CHAPTER 4SLOPE STABILITY ANALYSYS

Introduction: Slope Failures

Types of Slope

Causes of Failures

Types of Failures

Method of Analysis

Slope stabilization

Muhammad Azril Hezmi

Slope Failure

is the movement of mass on slope

(falls, slides, flows)

Landslide: involves an extensive area, mild slope (

TYPES OF SLOPE

� Natural Slopes

• Long term process

• Short process

� Man-made Slopes

� Excavated Slopes

� Slopes of Embankment and Earth Dam

CAUSES OF SLOPE FAILURE

Slope inclination

Additional load or Fill height

Excessive Pore water pressure

Loss of shear strength due to

Weathering

Liquefaction

Water (infiltration and seepage)

TYPES OF FAILURES

Wedge Failure is the soil mass movement dueto external force. This type of failure usually occur on a weak plane or weak joint

Circular Failure or non circular failure, Circular failure are associated with homogeneous soil conditions Non-circular slips are associated with non-homogeneous conditions

Translational Failures occur where the form of failure is influenced by the presence of weak layer. The failure surface tends to be plane and roughly parallel to the slope surface

TYPES OF FAILURES

Wedge Failure is the soil mass movement dueto external force. This type of failure usually occur on a weak plane or weak joint

TYPES OF FAILURES

Circular Failure or non circular failure, the shape of failure plane maybe circular or non-circular.

In general, circular slips are associated with homogeneous soil conditions while non-circular slips are

associated with non-homogeneous conditions

TYPES OF FAILURES

Translational Failures occur where the form of failure is influenced by the presence of weak layer. The failure surface tends to be plane and roughly parallel to the slope surface

Principle of Slope Stability

Analysis

Sliding will occur if the shear stress developed

exceeds the corresponding shear resistance of the

soil. In this case, failure is assumes at a certain

plane

W sinα Rs

Possible

failure

surface

FS natural slope = 1.25 to 1.4

FS man-made slope > 1.5

METHOD OF ANALYSISLIMIT EQUILIBRIUM METHODS

Factor of safety is the shear strength at the time of failure τf compared to the stress acting at that plane τm.

If FS = 1, then the slope is in critical condition.

At the time of failure, the shear strength of the soil is fully mobilized along the failure plane. The shear strength is represented by the Mohr-Coulomb criteria:

τ = cu (Total stress analysis)

τ = c’ + σ’ tan φ’ (Effective stress analysis)

1τ

τFS

m

f >=

� Linear Methods: Relatively simple• Infinite slope analysis• Linear Failure Plane • Analysis for the case of φu = 0 (undrained condition)• Wedge failure analysis

� Non- Linear Methods: Method of SlicesNecessary for irregular slope geometry, non-uniform soil condition, and seepagein soil.

METHOD OF ANALYSIS

INFINITE SLOPE ANALYSIS

1

m z cos2β β

z mzz

W

N or σ T or τ

µ

GWTT

Flow net

βγφγ

ββγ tan'tan'

sincos2 satsat z

cFS +=

INFINITE SLOPE ANALYSISThe shear strength along the failure plane

The expression for σ, τ, and µ are

σ = {(1-m)γ + m γsat} z cos2 βτm = {(1-m)γ + m γsat} z sin β cosβµ = m z γw cos2β

' φ tan )μ - σ ( c' τf +=

Substitute the above expressions to get F

( )mτ

φ'tanμ-σc'FS

+=

βtan

φ'tan

γ

γ'

βsinβcos zγ

cFS

sat2

sat

+=

For special case where c’ =0, βtan

'φtan

γ

γ'FS

sat

=

For the case where water table is far below the failure plane (m = 0)

tanβ

φ'tan FS =

Note that when c’ = 0, then factor of safety is independent of the height of the slope. The slope will be stable as long as slope angle β is less than the internal friction angle ϕ. If both cohesion and angle of internal friction angle is not zero, then the critical condition (FS = 1) will be achieved when

tanφβcosγ'c'

zz2cr

==

For a total stress analysis, the shear strength parameters cu and ϕu are used with a zero value of m

FINITE SLOPE WITH LINEAR FAILURE PLANE

H

β θ

C B

A

W

L

h

N =W cosθ T=W sinθ

Rs

θW

θWLc

θW

RFS s

sin

tancos

sin

φ+==

From the figure, line AC is the trial failure planeThe weight of soil (ABC) is:

βsin

θ)(βsinHLγ

2

1W

−=

The force that will cause the failure is T = W sinθ

and the resistance to sliding is given by Rs = cd L + W cosθ tanϕd

The factor of safety will be

θsinW

φtanθcosWLc

θsinW

RFS s

+==

Critical condition prevails when T = Rs.By substituting FS = 1, then

for critical failure plane θ = (β + φd)/2

Substituting θ, we get

And solving for H and replacing cd by c, then

Where Hcr is the safe depth of cut and β is the slope angle

−=

) φ - β ( cos 1

φ cos βsin γ

4c Hcr

Critical Conditions

( )

−−=d

dd

φcossinβ

)φ(θsinθβsinHγ

2

1c

( )

−−=d

dd

φcosβsin

φβcos1

4

Hγc

Same principal valid for condition where a slope consists of two layers where the upper layer is assumed to slide along the interface between the two layers

H D

β

θ

C

B

A

W

h

T = Wsinθ N= Wcosθ

Rs L

Circular slope failure

Defining a Failure surface for a toe circle

β α1 α211.3218.4326.5733.79

4560

252525262829

353535353740

Note: there other charts available as guidelines for finding the center of failure circle

zc

R

R B

d

W

θ

La

Pw

yc

b. with tension crack

A

R

R B

d

W

θ

La

a. No tension crack

Hydrostatic pressure in tension crack

SLOPE WITH CIRCULAR FAILURE PLANE(homogeneous cohesive soils, fu = 0)

Slope in Homogeneous Cohesive soils, φ = 0 analysis

FS

c

FS

τ uf ==mτ

au

ams LFS

cLτR ==

RLFS

cdW a

u=

dW

RLcFS au=

In the event of tension crack developing, then La is

shortened and hydrostatic force will act normal to the crack

if it is filled with water

cw

au

yPdW

R'LcFS

+=

The use of Charts

� Taylor’s stability number

� Janbu stability charts

� Bischop and Morgenstein charts for effective stress analysis

� Morgenstein’s graphs for rapid drawdown

Here we discuss the Taylor’s stability chart only

The Use of Charts, Taylor’s chart

H nd H β

METHOD OF SLICES

In this method, the potential failure surface is assumed to be a circular arc with center O and radius r (see figure).The soil mass (ABCD) above the failure surface (AC) is divided by vertical planes into a series of slices of width b. The base of each slice is assumed to be a straight line. For any slice, the inclination of the base to the horizontal line is αi and the height (measured at the centerline) is hi.

α

forces acting on a slice

Wi

1

87

6

9

54

32

α

β

Τι

Νι−µ

EiΕ ι−1

X i-1 Xi

Ο

METHOD OF SLICES

α

b

h

x

R

As before,

The factor of safety is defined as the ratio of the available

shear strength to the shear stress acting on the plane

The factor of safety is taken to be the same for each slice,

implying that there must be support between slices

(forces must act between slices)

m

f

τ

τFS=

Forces acting on a slice are

� The total weight of the slice, W = γbh� The total normal force on the base: the effective

normal force N’ = σ’l and the boundary water force U = µ l. where u is the p.w.p. at the center of the base and l is the length of the base

� The shear force on the base, T = τm l� The total normal forces on the sides, E1 and E2

� The shear forces on the sides, X1 and X2

� Any external forces must be included in the analysis.

Assumptions must be made regarding the inter-slice forces E and X

Taking moment about O, the sum of the moments of the shear forces T

on the failure arc AC must be equal to the moment of the weight of

the soil mass ABCD.

∑∑ = αsin/)τ( WFlf

( )∑

Σ+=α

φsin

'tan''

W

NLcF

a

∑∑ = αsinRWTR

∑∑=

ατ

sinW

lF

f

For analysis in terms of effective stress

( )∑

∑ +=α

φsin

'tan'σ'

W

lcF or

Where La is the arc length of AC

The Fellenius (Swedish) MethodFellenius assumed that the resultant of the inter-slice forces is zero, then

N’ = W cos α – ul

Hence the factor of safety in terms of effective stress is given by:

The components W cosα and W sinα can be determined graphically while angle a can be calculated or measured

For analysis in terms of total stress parameter or φu = 0, then

αsinWF

ΣΣ= au Lc

( )( )∑

∑ −+=α

φµβsin

'tancos'

W

lWlcFm

The Bischop (Routine) MethodBischop assumed that the resultant of the inter-slice forces are horizontal i.e. X1 –X2 = 0, then

)φ'tanN'l(c'F

1T +=

Resolving forces in the vertical direction:

αα sintanφF

N'sin

F

lc'cosαulcosαN'W ' +++=

+

−−=

F

luF

lcW

Nαφα

αα

sin'tancos

cossin'

' By replacing l = b secaAnd after some rearrangementWe obtain:

( ){ } ( )∑

+−+

Σ=

FaubWbc

aWFS

/'tantan1

sec'tan'

sin

1

φαφ

By replacing ru = u/γh = u/(W/b) then:

( ){ } ( )∑

+−+

Σ=

FarWbc

aWFS u

/'tantan1

sec'tan1'

sin

1

φαφ

The Bischop (Routine) Method (cont’d)

Since F appear in both sides of the equation, then use trial and error.

To simplify the calculation, the following chart could be used

+=F

' tantan 1 cos m

φaa

a

The Bischop (Routine) Method (cont’d)

( ){ }∑

−+Σ

=a

um

rWbcaW

FS1

'tan1'sin

1 φ

Then

To get FS from the equation,

can use computer program or

graph 1. Assume F right = 1, find mα

2. Find F left 3. Take the average of F

right and F left 4. Use this average F,

find mα5. Find new F left6. Repeat steps 3 and 4

until the difference between F right and F left is small enough (0.01)reroute to excell program for Bischop

COMMENT ON SLICES METHODS

Due to repetitive nature of the calculations and the need

to select the most critical failure surface, the method

of slices in particularly suitable for solution by

computer. More complex geometry and soil strata can

be introduced.

There are other methods of slices as shown in the following

Table. These methods use different assumption on inter-

slices forces.

Slices methods of analysis frequently used in practice.

MethodForce

equilibriumMoment

equilibriumShape of slip surface

Ordinary method of slices (Fellenius, 1927)

Does not satisfy horizontal or vertical forces equilibrium

Yes. Circular

Bishops Modified (Bishop, 1955)

Satisfy vertical force but not horizontal force equilibrium

Yes. Circular only. Non circular may have numerical problems.

Janbu’s simplified method(Janbu, 1956)

Yes No Any shape. More frequent numerical problems than other methods

Morgenstern and Price (Morgenstern and Price, 1965)

Yes. Permits side forces to be varied

Yes. Any shape.

Spencer’s Method (Spencer, 1967)

Yes. Side forces are assumed to be parallel

Yes. Any shape.

ASSIGNMENT 1:

SLOPE STABILITY ANALYSIS• Pick a problem and the CD + manual

• Analyze the problem using SLOPE/W student version (in this case you can use Bischop, Janbu or GLE methods available for Student version).

• Find the slip surface that gives the lowest factor of safety (critical failure surface)

• Sketch of your slope in graph paper and trace the critical failure surface you obtained from SLOPE/W on your graph

• Use method of slices to calculate the factor of safety either using Bischop or Fellenius method (you may make use of Excell for your calculation).

ASSIGNMENT 1:

SLOPE STABILITY ANALYSISDiscuss the results and write a report (Group). The report should include

� Introduction (the problem)

� Results of SLOPE/W � output including contour of FS and the critical failure

surface + analysis of 1 slice

� Results of your manual calculation (with the help of Excell program)

� Discussion and comparisons