = ’1
’h = ’3
X
X is on failure ’1
’3
effective stresses


’
’ c’
Failure envelope in terms
of effective stresses
c’ Cot’ ’ ’

’  ’




   2
2
 '

 '

 '
 '
c'Cot' 1 3 Sin'   1 3 
Therefore,
27

28. Mohr Coulomb failure criterion with Mohr circle
of stress



  2
2
 '
  '

 '
 '
c'Cot' 1 3 Sin'   1 3 
'
'
1 3
' '
1 3  2c'Cos'

   Sin'
'
3
'
1   1Sin' 2c'Cos'
 1Sin'
1 3
1Sin' 1Sin'
1 Sin' 2c'
Cos'
 '
 '










2

2

45
2
'
3
'
1
'
'
  2c'tan 45
  tan
28

29. 29

30. Typical Values of Drained Angle of Friction for Sands
and Silts
30

31. Determination of shear strength parameters of
soils (c,  or c’ ’
Laboratory
specimens
tests on
taken from
representative undisturbed
samples
Field tests
Most common laboratory tests
to determine the shear strength
parameters are,
1.Direct shear test
2.Triaxial shear test
Other laboratory tests include,
Direct simple shear test, torsional
ring shear test, plane strain triaxial
test, laboratory vane shear test,
laboratory fall cone test
1. Vane shear test
2. Torvane
3. Pocket penetrometer
4. Fall cone
5. Pressuremeter
6. Static cone penetrometer
7. Standard penetration test
31

32. Laboratory tests
Field conditions
z
vc
hc
hc
Before construction
A representative
soil sample
vc
z
vc + 
hc
hc
After and during
construction
vc + 
32

33. Laboratory tests
Simulating field conditions
in the laboratory
Step 1
Set the specimen in
the apparatus
apply the
and
initial
stress condition
vc
vc
hc
hc
Representative
soil sample
taken from the
site
0
0
0
0
Step 2
Apply
corresponding
the
field
stress conditions
vc + 
hc
hc
vc + 
vc


vc
33

34. Direct shear test
Schematic diagram of the direct shear apparatus
34

35. Direct shear test
Preparation of a sand specimen
Components of the shear box Preparation of a sand specimen
Porous
plates
Direct shear test is most suitable for consolidated drained tests
specially on granular soils (e.g.: sand) or stiff clays
35

36. Direct shear test
Leveling the top surface
of specimen
Preparation of a sand specimen
Specimen preparation
completed
Pressure plate
36

37. Direct shear test
Test procedure
Porous
plates
Pressure plate
Steel ball
Step 1: Apply a vertical load to the specimen and wait for consolidation
P
Proving ring
to measure
shear force
S
37

38. Direct shear test
Step 1: Apply a vertical load to the specimen and wait for consolidation
Step 2: Lower box is subjected to a horizontal displacement at a constant rate
P
Test procedure
Pressure plate
Steel ball
Proving ring
to measure
shear force
S
Porous
plates
38

39. Direct shear test
Shear box
Loading frame to
apply vertical load
Dial gauge to
measure vertical
displacement
Dial gauge to
measure horizontal
displacement
Proving ring
to measure
shear force
39

40. Direct shear test
Analysis of test results
Area of cross section of the sample
Normalforce(P)
  Normal stress 
  Shear stress 
Shear resistance developed at the sliding surface (S)
Area of cross section of the sample
Note: Cross-sectional area of the sample changes with the horizontal
displacement
40

41. r
aehS
,
ssert
s

Shear displacement
Dense sand/
OC clay
f
Loose sand/
NC clay
f
Change
in
height
of
the
sample
Expansion
Compression
Dense sand/OC Clay
Shear displacement
Loose sand/NC Clay
Direct shear tests on sands
Stress-strain relationship
41

42. f1
Normal stress = 1
Direct shear tests on sands
r
aehS
,
ssert
s

Shear displacement
f2
f3
How to determine strength parameters c and 
Normal stress = 3
Normal stress = 2
ehS
uli
af

f
Normal stress,
Mohr – Coulomb failure envelope

42

43. Direct shear tests on sands
Some important facts on strength parameters c and  of sand
Sand is cohesionless
hence c = 0
Direct shear tests are
drained and pore
water pressures are
dissipated, hence u =
0
Therefore,
 ’ =  and c’ = c = 0
43

44. Direct shear tests on clays
Failure envelopes for clay from drained direct shear tests
ert
s
r
aehS
,
er
uli
af

f
Normal force, 
Normally consolidated clay (c’ = 0)
’
In case of clay, horizontal displacement should be applied at a very
slow rate to allow dissipation of pore water pressure (therefore, one
test would take several days to finish)
Overconsolidated clay (c’ ≠ 0)
44

45. Interface tests on direct shear apparatus
In many foundation design problems and retaining wall problems, it
is required to determine the angle of internal friction between soil
and the structural material (concrete, steel or wood)
a
f
  c  'tan
Where,
a
c = adhesion,
 = angle of internalfriction
Soil
Foundation material
Foundation material
Soil
P
S
45

46. 46

47. Triaxial Shear Test
Soil sample
at failure
Failure plane
Porous
stone
impervious
membrane
Piston (to apply deviatoric stress)
O-ring
pedestal
Perspex
cell
Cell pressure
Back pressure
Pore pressure or
volume change
Water
Soil
sample
47

48. Triaxial Shear Test
Specimen preparation (undisturbed sample)
Sampling tubes
Sample extruder
48

49. Triaxial Shear Test
Specimen preparation (undisturbed sample)
Edges of the sample
are carefully trimmed
Setting up the sample
in the triaxial cell
49

50. Triaxial Shear Test
Sample
with
is covered
a rubber
membrane and sealed
Cell is completely
filled with water
Specimen preparation (undisturbed sample)
50

51. Triaxial Shear Test
Specimen preparation (undisturbed sample)
Proving ring to
measure the
deviator load
Dial gauge to
measure vertical
displacement
51

52. Types of Triaxial Tests
Is the drainage valve open?
yes no
Consolidated
sample
Unconsolidated
sample
Is the drainage valve open?
yes no
Drained
loading
Undrained
loading
Under all-around cell pressure c
c
c
c
c
Step 1
deviatoric stress
(  = q)
Shearing (loading)
Step 2
c c
c+ q
52

53. Types of Triaxial Tests
Is the drainage valve open?
yes no
Consolidated
sample
Unconsolidated
sample
Under all-around cell pressure c
Step 1
Is the drainage valve open?
yes no
Drained
loading
Undrained
loading
Shearing (loading)
Step 2
CD test
CU test
UU test
53

54. Consolidated- drained test (CD Test)
Step 1: At the end of consolidation
VC
hC
Total,  = Neutral, u Effective, ’
+
0
Step 2: During axial stress increase
’VC =VC
’hC =hC
VC + 
hC 0
’V = VC +
 =  ’1
’h = hC = ’3
Drainage
Drainage
Step 3: At failure
VC + f
hC 0
’Vf = VC + f = ’1f
’hf = hC = ’3f
Drainage 54

55. Deviator stress (q or d) = 1 –3
Consolidated- drained test (CD Test)
1 = VC +
3 = hC
55

56. Volume
change
of
the
sample
Expansion
Compression
Time
Volume change of sample during consolidation
Consolidated- drained test (CD Test)
56

57. d
Volume
change
of
the
sample
Expans
r
io
ot
n
ai
ve
D
,
ssert
s


Compression
Axial strain
Dense sand
or OC clay
Axial strain
Loose sand
or NC clay
Stress-strain relationship during shearing
Dense sand
or OC clay
d)f
Consolidated- drained test (CD Test)
Loose sand
or NC Clay
d)f
57

58. CD tests
r
ot
ai
ve
D
,
ssert
s


d
Axial strain
r
aehS
,
ssert
s

 or

’

Mohr – Coulomb
failure envelope
d)f
a
Confining stress = 3a
d)f
b
How to determine strength parameters c and 
d)f
c
Confining stress = 3c
Confining stress = 3b
3c 1c
3a 1a
(d)f
 3b 1b
( )
d fb
1 3
 =  +
(d)f
3
58

59. CD tests
Strength parameters c and  obtained from CD tests
Since u = 0 in CD
tests,  =  ’
Therefore, c = c’
and  =’
cd and d are used
to denote them
59

60. CD tests Failure envelopes
r
aehS
,
ssert
s

 or
’
d
Mohr – Coulomb
failure envelope
3a 1a
(d)f
a
For sand and NC Clay, cd = 0
Therefore, one CD test would be sufficient to determine d
of sand or NC clay
60

61. CD tests Failure envelopes
For OC Clay, cd ≠ 0

 or

’

3 1
(d)f
c
c
OC NC
61

62. Some practical applications of CD analysis for
clays
  = in situ drained
shear strength
Soft clay
1. Embankment constructed very slowly, in layers over a soft clay
deposit
62

63. Some practical applications of CD analysis for
clays
2. Earth dam with steady state seepage

Core
 = drained shear
strength of clay core
63

64. Some practical applications of CD analysis for
clays
3. Excavation or natural slope in clay
 = In situ drained shear strength

Note: CD test simulates the long term condition in the field.
Thus, cd and d should be used to evaluate the long
term behavior of soils
64

65. Consolidated- Undrained test (CU Test)
Step 1: At the end of consolidation
VC
hC
Total,  Neutral, u Effective, ’
= +
0
Step 2: During axial stress increase
’VC = VC
’hC =hC
VC + 
hC ±
u
Drainage
Step 3: At failure
VC + f
hC
No
drainage
No
drainage ±uf
±
’V = VC +  u
= ’1
h hC ±
’ =  u
= ’3
’Vf = VC + f ± uf
= ’1f
hf hC f
= ’3f
’ =  ±
u
65

66. Volume
change
of
the
sample
Expansion
Compression
Time
Volume change of sample during consolidation
Consolidated- Undrained test (CU Test)
66

67. ,
ssert
s


d
u
+
r
ot
ai
ve
D
-
Axial strain
Loose
sand /NC
Clay
Axial strain
Dense sand
or OC clay
Stress-strain relationship during shearing
Dense sand
or OC clay
d)f
Consolidated- Undrained test (CU Test)
Loose sand
or NC Clay
d)f
67

68. CU tests How to determine strength parameters c and 
r
ot
ai
ve
D
,
ssert
s


d
Axial strain
r
aehS
,
ssert
s

 or
’
d)f
b
3b 1b
3a 1a
( )
d fa
cu
Mohr – Coulomb
failure envelope in
terms of total stresses
ccu
1 = 3 +
(d)f
3
Total stresses at failure
d)f
a
Confining stress = 3b
Confining stress = 3a
68

69. (d
)f
a
CU tests
r
aehS
,
ssert
s

 or
’
3b 1b
(d)fa
cu
Mohr – Coulomb
failure envelope in
terms of total stresses
ccu
’1b
’3b
’3a
3a ’1a
1a
Mohr – Coulomb failure
envelope in terms of
effective stresses
’
C’ ufa
ufb
How to determine strength parameters c and 
’1 = 3 + (d)f -
uf
 ’ = 3 - uf
Effective stresses at failure
uf
69

70. CU tests
Strength parameters c and  obtained from CD tests
Shear strength
parameters in terms
of total stresses are
ccu and cu
Shear strength
parameters in terms
of effective stresses
are c’ and ’
c’ = cd and ’ =
d
70

71. CU tests Failure envelopes
For sand and NC Clay, ccu and c’ = 0
U test
r
aehS
,
ssert
s

 or

’
cu
Mohr – Coulomb
failure envelope in
terms of total stresses
3a
(d)f
Therefore, one Ca
would be sufficient to determine
cu and ’= d) of sand or NC clay
3a  
1a 1a
’
Mohr – Coulomb failure
envelope in terms of
effective stresses
71

72. Some practical applications of CU analysis for
clays
  = in situ
undrained shear
strength
Soft clay
1. Embankment constructed rapidly over a soft clay deposit
72

73. Some practical applications of CU analysis for
clays
2. Rapid drawdown behind an earth dam

Core
 = Undrained shear
strength of clay core
73

74. Some practical applications of CU analysis for
clays
3. Rapid construction of an embankment on a natural slope
Note: Total stress parameters from CU test (ccu and cu) can be used for
stability problems where,
Soil have become fully consolidated and are at equilibrium with
the existing stress state; Then for some reason additional
stresses are applied quickly with no drainage occurring

 = In situ undrained shear strength
74

75. Unconsolidated- Undrained test (UU Test)
Data analysis
C = 3
C
 = 3
No
drainage
Initial specimen condition
3
 +  d
3
No
drainage
Specimen condition
during shearing
Initial volume of the sample = A0 × H0
Volume of the sample during shearing = A × H
Since the test is conducted under undrained condition,
A × H = A0 × H0
A ×(H0 – H) = A0 × H0
A ×(1 – H/H0) =A0
A
1z
A  0
75

76. Unconsolidated- Undrained test (UU Test)
Step 1: Immediately after sampling
0
0
= +
Step 2: After application of hydrostatic cell pressure
uc = B3
 = 
C 3
C = 3 uc
’3 = 3 - uc
’3 = 3 - uc
No
drainage
Increase of pwp due to
increase of cell pressure
Increase of cell pressure
Skempton’s pore water
pressure parameter, B
Note: If soil is fully saturated, then B = 1 (hence, uc =  3)
76

77. Unconsolidated- Undrained test (UU Test)
Step 3: During application of axial load
3 + d
3
No
drainage
c
’1 = 3 + d - u ud
 ’3 =  3 - uc ud
ud =ABd
uc ± ud
= +
Increase of pwp due to
increase of deviator stress
of deviator
Increase
stress
Skempton’s pore water
pressure parameter, A
77

78. Unconsolidated- Undrained test (UU Test)
u = uc + ud
Combining steps 2 and 3,
uc =B 3 ud =ABd
Total pore water pressure increment at any stage, u
u = B [3 + Ad]
Skempton’s pore
water pressure
equation
u = B [3 + A(1 – 3]
78

79. Unconsolidated- Undrained test (UU Test)
Step 1: Immediately after sampling
0
0
Total,  Neutral, u Effective, ’
= +
-ur
Step 2: After application of hydrostatic cell pressure
 ’V0 =ur
’h0 =ur
C
C
-ur   uc =
-ur   c
(Sr = 100% ; B= 1)
Step 3: During application of axial load
C + 
C
No
drainage
No
drainage
r c
-u   ±
u
’VC = C + ur - C = ur
 ’h =ur
Step 3: At failure
V C r
’ =  +  + u -  c
u
r
’h = C + u -  u
c
= ’3f
c
’Vf = C + f + ur -  uf = ’1f
’hf = C + ur - c uf
r c
-u   ± uf
C
C + f
No
drainage
79

80. Unconsolidated- Undrained test (UU Test)
Total,  Neutral, u Effective, ’
= +
Step 3: At failure
= ’3f
c
’Vf = C + f + ur -  uf = ’1f
’hf = C + ur - c uf
-ur  c ± u f
C
C + f
No
drainage
Mohr circle in terms of effective stresses do not depend on the cell
pressure.
Therefore, we get only one Mohr circle in terms of effective stress for
different cell pressures

’
’3 ’1
 80

81. 
3a
3
b

1a
1
b
f
’3 ’1
Unconsolidated- Undrained test (UU Test)
Total,  Neutral, u Effective, ’
= +
Step 3: At failure
= ’3f
c
’Vf = C + f + ur -  uf = ’1f
’hf = C + ur - c uf
r c
-u   ± uf
C
C + f
No
drainage
 or
’
Mohr circles in terms of total stresses

ua
ub
Failure envelope, u =0
cu
81

82. Unconsolidated- Undrained test (UU Test)
Effect of degree of saturation on failure envelope
3a b a
3c 3b c

 or
’
S < 100% S > 100%
82

83. Some practical applications of UU analysis for
clays
  = in situ
undrained shear
strength
Soft clay
1. Embankment constructed rapidly over a soft clay deposit
83

84. Some practical applications of UU analysis for
clays
2. Large earth dam constructed rapidly with
no change in water content of soft clay

Core
 = Undrained shear
strength of clay core
84

85. Some practical applications of UU analysis for
clays
3. Footing placed rapidly on clay deposit
 = In situ undrained shear strength
Note: UU test simulates the short term condition in the field.
Thus, cu can be used to analyze the short term
behavior of soils 85

86. Unconfined Compression Test (UC Test)
1 = VC +
3 = 0
Confining pressure is zero in the UC test
86

87. Unconfined Compression Test (UC Test)
1 = VC +
f
Shear
stress,

Normal stress, 
3 = 0
qu
τf = σ1/2 = qu/2 = cu
87

88. Various correlations for shear strength
For NC clays, the undrained shear strength (cu) increases with the
effective overburden pressure, ’0
cu
 0.11 0.0037(PI)
 '
Skempton (1957)
0
Plasticity Index as a %
For OC clays, the following relationship is approximately true
0.8
'
'
 (OCR)
 

0 Overconsolidated 0 Normally Consolidated
 

 cu   cu 


Ladd (1977)
For NC clays, the effective friction angle (’) is related to PI as follows
Sin' 0.814  0.234log(IP) Kenny (1959)
88

89. Shear strength of partially saturated soils
In the previous sections, we were discussing the shear strength of
saturated soils. However, in most of the cases, we will encounter
unsaturated soils
Solid
Water
Saturated soils
Pore water
pressure, u
Effective
stress, ’
Solid
Unsaturated soils
Pore water
pressure, uw
Effective
stress, ’
Water
Air Pore air
pressure, ua
Pore water pressure can be negative in unsaturated soils
89

90. Shear strength of partially saturated soils
Bishop (1959) proposed shear strength equation for unsaturated soils as
follows
 f  c'(n ua )  (ua uw )tan'
Where,
n – ua = Net normal stress
ua – uw = Matric suction
= a parameter depending on the degree of saturation
( = 1 for fully saturated soils and 0 for drysoils)
Fredlund et al (1978) modified the above relationship as follows
 f  c'(n  ua ) tan'(ua uw ) tan b
Where,
tanb = Rate of increase of shear strength with matric suction
90

91. Shear strength of partially saturated soils
 f  c'(n ua ) tan'(ua uw ) tan b
Same as saturated soils Apparent cohesion
due to matric suction
Therefore, strength of unsaturated soils is much higher than the strength
of saturated soils due to matric suction

’
 - 91

92. 
 -
ua
possible
How it become
build a sand castle
uw ) tan b
 f  c'(n ua ) tan'(ua
Same as saturated soils Apparent cohesion
due to matric suction
’
Apparent
cohesion
92

93. 93