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Effect of Stress Dependent Angle of Shearing Resistance in theDesign of Geotechnical Structures
Citation preview
Effect of Stress Dependent Angle of Shearing Resistance in the
Design of Geotechnical Structures
GeoPractices- 2015
Indian Institute of TechnologyHyderabad
Presented by
B. Umashankar, Asst. Prof., IIT Hyderabad
Presentation Overview 2
� Introduction
� Sand Dilatancy
� Problem Definition
� Approach
GeoPractices-2015
� Approach
� Results
� Conclusions
3
�Earth retention has gained major importance with rise in infrastructure
development
�Geotechnical Problems - Slope stability analysis and design of high retaining
walls
Introduction
GeoPractices-2015
Images Source: Retainingwallexpert.com and Feat
Codal Provisions
� In the BS 8006- 1995 code of practice use is made of ϕp (Peak friction angle) for walls,
abutments and steep slopes constructed with frictional fill, and ϕcv (Critical state friction
angle) for fill to shallow slopes and embankments founded on weak foundations.
� According to BS 8002-1984, in the working state, the design values of lateral earth pressure
are intended to give an overestimate of the earth pressure on the active or retained side and
an underestimate of the earth resistance on the passive side for small deformations of the
structure as a whole.
4
GeoPractices-2015
structure as a whole.
� Earth pressures reduce as fully active conditions are mobilized at peak soil strength in the
retained soil, under deformations larger than can be tolerated for serviceability. As collapse
impends, the retained soil approaches a critical state, in which its strength reduces to that of
loose material and the earth pressures consequently tend to increase once more to active
values based on critical state strength.
� In FHWA (2001), ϕpeak was used in calculating the F* (Pull out resistance factor).
Laboratory Testing to Determine φ 5
Direct shear tests done for the normal
stress in the range 50-200 kPa
GeoPractices-2015
τ
σ
Is φ a constant ?
ϕp depends on
� Relative density
� Confining stress
Inter-particle friction angle
6
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� Inter-particle friction angle
� Loading path
� Soil fabric, cementation, particle angularity,
gradation, etc.
Effect of Confining Stress on φp7
ϕp = ϕc+ ψ
GeoPractices-2015
ϕp = ϕc+ ψϕp – peak friction angle
ϕc – critical state friction angle
ψ – dilatancy angle
Sand Dilatancy
� Dilatancy plays a crucial role in determining the
mechanical behavior of sand due to its discrete particulate
nature.
� Newland and Alley (1957) followed by Rowe (1962)
derived stress dilatancy relationships.
8
GeoPractices-2015
derived stress dilatancy relationships.
� Marsal (1967), Leps (1970), and Marachi et al. (1972)
found that granular materials exhibit a peak angle of
shearing resistance as a function of effective stress.
� Bolton (1986) studied the concepts of friction and
dilatancy angles in relation to strength of the soil.
Influence of Sand Dilatancy
� Houlsby (1991) studied the relationships between the friction angle, dilation angle,
density and pressure in a granular material.
� Manzari and Nour (2000) analyzed the stability of slopes by considering the influence
of soil dilatancy using finite-element approach and observed that soil dilatancy may
have a significant effect on the stability of slopes.
9
GeoPractices-2015
have a significant effect on the stability of slopes.
� Chakraborty and Salgado (2010) performed triaxial and plane-strain compression tests
on Toyoura sand to establish a correlation between peak friction angle, critical state
friction angle and dilatancy for confining pressures varying from 4 kPa to 196 kPa.
� Umashankar and Madhav (2011) studied the influence of effective confining stress on
angle of shearing resistance in high retaining walls.
10Leps Curve
� Triaxial tests on gravels and
cobbles, at effective stresses in the
range 40– 3500 kPa.
� Upper limit for dense, well-graded
GeoPractices-2015
� Upper limit for dense, well-graded
particles.
� Lower limit for loose, poorly-
graded particles.
11Problem Definition
Considering the influence of mean confining stress on the mobilized angle of shearing
resistance along the slip plane
1. Determining the factor of safety for various embankment heights
2. Determining the earth pressures for a wall of height H with a level backfill
GeoPractices-2015
H
Fill Properties
Unit weight- γRelative density - DR
Critical state fricton angle - φc
2. Determining the earth pressures for a wall of height H with a level backfill
ϕ
ϕ1
ϕ2
ϕ3
ϕ4
ϕ1
ϕ2
ϕ3
Soil Properties 12
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17.4
Soil Properties 13
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Grain size distribution curveMorphological features of sand particles
using SEM.
Peak Friction Angle Vs. Normal Stress 14
� Direct shear tests - at effective
stresses in the range 5 – 400 kPa.
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� Linear relationship with
logarithmic X-axis .
Dilation behavior of sand 15
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Normal stress of 5 to 35 kPa Normal stress of 100 to 400 kPa
Coulomb’s Theory 16
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N: Normal thrust on the plane
S: Shearing force along slip plane
W: Weight of sliding wedge
Pminθc
Pmaxθc
N: Normal thrust on the plane
S: Shearing force along slip plane
W: Weight of sliding wedge
Active Earth Thrust Passive Earth Thrust
Approach 17
P
Vi
Hi
SH
⁄⁄⁄⁄
.
.
.
1
2
3
P
Vi
Hi
S
• Vi and Vi+1 = Vertical interslice forces,
• Hi and Hi+1 = Horizontal interslice forces,
• Wi = Weight of the slice,
GeoPractices-2015
Pi
Vi+1
Hi+1
Ni
Si
θWi
θ
H
⁄⁄⁄⁄
.
.
.
i
n
Pi
Vi+1
Hi+1
Ni
Si
θWi
Horizontal slice method: (a) backfill behind the wall height divided into n horizontal slices, and
(b) forces on a slice for active case (c) forces on a slice for passive case
(a) (b) (c)
• Ni = Normal force at the base of the slice,
• Si = Shear strength of the backfill,
• c = Cohesion of the backfill,
• bi = Base width of the slice along the slip plane,
• ϕi = Mobilized friction angle of the backfill material.
Active Case Passive Case
Approach 18
10 sin cos 0
y i i i iF V V S Nθ θ+= ⇒ − + + =∑
tanS cb N φ= +
0cossin0 1 =+−−⇒= +∑ θθ iiiiyNSVVF
tanS cb N φ= +
The Horizontal Slice Method proposed by Shahgoli et al. (2001) is modified
tani i i iS cb N φ= +
1 sin
tan sin cos
i i i ii
i
V V W cbN
θ
φ θ θ+− + −
=+
1 1
0 ( sin ) ( cos )n n
x i i
i i
F P N Sθ θ= =
= ⇒ = −∑ ∑ ∑
tani i i iS cb N φ= +
θθφ
θ
cossintan
sin1
−
−−−= +
i
iiii cbWVVNi
∑ ∑∑==
+=⇒=n
i
i
n
i
ix SNPF11
)cos()sin(0 θθ
Approach
�The stress state of the soil element along the slip plane for each slice is determined
assuming the horizontal and vertical planes to be the principal planes.
�For an assumed value of ϕi, normal stress acting on the slice was determined and
this computed normal stress will be compared with the normal stress obtained from
the developed relationship between the normal stress and peak friction angle.
19
GeoPractices-2015
the developed relationship between the normal stress and peak friction angle.
� If the normal stresses are not equal, ϕi corresponding to computed normal stress
will be used in the further iterations until the computed normal stress and
theoretical normal stress are equal.
Results 20
P*
=P
/γH
2
P*
=P
/γH
2
P* increases as wall height increases
GeoPractices-2015
Variation of P* with slip plane θ for H=5m-40m, γ=17.4 kN/m3 (a) Active case and (b) Passive case
(a) (b)
P*
=P
/
P*
=P
/
P* decreases as wall height increases
Results 21
P* increases as wall height increases
P*
=P
/γH
2
γH2
GeoPractices-2015
Variation of P* with slip plane θ for H=5m-40m, γ=17.4 kN/m3 (a) Active case and (b) Passive case
(a) (b)
P* decreases as wall height increases
P*
=P
/
P*
=P
/γ
Results 22
GeoPractices-2015
Variation of critical slip surface θc with the wall height
�θc increases from 23o to 28o as the wall
height increases from 5 m to 40 m
�Coulomb’s theory gives a value of 22o
(= 45 - φ/2, , φ = 46o)
�θc decreases from 72.5o to 66o as the
wall height increases from 5 m to 40 m
�Coulomb’s theory gives a value of
68o (= 45 + φ/2, φ = 46o)
Results 23
10
5
0
Dep
th (
m)
10
5
0
Variation of mobilized friction angle along
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45 50 55 60 65 70 75
20
15
10
Active case
Dep
th (
m)
Mobilized phi (o)
36 39 42 45 48 51
20
15
10
Passive case
Variation of mobilized friction angle along
critical slip plane with depth for
wall height H=20 m.
Results 24
Variation of mobilized friction angle along
GeoPractices-2015
Variation of mobilized friction angle along
critical slip plane with depth for
wall height H=40 m.
Mobilized Phi (°)
Results 25
Variation of coefficient of lateral earth pressure with
depth for wall height H=30 mm
10
5
0
Dep
th (
m)
10
5
0
Constant ϕ
Constant ϕ
GeoPractices-2015
depth for wall height H=30 mm
0.00 0.05 0.10 0.15 0.20
30
25
20
15
Active case
Dep
th (
m)
Earth pressure coefficient
4 6 8 10
30
25
20
15
Passive case
Results 26
Variation of coefficient of lateral earth pressure with
depth for wall height H=40 m
Constant ϕ
GeoPractices-2015
depth for wall height H=40 m
Constant ϕ
Results 27
GeoPractices-2015
Variation of normalized total lateral thrust with wall height
Comparison
Height
(m)
Active case
Change
(%)
Passive case
Change
(%)
Coulomb’s
Theory Value
(kN)
Stress
Dependent
Value
(kN)
Coulomb’s
Theory
Value
(kN)
Stress
Dependent
Value
(kN)
28
GeoPractices-2015
(kN) (kN) (kN)
5 35.50 13.83 61.00 1334.36 1550.52 -16.20
10 142.02 77.29 45.57 5337.42 5225.88 2.10
20 568.06 408.67 28.06 21349.69 17794.66 16.65
30 1278.147 1063.39 16.80 48036.81 36591.03 23.83
40 2272.26 2082.63 8.34 85398.77 61125.87 28.42
29
Slope Stability Analysis
Properties of Slope 30
1H
B� Embankment Soil Properties:
� Granular fill
� Unit weight of soil = 18 kN/m3
� Slope = 1.5 H to 1 V
� Top width = 10 m
GeoPractices-2015
Granular Fill1.5
H
Foundation Soil
� Top width = 10 m
� Height = 5 m, 10 m and 15 m.
� Foundation Soil Properties:
• Angle of Shearing resistance (ϕ) = 44°
• Unit Weight = 18 kN/m3
31
• An embankment with required dimensions
was generated using Geostudio/Slope-W.
• Morgenstern-Price analysis was adopted
for calculating the factor of safety.
Procedure
GeoPractices-2015
for calculating the factor of safety.
• The material properties were assigned to
the embankment.
• The critical slip plane and corresponding
factor of safety were determined.
• The coordinates of the critical slip plane
were determined
32
• The embankment was divided into number of layers.
• Suitably assumed ϕ values were assigned to the layers of
the embankment.
• ‘Fully Specified’ slip plane option was selected while
defining the slip plane.
Procedure
GeoPractices-2015
defining the slip plane.
• The coordinates of the critical slip plane were inputted
to generate the same critical slip plane.
• The factor of safety corresponding to
assumed ϕ values was determined.
33
• Normal stress on each slice can be verified.
• The ϕ corresponding to the normal stress acting
on the slip plane was determined using Leps
lower limit curve.
• The assumed and calculated ϕ values were
Procedure
GeoPractices-2015
• The assumed and calculated ϕ values were
compared and checked for convergence.
• If the convergence was not achieved, new value
of ϕ equal to the calculated was assumed and
process was repeated.
• The factor of safety was determined for the critical slip plane,
once convergence was achieved
Results 34
H = 5 m
Z=0.9 m
Z=2.4 m
GeoPractices-2015
Homogeneous ϕ case ϕ = f(σm) case
Z=2.4 m
Z=3.9 m
Z=5.0 m
35Results
H = 10 m
Z=1.4 m
Z=3.2 m
GeoPractices-2015
Homogeneous ϕ case ϕ = f(σm) case
Z=3.2 m
Z=6.8 m
Z=10.0 m
36Results
H = 15 m
Z=2.1 m
Z=5.2 m
GeoPractices-2015
Homogeneous ϕ case ϕ = f(σm) case
Z=5.2 m
Z=8.6 m
Z=12.8 m
Z=15.0 m
Height (m)Factor of Safety
% DifferenceUniform ϕ Varying ϕ
5 2.358 2.910 24
37Results
GeoPractices-2015
10 1.658 1.919 16
15 2.064 2.317 12
Conclusions
� Peak friction angle of sand reduced 54% as the confining stress increases from 0 to 400kPa.
� For an increment of wall height from 5m to 40m the inclination of critical slip plane with
horizontal is decreased from 72.5o to 66o for active case and increased form 23o to 28o for passive
case.
� The active thrust estimated with the consideration of stress dependent angle of shearing resistance
was lower than that obtained from Coulomb’s theory by 61% to 8.34% as the wall height increases
from 5 m to 40 m.
38
GeoPractices-2015
from 5 m to 40 m.
� The lateral passive thrust estimated with the consideration of stress dependent angle of shearing
resistance was lower than that obtained from Coulomb’s theory by -16.20% to 28.42% as the wall
height increases from 5 m to 40 m.
� The factor of safety increases by considering variation in maximum angle of shearing resistance
(ϕ) with the vertical effective stress.
� The rate of increase in factor of safety decreases from 24% to 12% with the increase in height of
the embankment from 5m to 15m.
References
� Marsal, R.J. (1967), Large-scale testing of rockfills materials, Journal of Soil Mechanics
and Foundation Engineering Division of ASCE, 93, No. 2, 27–44.
� Leps, T.M. (1970), Review of shearing strength of rockfill, Journal of Soil Mechanics and
Foundation Engineering Division of ASCE, 96, SM4:1159–1170.
� Marachi, N. D., Chan, C. K. & Seed, H. B. (1972), Evaluation of properties of rockfill
Materials, Journal of Soil Mechanics and Foundation Engineering Division of ASCE, 98,
39
GeoPractices-2015
Materials, Journal of Soil Mechanics and Foundation Engineering Division of ASCE, 98,
SM1:95–114.
� Bolton, M.D. (1986), The strength and dilatancy of sands, Geotechnique, Vol. 36, No. 1,
pp: 65-78.
� Manzari, M.T. and Nour, M.A. (2000), Significance of soil dilatancy on slope stability
analysis, Journal of Geotechnical and Geoenvironmental Engineering, Vol. 126, No.1, 75-
80.
40