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

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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.

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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.

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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.

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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

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� 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.

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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

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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)

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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

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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.

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• 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

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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.

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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,

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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.

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