NPGeoCalc Stability SLOPE2000 Theory Manual 1.7

8/17/2019 NPGeoCalc Stability SLOPE2000 Theory Manual 1.7

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SLOPE 2000

Theory Manual

Copyright: Dr. Y.M. Cheng

Department of Civil and Structural Engineering

Hong Kong Polytechnic University

December, 2005

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1. Preface

Slope 2000 Copyright:

Dr. Y.M. Cheng

Department of Civil and Structural Engineering

Hong Kong Polytechnic University

December, 2005

The theory of the methods of wedges and slices as adopted in SLOPE 2000 are

covered in this theory manual. The reader should note that the formulae in this manual

may be different from that in the references. Actually the author has included more

detailed on the internal forces calculation in Janbu’s rigorous and Sarma’s method to

allow for the presence of external loads. Such extensions are done by the author and

are not included in the original papers by Janbu or Sarma.

For the theory on locating the critical failure surface under general conditions, the

mathematics required is only briefly included in this manual. The background is too

tedious to be included in this manual and the reader is advised to consult the

references as suggested by the author if he wants to understand the details of the

background mathematics.

Dr. Y.M. Cheng

Dec. 2005

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1. General Introduction

2. hermes

2.1. Slope Stability Analysis

2.2. Method of Wedges

2.3. Method of slices

2.3.1. Bishop’s Method

2.3.2. Janbu’s simplified method:

2.3.3. Janbu’s Rigorous method

2.3.4. Spencer’s Method (1967)

2.3.5. Sarma’s Method

2.3.6. Morgenstern-Price’s Method

2.3.7. Miscellaneous Consideration on Slope Stability Analysis

2.4. Three-Dimensional Slope Stability Analysis

2.5. Location of Critical Failure Surface

3. References

4. Appendix A

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1. General Introduction

Slope 2000 has two major groups of methods: namely wedge method and method of slices.

The wedge method that has been incorporated is actually developed for wedges with inclined

sides and is a commonly used method in China. The author adopts this China type wedge

method but uses the method of slices with vertical sides in the calculation. It can hence be

viewed as a special form of the wedge method. Slope 2000 has also incorporated several 3D

slope analysis methods based on the extension by Hungr (1987,1989). Actually all slope

failures are 3D in nature, but 3D analysis in not commonly adopted because it is more

complicated to consider such analysis and no commercial software allow such an option. It is

hoped that 3D analysis which can gives slightly higher factor of safety as compared with 2D

analysis will be adopted by the construction industry in the future.

The basic theory of the wedge method will be introduced first in the theory section. Wedge

method is fundamentally different from classical method of slices like Bishop or Janbu and all

the necessary mathematical details are included in the following sections. It is expected that

the reader may not any previous knowledge on the method of wedge. The theory of

generalized method of slices and the reduction of the generalized approach to various

methods like Bishop, Janbu etc. will follow the section on wedge method. For this section, the

readers are assumed to have the basic background on soil mechanics and simple slope

stability analysis method like the Fellenius and Swedish method. Readers without such

background should consult any classical textbooks [1,4,20,22] before reading this section.

Finally, the basic theory for 3D analysis will be presented. The knowledge for this section is

built on the foundation of the previous sections and the readers may find this section to be

difficult to understand. If the readers are not going to perform 3D analysis, they may skip this

section with problem.

Based on the author’s teaching experience and HKIE/ICE examiner’s experience, many

students and engineers use computer software, design codes and procedures without a clear

understanding of the basic theory of the problem. An even more critical but common mistake

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is that many engineers have not check clearly the accuracy and acceptability of the input data

and fully assess the suitability of the output results. When problems come out, they may

blame the computer software, design codes or procedures instead of themselves. This is poor

attitude but is commonly found among many young engineers. Although the author has

incorporated many check for the accuracy of the input data, the user is actually totally

responsible for the checking of the input data. I prepare the checking subroutines (sufficient

for most cases) purely for the inexperience engineers and I do not have the plan to greatly

expand the checking of the input data. I hope the users should be more careful in preparing

the input data and should develop such attitude in performing all kinds of engineering

analysis. Finally, many engineers do not check the output information carefully or even do not

know how to assess the acceptability of the output result. The users should fully understand

the basic theory of analysis before performing the analysis. I have included a section

Important Notes to User Guide on Slope Stability Analysis in the User Guide with some of

my experience and guidelines in performing a proper analysis. These experience are gathered

from my experience in developing Slope 2000 and may be useful if the users come across

some special cases.

2. hermes

2.1 Slope Stability Analysis

Slope stability analysis can be carried out by limit equilibrium method, limit analysis method,

finite element method or finite difference method (by using Flac or equivalent). By far, most

of the engineers are still using the limit equilibrium method which they can familiar with. For

the other methods, they are not commonly adopted. In the conventional limiting equilibrium

method, failure is assumed to occur along a prescribed failure surface which is practically an

upper bound approach. They driving force or moment under the limiting condition is

compared with the available shear strengths of the soil which will give an average factor of

safety along the failure surface.

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The shear strength mτ which can be mobilized along the failure surface is give by:

F f m

/τ τ = (1.1)

where F is the factor of safety (FOS) with respect to the shear strength f τ which is given by

the Mohr-Coulomb relation as

φ σ τ ′′+′= tann f c (1.2)

Where cohesion,=′c =′nσ effective normal stress and =′φ angle of internal friction

In the equilibrium procedures for stability analysis, F is usually assumed to be constant along

the entire failure surface. Therefore, an average value of F is obtained along the slip surface

instead of the actual factor of safety which varies along the failure surface. At present, there

are many types of slope stability analysis methods based on different assumptions on the

internal forces distribution. In general, the differences between the different methods of

analysis are not great. In Slope 2000, most of the popular methods of analysis are

incorporated which will be sufficient for most purposes. More methods of analysis may be

incorporated into the future release of Slope 2000 if necessary.

2.2 Method of Wedges

In slope stability analysis, the slip surface is normally not a cylindrical one but is a nonlinear

curve. The slip surface may even consists of several planes, especially when a weak stratum

within or below the slope is encountered. For this type of problem, the wedge method is the

most appropriate. In China, a wedge type method is sometimes used and this method has been

incorporated Slope 2000. Consider the sliding mass in Figure 1:

Where:

Wi = weight of wedge i

Qi = horizontal surcharge on wedge i

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Ni = normal force on base i

Ui = water pressure on base I

f i = iφ ′tan at base i

C i = cohesion at base i

l i = length of base i

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Fig. 1 Wedge analysis as used in China

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f i,i+1 = 1,tan +′iiφ at interface i, i+1

Ei,i+1 = normal force at interface i, i+1

Ci,i+1 = cohesion at interface i, i+1

l i,i+1 = length of interface i, i+1

ui,i+1 = water pressure at interface i, i+1

iα = inclination of base i

F = factor of safety on material strength

Consider the force equilibrium of wedge ( 1 ):

F

lC

F

E f U

F

lC

F

N f N W 12121212111

1111111 cossincos +++⎟

⎠

⎞⎜⎝

⎛ ++= α α α (2.1a)

1111111

1111212 sincossin α α α U F

lC

F

N f N Q E U +⎟

⎠

⎞⎜⎝

⎛ +−=−+

(2.1b)

Rearrange gives:

F

lC

F

lC U W

F

f N

F

f E 12121

111111

111

1212 sincossincos −−−=⎟

⎠

⎞⎜⎝

⎛ ++ α α α α (2.1c)

( ) 111

21121111112 cossinsincos α α α α F

lC U U Q f N E −+−=−+ (2.1d)

Similarly, for wedge ( 2 ):

=⎟ ⎠

⎞⎜⎝

⎛ ++ 2

222

2323 sincos α α

F

f N

F

f E

F lC

F E f

F lC

F lC U W 1212121223232

22222 sincos ++−−− α α (2.2a)

12222

2223122222

223 cossinsincos E F

lC U U U Q

F

f N E +−+−+=⎟

⎠

⎞⎜⎝

⎛ −+ α α α α (2.2b)

For wedge ( i ):

iiiii

ii

ii

ii U W F

f N

F

f E α α α cossincos

1,

1, −=⎟ ⎠

⎞⎜⎝

⎛ ++++

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F

lC

F

E f

F

lC

F

lC iiiiiiiiiiiii

ii 1,21,21,21,21,1,sin −−−−−−−−++ ++−− α (2.3a)

⎟ ⎠

⎞⎜⎝

⎛ −++ ii

iiii

F

f N E α α sincos1,

iiiii

iiiiiii E F

lC U U U Q ,11,,1 cossin −+− +−+−+= α α (2.3b)

For the last wedge ( n ):

⎟ ⎠

⎞⎜⎝

⎛ + n

nnn

F

f N α α sincos

1,

,1,1,1,1sincos +

−−−− +++−−= nnnnnnnnnn

nnn

nnn U F

lC

F

E f

F

lC U W α α (2.4a)

⎟ ⎠

⎞⎜⎝

⎛ − nn

nn

F

f N α α sincos

1,,1,1 sincos +−− −++−+= nnnnnnnnn

nnn U E U F

lC U Q α α (2.4b)

where: water pressure at toe of the slip surface.=+1,nnU

Generalizing the above equations gives:-

For wedge ( 1 ): 131121211 a N a E a =+

161151214 a N a E a

(2.5a)

=+ (2.5b)

These two equations have two unknowns E12 and N1, where:

F

f a 1211 = 1

1112 sincos α α

F

f a +=

F

lC

F

lC U W a 12121

1111113 sincos −−−= α α

114 =a 111

15 sincos α α −=F

f a

1

11

2112116

cossin α α F

lC U U Qa −+−= (2.6)

for wedge (i):

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3,2,1,1, iiiiii a N a E a =++ (2.7a)

6,5,1,4, iiiiii

a N a E a =++ (2.7b)

where:

F

f a

ii

i

1,

1,

+= α α sincos2,F

f a iii += (2.8a)

F

lC

F

E f

F

lC

F

lC U W a

iiiiiiiiiiii

iii

iii

,1,1,1,11,1,

3,1 sincos −−−−++ ++−−−= α α (2.8b)

14, =ia iii

iF

f a α α sincos5, −= (2.8c)

iiiii

iiiiiiii E F

lC U U U Qa ,1,1,16, cossin −−− +−+−+= α α (2.8b)

For the last wedge(n):

3,2,1, nnnnn a N a E a =+ (2.9a)

6,5,4, nnnnn a N a E a =+ (2.9b)

where 01, =na nn

nnF

f a α α sincos2, += (2.9c)

1,

,1,1,1,1

3, sincos −−−−− ++−−−= nn

nnnnnnnn

nnn

nnnn U F

lC

F

E f

F

lC U W a α α (2.9d)

04, =na nnn

nF

f a α α sincos5, −= (2.9e)

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nnnnnnn

nnnnnn U E F

lC U U Qa ,1,1,16, cossin −−− −+−++= α α (2.9f)

To obtain the factor of safety, the problem can be solved from wedge number one to n-1, then

E1, E2……En-1 can be obtained. During the solution for wedge n, two N n values can be

obtained from equation (2.9a) and (2.9b). If the two Nn are the same, the F value obtained is

the required factor of safety. However, if different values are obtained, another F value should

be assumed and the calculation should be repeated from the first step until the two N values

are the same. By ignoring the interface cohesion, the above equation would give a lower

bound solution (i.e. % mobilized = 0 %) while 100 % mobilization will be the upper bound.

In actual computation, it is found that multiple solutions for the safety factor may be found,

but it is known that only the highest solution in physically acceptable (ref. 21). This result is

well known among the engineers in China for using the wedge method. The author has found

that a converged result is not necessarily a correct answer. Through the investigation into the

internal forces of the slices, the author has found that an incorrect factor of safety is

associated with incorrect internal force distribution. In Slope 2000, the internal forces are

examined and re-analysis with an automatic change of initial factor of safety is performed

until the internal force distribution is correct. Through such re-analysis, the author has

eliminated the multi-value problem associated with the wedge method.

From the above equations, it is noted that the wedge method is mainly based on force

equilibrium which is very similar to that as given by Chowdhury. In Chowdhury’s book [7],

the interface shear is combined with the interface normal force to give a resultant force acting

on the interface between slices with a inclination θ . He suggested that changing the

inclination θ of this resultant force can give different factors of safety, but he did not give

any guidance on the choice of the inclination of the resultant force. Actually, interface friction

and cohesion are both exist and the approach by Chowdhury may not satisfy Mohr-Coulomb

relation along the interface. For the interface cohesion, it may be in a state that is not fully

mobilized. Therefore, a mobilized interface cohesion or adhesion factor instead of changing

the inclination of interface resultant force is adopted by the author which should be better than

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that as used by Chowdhury. If the mobilized cohesion force is equal to zero, it will be

equivalent to a horizontal interface resultant as in the Chowdhury’s method.

2.3 Method of slices

At present, most of the engineers are still using method of slices which has proved to be

reasonable for most cases. The theory presented here start form a general formulation which

will then reduce to individual methods of analysis. Consider the static equilibrium of the soil

mass (of unit thickness) in Figure 2. The variables for slice i are listed below:

l = length of slice base

= width of the slicei X Δ

= externally applied vertical forcesiV Δ

= weight of the sliceiW

= externally applied horizontal forcesiQΔ

iα = inclination of the base plane

= average height of the slicei H

= interslice normal forces on i and i + 1 interface1, +ii E E

= interslice shear forces on i and i + 1 interface1, +ii X X

= normal reaction on the base of the sliceiP

= shear force reaction on the base of the sliceiS

= pore water force on the base of the sliceiu

= Distance between horizontal loadQ y QΔ and Ω

= Distance between vertical load V andV x Ω

Ω = arbitrary point for considering moment equilibrium

= horizontal distance between base centre and parallel to the x′ Ω

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slice base

y′ = vertical distance between base centre and Ω perpendicular to

the slice base

Based upon static equilibrium conditions and the concept of limit equilibrium, a number of

equations and unknown variables are summarized in tables 2 and 3.

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Fig.2 Slope stability by method of slices

Fig.3 Free body diagram of slice i

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Table 2: Summary of system of Equation

Equation Condition

n Moment equilibrium for each slice

2n Force equilibrium in X and Y directions for each slice

n Mohr-Coulomb failure criterion

4n Total number of equations

Table 3: Summary of Unknowns

Unknowns Description

1 Safety factor

n Normal force at the base of slice, Pi

n Location of normal force at base of slice

n Shear force at base of slice, Si

n-1 Interslice horizontal force, Ei

n-1 Interslice tangential force, Ti

n-1 Location of interslice force (line of thrust)

6n-2 Total number of unknowns

From the above tables, the problem is statically indeterminate of 6n-2-4n = 2n-2. In other

words, we have to introduce additional (2n-2) assumptions to solve the problem. The author

has noticed that many engineers have the wrong concept that methods which satisfy both

force and moment equilibrium are accurate or even exact. This is actually a wrong concept as

all methods of analysis require some assumptions to make the problem statically determinate.

In this respect, no method is particularly better than others, though methods which have more

careful consideration of the internal stresses will usually be better than the others. The author

as well as many other researchers have found that most of the commonly used methods of

analysis give results which similar to each other. To begin with the generalized formulation,

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let us ignore any assumptions made in the problem and consider the equilibrium of force and

moment for a general case.

Equilibrium of Force

Resolving the force acting on slice i with respect to the direction parallel to P i

α α α sinsincos)( Q E X W P Δ−Δ+Δ−= (2.10)

where V W W s Δ+=

ii E E E −=Δ +1

ii X X X −=Δ +1

Similarly, in the direction perpendicular to P i:

α α α coscossin)( Q E X W S Δ+Δ−Δ−= (2.11)

Introducing the concept of mobilized shear force at the base of slice i (i.e. mobilized shear

force )/}tan)({ F ulPlclS mm φ τ ′−+′== ), equation (2.11) gives

}tan)({/1coscossin)( φ α α α ′−+′=Δ+Δ−Δ− ulPlcF Q E X W (2.11a)

( ) tan 1/ { ( ) tan }sec E W X F c l P ul Qα φ α ′ ′Δ = − Δ − + − + Δ (2.11b)

Substituting Equation (2.10):

Q X mF

ullcW FW E m Δ+−Δ+

′+

′−′−=Δ )tan(

cos)

cos

tan

costantan( α φ

α

α

φ

α φ α α (2.11c)

where: F m /tantan φ φ ′=′ and )/tantan1/(sec F m φ α α α ′+= . The summation of the

increments of horizontal interslice force throughout the whole soil mass gives

}]tantancos[1

{]sin[ α α φ φ α α mullcW F

mW E i ′−′+′∑−∑=Δ∑

Qi X m Δ∑+−Δ∑+ )]tan([ α φ (2.11d)

The factor of safety with respect to force F f is equal to:

)tan(]sin[

]}tan)cos([{

α φ α

φ α

α

α

−′Δ+Δ∑+Δ∑+∑

′−+′∑=

mii

f X Q E mW

mulW lcF (2.12)

Equilibrium of Moments

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The moments of all the forces acting on the system of slices are expressed with respect to an

arbitrary point Ω

( ) ( ) ( )i Q i i

W x Qy Px Sy′ ′∑ − ∑ Δ − ∑ + ∑ = 0 (2.13a)

where: cos sini i i i i

x x yα α ′ = +

sin cosi i i

y xi i yα α ′ = +

=i

x horizontal distance measured from mid-point of slice base to Ω

=i y vertical distance measured from mid-point of slice base to Ω

Consider clockwise moment as positive:

'( ) ( ) [( ) ] (i Q i i

W x Qy ulx P ul x Sy )i

′ ′∑ − ∑ Δ − ∑ = ∑ − − ∑ (2.13b)

Equilibrium of vertical forces in a single slice yields:

α α sincos S X W P −Δ−= (2.14)

α

α

α cos

sin

cos

S X W P −

Δ−=

Consider the shear force at base as mobilized or S = S m

α φ α

tan]tan)([1

cos′−+′−

Δ−= ulPlc

F

X W P

F ulP

F

lc X W P

α φ α

α

tantan)(tan

cos

′−−

′−

Δ−=

ulF

ulPF

lc X W ulP −

′−−

′−

Δ−=−

α φ α

α

tantan)(tan

cos

ulF

lc X W

F ulP −

′−

Δ−=

′+− α

α

α φ tan

cos)

tantan1)((

α α α α α φ

cos}tancos{cos)tantan

1)(( ulF

lc X W

F ulP −

′

−Δ−

=

′

+−

substituting the symbol α m as

F

mα φ

α α tantan

1

sec

′+

=

yields:

α

α α m

F

ul

F

lc X W P }

cossin{ −

′−Δ−= (2.15)

Similarly, an expression for S can be obtained by substituting (2.15) into lS mm τ = and

eliminating P:

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}tan)({1

φ τ ′−+′== ulPlcF

lS mm

}tan]cossin[{1

φ α α α ′−′

−Δ−+′= mulF

lc X W lc

F S m

Consider the following terms in above equation:

φ α

α ′′−′ tan

sinm

F lclc

}tantantan

1

secsin1{ φ

φ α

α α ′

′+

−′=

F

F lc

)tantan

1/()tantan

(1{F F

lc φ α φ α ′

+′

−′=

α α

φ α mlc

F

lc costantan1

1′=

′+

′=

α φ α α mul X W lcF

S m }tan]cos[cos{1

′−Δ−+′=∴ (2.16)

Substituting (2.15) and (2.16) into (2.13) and rearranging gives:

−′−′′Δ∑+′−∑=∑−Δ∑−∑ ])tan([])cos[()()( α α φ α m x y X xmulW ulxQy xW iQi

}]tan)cos()tan(cos[1

{ α φ α α α m yulW y xlc

F

′′−+′+′′∑

The Factor of Safety with respect to moment is given by:mF

mmQ

m I xulQy xmulW xW

m yulW ylcF

+′∑+Δ∑+′−∑+∑−

′′−+′∑=

)(][])cos[()(

}]tan)cos({[

α

α

α

φ α (2.17)

where: ])tan([ α φ m x y X I mmm ′−′′Δ∑=

Euations (2.12) and (2.17) are the generalized safety factor equations with respect to force

and moment respectively, and are statically indeterminate in the order of 2n-2. For the case of

a circular slip surface of radius R, the centre of moments Ω coinciding with the centre of

circle (i.e. α sin R x −= ; α cos R y = ). In this case, 0=′ x and R y =′ which results in

a simplified form of Fm and Imm.

In addition to the above equations, considering the moment equilibrium of each slice such

that:

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(i)

Width of slice becomes very small ( 0≈Δ x )

then ;)(1 x X X X ii == + )(1 x E E E ii == +

(ii)

Let h be the height of the line of thrust

(iii)

Contributions of P and W to the sum of moments with respect to mid-point of slice

base are negligible in comparison to other forces acting on slice i; which gives:

Qhdx

dQ

dx

xdM x E x X +−=

)(tan)()( α (2.17a)

where: )()()( xh x E x M =

This is the general equation describing the interslice forces. Now we will introduce some

assumptions to make equation (2.12) and (2.17) statically determinate.

Through many numerical studies, it is found that F f is more sensitive to the magnitude of

interslice forces than Fm. This is because the Imm factor occur in equation F m is small compare

with other terms in the equation. However, the value of Fm is much more sensitive to the

change in shape of failure due to the values of x′ , y′ , x and y in the equation Fm.

The current analytical methods of slices can be grouped into three categories based on the

hypotheses used to describe the internal forces, namely:

(a)

Direction of internal forces; (e.g. Spencer assume parallel interslice forces)

A relationship between the internal force distributions, E(x) and X(x), is assumed. The

mathematical function used to relate the internal tangential forces X(x) to the horizontal

internal force E(x) is expressed as:

=)( x X )()(11 x E x f λ

where: 1λ = a dimensionless scaling parameter

= a scalar function)(1 x f

Recently, Chen and Morgenstern (1983) proposed a more general form to take into account of

stress condition as slope boundary:

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)()]()([)( 11 x E x f x f x X o λ +=

where is a function which is chosen such that the boundary conditions on the ratio

X(x)/E(x) is satisfied.

)( x f o

(b) Height of the line of thrust; (e.g. Janbu’s rigorous method)

Form equation (2.17a), M(x) = E(x)h(x) and h is assumed to be known which is given by:

=)( xh )()(22 x H x f λ

where: 2λ is a scaling factor for function )(2 x f

= scalar function (for Janbu’s method = 1))(2 x f )(2 x f

Then equation (2.17a) is reduced to:

hQdx

dQ

dx

df x E x X +−= 22tan)()( λ α (2.17b)

(c)

Shape of the distribution function of the internal shear force

(e.g. Sarma method)

The shape of the internal tangential force distribution X(x) can be written as

)()( 33 x f x X λ = (2.17c)

Where: 3λ = a scaling factor and = a scalar function)(3 x f

2.3.1 Bishop’s Method

In 1955, Bishop proposed a method of analysis by assuming a circular slip surface and failure

is assumed to occur by rotation of infinite extent on a cylindrical slip surface centered at O

which is a special case of Spencer (1967) method and hence in category (a).

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Fig. 4 Circular failure surface for Bishop’s method

Bishop theory is based on the following assumptions:

(a) circular slip surface and rotating about centre of circle

(b) X Δ equal to zero for any slice, in other words, interslice shear forces could be

ignored.

By using equation (2.17) and considering moment equilibrium, the following equations can

be obtained as α cos R y = , α sin R x −= , 0=′ x and R y =′

)(sin

]tan)cos(cos[

Q

mQyWR

m RulW lRcF

Δ∑+∑

′−+′∑=

α

φ α α α (2.18)

From the above equation, it is clear that the Bishop’s method only satisfy the overall moment

equilibrium condition but not moment equilibrium of an individual slice or force equilibrium.

It does not satisfy horizontal force equilibrium and should be used with care for the case of a

slope with large horizontal force. It is however one of the most popular slope stability method

in the world and possesses the advantage of very fast convergence under most cases.

2.3.2 Janbu’s simplified method:

Janbu et al (1956) proposed a relatively simple analysis for generalized slip surface which

- 21 -


23/48

again neglects the interslice force in the expression (2.15).

α

α α m

F

ul

F

clW P )

cossin( −−= (2.15a)

For overall force equilibrium:

α φ α sec}tan){(1

tan0 ′−+′∑−∑+∑==Δ∑ ulPlcF

QW E (2.19a)

Therefore, the equation of force equilibrium can be written as follows:

QW

ulPlcF o

∑+∑

′−+′∑=

α

α φ α

tan

sec}tan)(cos{ (2.19b)

Substituting P into the above equation gives:

QW

mulW lcF o ∑+∑

′−+′∑= α

α α α

tan

}tan)cos(cos{ (2.19c)

Where α cosl = width of slice. Janbu has performed some numerical tests with both his

simplified and rigorous methods and has proposed to improve the solution from the simplified

method by introducing a correction factor which takes interslice shear force into account.o f

The corrected factor of safety is given by

oo f F f F = (2.19d)

The correction factor is given by the following equations [1]:o f

For c , 0>φ , ])/(4.1/[5.01 2 Ld Ld f o −+≈

For, ,0=c ])/(4.1/[31.01 2 Ld Ld f o −+≈

For, 0=φ , ])/(4.1/[69.01 2 Ld Ld f o −+≈

Where: d = depth of the failure mass

L = length of the failure mass

This method is also a special case of category (a) and it satisfied the conditions of force

equilibrium that the interslice force X(x) are ignored by setting )(11 x f λ to zero.

2.3.3 Janbu’s Rigorous method

With the consideration of the interslice shear forces X i in the analysis, Janbu’s simplified

- 22 -


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solution is improved to Janbu’s rigorous method. In the rigorous method, the resultant

interslice forces Ei were assumed to be acting on a ‘line of thrust’ and β = inclination of line

of thrust to horizontal at interface.

By taking moments about the base centre of each slice and assuming width b is small, the

following equations can be obtained:

0)()()2

tan)((

2

)(tan 1

11 =−Δ+−Δ−−−+

++ +

++ iQiviii

iii y yQ x xV

bh E E

b X X b E

α β

1 1 11

tan tantan ( ) (

2 2 2

i i i i i i i ii v

X X E h E Eh E V Q E x

b b b b)i Q i x y y

α α β + + ++

+ Δ Δ=− − + + − + − − − (2.20a)

where α cos1=b and h = height of the line of thrust and β is the slope of the line of

thrust. If the width of a slice is small, 1+≈ ii X X and 1+≈ ii E E , can be approximated asi X

1 1 tan ( ) ( )i i v i QdE dV dQ

i X E h x x ydX dx dx

β + += − − + − − − y (2.20b)

For the requirement of overall horizontal equilibrium:

0)( 1 =Δ∑=−∑ + E E E ii

Summation of E Δ is given by eq. (2.11b) as:

QulPlcF

X W E Δ∑=′−+′∑−Δ−∑=Δ∑ α φ α sec]tan)([1

}tan]{[)( (2.20c)

Q X W

ulPlcF f

∑+Δ−∑

′−+′∑=

α

α φ

tan)(

sec]tan)([

Q X W

mub X W bc

∑+Δ−∑

′−Δ−+′∑=

α

α φ α

tan)(

sec}tan])[({ (2.21)

In 1973, Janbu found that the factor of safety is relatively insensitive to the assumed location

of the line of thrust. Previous study by the author has shown that the difference between the

largest and smallest factor of safety is not more than 5% [23].

Actually, the location of the line of thrust is unknown except for the fact that it is seldom

located adjacent to the ground profile or slice base. It is always located around one third of

the interface length measured from slice base (i.e. 0.3H). However when , in most0=′c

- 23 -


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cases it will occur slightly above 0.3H in the passive zone (i.e. Lower part of the sliding mass).

On the other hand, it will occur slightly below 0.3H in the active zone.

2.3.4 Spencer ’s Method (1967)

In 1967, Spencer proposed that the inclination θ of the resultant interslice forces Q is

constant, therefore:

θ λ tan)(

)(11 == f

x E

x X (2.22)

That means θ λ tan;1 11 == f in this case and

=== ++ 11 //tan iiii E X E X θ constant for all slices

θ tan)/()( 11 =−−⇒ ++ iiii E E X X

Considering overall moment equilibrium, similar to Bishop’s method, failure is also assumed

to occur on a cylindrical slip surface with a centre and the factor of safety with respect to

moment is actually given by eq. (2.18). For the horizontal force equilibrium, it is given by

α φ α cos]tan)([1

sin)( 1 ′−+′∑−∑+∑=−∑ + ulPlcF QP E E ii

QP

ulPlcF f

∑+∑

′−+′∑=

α

α φ

sin

cos]tan)([ (2.23)

Therefore the Spencer’s method satisfy both force and moment equilibrium conditions.

Normally, different values can be obtained by the equations of Fm and Ff . The value of θ tan

is varied until Fm is equal to Ff and a unique factor safety can then be obtained.

2.3.5 Sarma’s Method

In 1973, Sarma proposed a completely different approach to compute the factor of safety. His

suggestion is based on the critical acceleration that is required to bring a soil mass to a state of

limiting equilibrium. He also assumed the slope of internal tangential force distribution X(x)

is known and is similar to equation (17c) and is given by.

]tan2

)()()[()(

2

3 avg

avg

uavgavg H RK x H c x f x X φ γ λ −+= (2.24)

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where: = interslice forces parallel to slice interface)( x X

λ = scaling factor determined from eq.(2.36)

= scaling function determined by the user, usually chosen as 1.0)(3

x f

, , , tanavg avg avg avg

c K γ φ = average soil strength parameter along interface of

the slice (see eq. 2.44)

H = height of slice

i Ru = pore pressure parameter equal to the ratio of pore water pressure to the

total overburden pressure

Considering the vertical and horizontal equilibrium of the slice i with horizontal and vertical

surcharge Q and V, the vertical and horizontal equilibrium equations are obtained as:

iiiiii X V W T P Δ−+=+ α α sincos (2.25)

iiiiii Q E KW PT Δ+Δ+=− α α sincos (2.26)

where T = shear force at slice base. It is assumed that under the action of KW, the full shear

strength of soil is mobilized. Hence

α φ φ σ sectan)(tan)( iiii bcU PT ucS ′+′−==>′−+′= (2.27)

From equations (2.25) and (2.27):

iiiiiiiiiii X V W U PbcP Δ−Δ+=′−′+′+ α φ φ α α sin)tantansec(cos

))]/[(cos(cos]sintantan[ iiiiiiiiiiii U bc X V W P α φ φ α φ α −′′+′−Δ−Δ+==>

))]/[(cos(cos]sintantan[ iiiiiiiiiiiii W Rubc X V W P α φ φ α φ α −′′

+′

−Δ−Δ+==> (

i W T

2.28)

iiiiii P X V α α sin/)cos−Δ(= −Δ+

Substitute by (2.28) gives:iP

sin)[( X V T )]/[cos(]sincos iiiiiiiiiiiii W RubcW α φ φ φ φ −′′−′′+′Δ−Δ+= (2.29a)

ut in eq.(2.26),iiiiiii KW DQ E X −=Δ+Δ+−′Δ )tan( α φ p (2.29b)

)cos(

sinsec U cos

)tan()(ii

iiiiiiiiiiii

bc

V W Q D α φ

φ α φ

α φ −′

′−′′

+−′Δ+=Δ−

- 25 -


27/48

)cos(

sec)sincos()tan()(

ii

iiiiiiiiiiii

W RubcV W Q

α φ i D

α φ φ α φ

−′

′′ ′ −+−′Δ+=Δ (2.30)

Consider the horizontal equilibrium of the whole mass where

−

i E ΣΔ cancel out:

iiiiii DKW X Q Σ=Σ+−′ΣΔ+ΣΔ )tan( α φ (2.31a)

For the moment equilibrium, we can take moment about the centre of gravity of the whole

mass (including vertical surcharge) to

vertical surcharge terms

sliding soil eliminate the weight term W and thei

iV Δ .

( ) ( cos sin )( ) ( cos sin )( ) 0i Q g i i i i g i i i i i g i

Q y y T P y y P T x xα α α α ΣΔ − +Σ − − −Σ + − = (2.31b)

Where = coordinates of the centre of gravity of the whole soil mass

= coordinates of the mid-point of the slice base

Equations (2.25), (2.26), (2.29b) and (2.31b) give

),( gg y x

),( ii y x

0))(())(()( =−Δ−Δ+Σ−−Δ+Σ+−ΣΔ x x X V W y yQ E KW y yQ Δ+ igiiiigiiigQi

0))(())](tan([)( =−Δ−Δ+Σ−−−′Δ−Σ+−ΣΔ igiiiigiiiigQi x x X V W y y X D y yQ α φ

)()()()]()tan()[( gQigiiigiigiigii y yQ y y D x xW x x y y X −ΣΔ−−Σ+−Σ=−+−′−ΣΔ α φ (2.32)

Sarma then assumed that the resultant shear force can be expressed as eq.(2.33) and f 3 is

assumed to be known.

eq.2.24. This is equivalent to assuming the shape of the distribution of

(2.33)

where F3 is found from

the interslice shear force. However, the magnitude of the interslice shear force is not assumed

to be known. Therefore, substituting equation (2.33) into (2.31) and (2.32), it gives

iiiii DW K F Q Σ=Σ+−′Σ+ΣΔ )tan(3 α φ λ (2.34)

)()()]()tan()[(3 giiigiigiigi y y D x xW x x y yF −Σ+−Σ=−+−′−Σ α φ λ

)( gQi y yQ −ΣΔ− (2.35)

With an estimated value of F3, equations (2.34) and (2.35) can be solved to obtain lambda and

below:K which can be expressed as

- 26 -


28/48

32 / S S =λ (2.36a)

iW S Σ/)4S K −= ( 1 λ (2.36b)

Where: ii Q DS ΣΔ−Σ=1 (2.37a)

)()()(2 gQigiiigi y yQ y y D x xW S −ΣΔ−−Σ+−Σ= (2.37b)

)]()tan()[(33 igiigi x x y yF S −+−′−Σ= α φ (2.37c)

)tan(34 iiF S α φ −′Σ= (2.37d)

The lever arm of the interslice normal force is given by

1111 0)(tan5. ++++ /)]()()(5.0[ −Δ+−Δ−+−+ iiV iiQiiiiiiiiiii E E −= E X X V y yQ X X h E h α (2.38)

This K value will gives the critical acceleration for the soil mass under analysis while the

lambda give the change of the interslice shear force by equation (2.33). In order to det

as carried out detailed study and

hat a value of 1 is applicable for m cases. If there are some local sections where X

terslice surface i in soil layer j

hich can be used as the start for estimating E in the iteration analysis is given by

ermine

the factor of safety for the soil mass, an adjusted factor of safety is applied to the properties of

the soil in the calculation of the mobilized shear force at the slice base by the equation of

(2.27) until the value of K is zero (i.e. acceleration = 0).

For the choice of the assumed function )(3 x f , Sarma h

found t ost

and E violate the failure criteris, can be slightly reduced.

For non-homogeneous soil, the normal force acting on the in

3 f

w

2

, , , , 1( )i j i j i j i ja h hγ +−, , , , , 1 , , , 1 ,( ) ( )

2i j i j i j i j i j i j i j i j i j wij E a W h h b h h d P+ += − + + − + (2.39)

where:

ji ji

ji ji

jia

,,

,,

,sin)sin(1

sin)sin(1

φ β

φ β

′+

′−= (2.40a)

- 27 -


29/48

ji ji

i ji ji

ji

cb

,,

,,

,sin)sin(1

sin(cos2

φ β

′ j, ) β φ ′−

′+= (2.40b)

ji ji

ji ji

ji

d ,,

,,

, sin)sin(1

)sin(sin2

φ β

β φ

′+

′= (2.40c)

ji ji ji ,,, 2 φ α β ′−=

= weight per unit area at the level j along the i th interslice

surface

= (2.41)

the piezometric height at level j along i

=

(2.40d)

jiW ,

∑ +− 1,,, )( j

k

k ik ik i hhγ −1

=1

jiV . = nterslice surface i.

wijP ))((2

1,1,1,, ji ji ji jiw hhV V −+ ++γ (2.42)

= the y-coordinate of j th soil boundary at the i th slice. ji,h

ji,γ = density of the soil layer j at slice i.

w

γ = density of water.

The shear force acting on the interslice surface i for nonhomogeneous case is given by:

]tan)2

,,,,3 iaveiaveiaveiaveii )()[((

2

, iavei H c

H RuK x f X +′−= φ λ (2.43)

where:

γ

i

ji ji j

avei H

hh )( 1,,,

+−∑=

γ γ (2.44a)

, 2

, / 2

wij

i ave

i ave i

P Ru

H γ

∑= (2.44b)

2/2,

,

,

iavei

ji

avei H

E K

γ

∑= (2.44c)

,

,

,

( ) tan j

tan( )

i j wij

i ave

i j wji

E P

E P

φ ′∑φ

−′ =

∑ − (2.44d)

- 28 -


30/48

i

ji ji j

avei H

hhcc

)( 1,,,

+−′∑=

= Height of interslice i.

2.3.6 Morgenstern-Price’s Method

orgenstern-Price’s method is actually an extension of Spencer’s method by assuming the

ormal E by the relation

(2.44e)

i H

M

E x f X )(λ =interslice shear force X related to the interslice n . In

ethod.

_x(1) and

e section with respect to the left exit end

by the user.

Slope 2000, 6 types of f(x) are incorporated for the 2D and 3D options which are:

Type 1: f(x) = constant (1 actually). Actually this case is just the Spencer m

Type 2: f(x) = sin(x)

Type 3: f(x) = trapezoidal shape defined by two normalized parameters tan_phi

tan_phi_x(2). Tan_phi_x = (horizontal distance of th

of the failure surface)/(total horizontal length of the failure surface). f(x) will takes a

maximum value of 1.0 when x is smaller than tan_phi_x(2) and greater than tan_phi_x(1).

Linear relation is assumed when x is smaller than tan_phi_x(1) or greater than tan_phi_x(2).

The user is required to define the two normalized parameters for the trapezoidal shape. The

extreme values of 0.0 and 1.0 means that Type 3 will reduce to Type 1.

Type 4: f(x) = error function or Fredlund-Wilson-Fan force function which is in the form of

f(x) = )5.0exp( nnc η −Ψ and Ψ , c and n have to be defined η is a

normalized dimensional factor which has a value of -0.5 at left exit end and =0.5 at right exit

end of the failure surface. η varies linearly with the x-ordinates of the failure surface. This

error function is actually based on the finite element study by Fredlund.

For the four types of functions as shown above, moment and force equilibrium can be

atisfied together.

Engineers interslice force function. f(x) is assumed to be constant and is

qual to the slope angle defined by the two extreme ends of the failure surface.

s

Type 5: Corps of

e

- 29 -


31/48

Type 6: Lowe-Karafiath interslice force function. f(x) is assumed to be the average of the

lope angle of the ground profile and the failure surface at the section considered. For Type 5

tability Analysis

esides the theory of the various methods of analysis, there are some further considerations in

ternal

ered soil system,

perch water table may exist together with the presence of a water-table if there are great

s

and 6, only force equilibrium can be satisfied. These two options can be chosen directly form

the menu or as a special case of Morgenstern-Price’s analysis.

2.3.7 Miscellaneous Consideration on Slope S

B

developing Slope 2000. Earthquake loading is simulated in Slope 2000 as an ex

horizontal load applied at the centroid of the sliding mass and the value is given by the

earthquake acceleration factor k h multiply with the weight of the soil mass. This horizontal is

assumed to act at the centroid of each slice/wedge which is a reasonable assumption and is

also commonly adopted approach. This quasi-static simulation of earthquake loading is

simple in implementation but should be sufficient for most design purposes.

Another point to consider is the presence of perch water table. In a multi-lay

a

differences in the permeability of the soil. Consider Fig.5, a perch water table may be present

in soil layer 1 with respect to the interface between soil 1 and 2 due to the permeability of soil

2 being 10 times less than that of soil 1. For slice base between A and B, it is subjected to the

perch water table effect and water pressure should be included into the calculation. For slice

base between B and C, no water pressure is required in the calculation while the water

pressure at slice base between C and D is calculated using the water table only.

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32/48

Fig.5 Perch water table

- 31 -


33/48

2.4 Three-Dimensional Slope Stability Analysis

All slop failures are three-dimensional in nature but most of the engineers adopt

two-dimensional analysis for simplicity. A true three-dimensional slope failure analysis is

difficult to perform and it appears that no commercial software can consider this at present.

Since it in complicated to define a general non-circular failure surface, Slope 2000 at present

adopt only the circular three-dimensional failure. In the second release of Slope 2000, a

general non-circular failure surface will most probably be available. The theory adopted by

the author is based on that by Hungr (1987) which is an extension of Bishop’s simplified

method of slope stability analysis to three-dimension. The advantage of Hungr’s approach is

that method of columns retains the conceptual and mathematical simplicity of Bishop’s

original simplified method.

The key formulae of the algorithm are derived by adopting literally the two assumptions of

Bishop (1955):

1. The vertical shear forces acting on both the longitudinal and the lateral vertical faces of

each column can be neglected in the equilibrium equations.

2. The vertical force equilibrium equation of each column and the summary moment

equilibrium equation of the entire assemblage of columns are sufficient conditions to

determine all the unknown forces.

It is implicit in the second assumption that both the lateral and the longitudinal horizontal

force equilibrium conditions are neglected, as they are in the two-dimensional model.

With reference to Figure 6, the total normal force N acting on the base of a column can be

derived from the vertical force equilibrium equation:

y z z zF

cA

F

uA N N S N W α

φ γ sin]

tan)([cos +

−+=+= (2.45)

Where W is the total weight of the column, u is the pore pressure acting in the centre of the

- 32 -


34/48

column base, A is the true base area. From this:

α

α φ α

m

F uAF cAW N

y y /sintan/sin +−= (2.46)

Where:

)cos

tansin1(cos

z

y

zF

mγ

φ α γ α += (2.47)

The true area of the column base, A, has been derived by Hovland (1977) as:

y x

y x y x A

α α

α α

coscos

)sinsin1( 2/122−ΔΔ= (2.48)

The angel zγ between the direction of the normal force N and the vertical axis can be

derived from geometrical consideration as:

2/1

22)

1tantan

1(cos

++=

x x

zα α

γ (2.49)

The plan area of the sliding body is now divided into a series of columns arranged in rows of

uniform width. Their bases lying on a rotational surface related to a unique axis of rotation. A

moment equilibrium equation for an assemblage of j columns can be written as follows (all

the intercolumn forces cancel out against their respective reactions):

- 33 -


35/48

Figure 6 Forces acting on a single column

- 34 -


36/48

∑∑==

=+− j

i

y

j

i

W F

cA

F uA N

11

sintan

)( α φ

(2.50)

From this, with the substitution of the equation of N, the factor of safety is obtained as:

(2.51)∑∑==

+−= j

i

yii

j

i

z z Wr mr cAuAW F 11

)sin/(/]costan)cos[( α γ φ γ α

where r i is the radius for the section under consideration. It should be noted that eq.(2.51) is

slightly different from the original equation by Hungr because Hungr has adopted the global

radius instead of the local radius in the analysis which is not correct. The moment should be

considered about an axis passing through the centre instead of the centre itself. For 0= xα

(i.e. a cylindrical failure surface), equation (2.51) will reduce to the well-known formula of

two–dimensional Bishop’s simplified method. 3D Bishop’s analysis is hence a simple

extension of the corresponding 2D Bishop’s analysis. The development of the extension of

Janbu’s simplified method of slope stability analysis to three dimensions is similar to the

extension of Bishop’s simplified method and is proposed by the author as an extension of the

Janbu’s simplified method of analysis.

Similar to the two-dimensional Janbu’s simplified method, the interslice shear force are

assumed to be zero. By examining overall horizontal force equilibrium a value of the factor of

safety Fo can be obtained. A horizontal force equilibrium equation for an assemblage of j

columns can be written as follows:

∑∑==

=+− j

i

y z

j

i

W F

cA

F uA N

11

tansec]tan

)[( α γ φ

(2.52)

From this, with the substitution of the equation of N, the factor of safety is given by:

∑∑==

+−= j

i

yi z

j

i

z z Wr mcAuAW F 11

0 )sin/(cos/]costan)cos[( α γ γ φ γ α (2.53)

To allow for the effect of the interslice shear forces, a correction factor f o adopted from

two-dimensional analysis is applied, thus the final factor of safety of the slope with respect to

force equilibrium is given

00 f F F f ×= (2.54)

- 35 -


37/48

The normal intercolumn force P and horizontal shear force T are not neglected in the analysis.

They do not enter the equations and neither their magnitudes nor the position of their points

of application need be known. The factor f o is taken as that for two-dimensional case while

the depth of the failure surface is taken at the middle section. Although this approach is

empirical, it should be good enough as the d/l ratio will not have a great change at different

section. Furthermore, f o is a small number which is between 1.0 and 1.1 (mostly below 1.05),

the present approach should be good enough for design purpose.

2.5 Location of Critical Failure Surface

The factor of safety is a function of the topography, soil parameters, external loads, water

table and shape of the failure surface. This function is multi-dimension (dimension refer to the

number of variables defining the problem) and can be a very complex non-smooth function

which is most-likely non-convex in nature. For a user prescribed failure surface, the use of

slope stability analysis method is relatively simple. The user needs only to define the problem

in accordance to the Slope 2000 User Guide and perform the analysis. The user is required to

read all the output information to assess the acceptability of the results of analysis. Although

Slope 2000 has built-in algorithm to assess the suitability of the internal force distribution and

perform re-analysis automatically, the user is still strongly advised to carry out his assessment

as well.

In the analysis of a slope, the most critical location of the failure surface and the

corresponding factor of safety should be determined so that preventive measure can be

performed effectively. The location of the critical failure surface is however usually very

difficult as the factor of safety function is a very complex function. In most of the commercial

software, only search for critical circular failure surface is available. Under this condition, the

objective function (factor of safety) is controlled by 3 variables which the x/y coordinates of

the centre of rotation and the radius of rotation. A very obviously method to locate the critical

circular failure surface is to define a pattern of centre of rotation (usually in form of grid) and

the radius of rotation will be varied for each centre of rotation. If the grid is fine enough,

- 36 -


38/48

accurate solution can be obtained. A difficult question faced by many engineers is how to

define a suitable grid which can cover all possible locations. In practice, factor of safety

function is well known to possess many local minima in the solution domain (see chapter 5 of

the User Guide).

For non-circular failure surfaces, very few commercial programs address this problem. Some

computer programs address this problem by a random generation of non-circular failure

surfaces and the one with lowest factor of safety is then chosen as the critical cases. From the

theory of probability, it is clear that some procedures require tremendous amount of trials

before a high level of confidence can be attained in locating the critical failure surface.

Furthermore, the random generation scheme in these computer programs are generally not

satisfactory. The author view that using these commercial softwares with random generation,

the critical failure surface may not be achieved in some cases (eg. existence of horizontal soft

band etc.) due to the limitations of the generation scheme used. In most of the commercial

programs, the search for non-circular failure surface option is not present at all because of

such difficult. The engineers have to rely on their experience to draw up 5 to 10 non-circular

failure surfaces and the lowest one is accepted as the solution. This approach is also not

satisfactory as some of the shape and location of the critical non-circular failure surface as

found from Slope 2000 will practical be not attempted by most of the engineers. Actually, in

the beta test of Slope 2000 to some actual problems, it is always found that Slope 2000 gives

values lower than that based on the engineer’s experience and the difference can be major in

some cases.

Slope 2000 attempt this problem from a totally different way. In Slope 2000, the factor of

safety is defined as an objective function controlled by some variables. For circular failure

surface, there will be three variable which are the x/y coordinates of the centre of rotation and

the radius of rotation. For non-circular failure surface, the number of unknowns will be more

than three. Using the default value of No_Slice as 15 (see User Guide), the number of control

variables will be 15. The first two variables are the x-ordinates of the left and right exit end

(Xleft and Xright) of the failure surface. The y-ordinates are not the control variable as they

can be interpolated from the ground profile and Xleft and Xright. Based on the default

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number of slice equal to 15, the x-ordinates of the inter-slices can be obtained by a uniform

division. The outstanding control variables are the y-ordinates of the failure surfaces and there

will be No_Slice-1 variables. Totally there will be No_Slice+1 variable or 16 with the default.

Hence, the factor of safety function (objective function) is cast into a mathematical function

which is controlled by at least three control variables. These variables are subjected to the

following constraints:

1.

Slip surface must be below the ground level.

2.

The shape of the slip surface must be concave (except for mode 2 which allow a

non-concave profile to be tried).

3.

The slip surface cannot penetrate into rock stratum.

4.

The slip surface should be kinematically acceptable. It should not cut the ground profile

at more than 2 points.

The remaining task is to perform optimization analysis of the objective function subjected to

the constraints as stated above.

In the search for critical solution of an objective function, the commonly used technique is

actually the gradient type method. Consider an objective function f which is a function of

several variables. The maximum or minimum point has the property of

(2.55)0=∇ f

where ∑∂

∂=∇

n

i

i

dx x

f f

1

(2.56)

The gradient type method is fast in convergence and has actually been tried by the author in

the past. The most critical limitation of the gradient type method is that different trial value

will give different maximum of minimum. Actually within a feasible solution domain, there

are multiple maxima or minima even for a very simple slope. The gradient type method is

suitable only for the local maximum or minimum instead of the global maximum or minimum.

Furthermore, the most critical solution is not necessarily given by the condition of .

Since the objective function is in general non-smooth and non-convex, the gradient type

method may not be able to locate the critical value for real problems and it will actually fail

0=∇ f

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for the non-convex function. With the experience in the gradient type method, the author later

adopts a totally different approach. There are several new mathematical methods which may

be useful to the present problem and that include the genetic algorithm, simulated annealing

technique and tuba search. Since the author has used the simulated annealing in the pile

driving signal matching analysis (similar to the CAPWAP analysis in PDA) previously and

has obtained good result from it, the author has also chosen the simulated annealing technique

in Slope 2000.

Simulated annealing method is a combinatorial optimization technique based on random

evaluation of an objective function in such a way that transitions out of a local

maximum/minimum are possible [15]. It is a general adaptive heuristic and belongs to the

class of nondeterministic optimization algorithm and is now commonly used for N-P type

difficult optimization problem. It is a relatively new method which is probably not covered in

any civil engineering courses, but has been used successfully in many real problems. The user

can find the detailed theory of this method in references 2, 14, 15, 16.

Simulated annealing method will find the global maximum/minimum solution with a high

probability even for ill-conditioned functions with numerous local maximum/minimum.

Actually, the probability of finding the global maximum/minimum will increase towards 1

when the numbers of trial keep on increasing. In practice, only finite precision of the global

minimum is required, and the numbers of trial required in the simulated annealing analysis

will then finite. This technique has been used in various types of problems with great success.

The term annealing refers to heating a solid to a very high temperature and then slowly

cooling the molten material in a controlled manner until it crystallizes [15]. By cooling the

metal at a proper rate, atoms will have an increased chance to regain proper crystal structure

with perfect lattices. During this annealing process, the free energy of the solid is minimized.

Simulated annealing algorithm is actually an algorithm to simulate the evolution of solid in a

heat bath to its thermal equilibrium. During annealing, the atoms have a greater degree of

freedom to move at higher temperature than at lower temperature. The movement of atoms is

analogous to the generation of new neighborhood in an optimization process.

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Let the optimization problem be stated as follows:

Minimize f(X) subjected to (2.57)u

iii x x x ≤≤1

where X is the solution vector (x1, x2, … n) and , are the upper and lower bounds of

. Starting from an initial trial vector X

1

i x

X i

u

i x

X

i x 1, the algorithm generates successively improved

solution X2, X3 etc. moving towards the global minimum. If X i denotes the current points,

random moves are made along each coordinate direction, in turn. The new coordinate values

are uniformly distributed around the corresponding coordinate of X i. On half of these

intervals along the coordinates are stored as the step vector S i. If the point falls outside the

range as given by eq.2.47, a new point satisfying eq.2.47 is found. A solution vector X is

accepted or rejected according to a criterion, known as the metropolis criterion. If is less

than or equal to 0, accept the new point and set

f Δ

=+1 . Otherwise, accept the new point

with a probability of

(2.48)kT f

e f P/)( Δ−=Δ

where , k is a scaling factor called Boltzmann’s constant and T is a

parameter called temperature. The simulated annealing algorithm starts with a high

temperature T

)()( 1 ii X f X f f −=Δ +

0. A sequence of trial vectors is then generated until equilibrium is reached (or

the average value of f reaches a stable value as i increase). During this phase, the step vector

S is adjusted periodically to better follow the function behaviour. The best point reached is

recorded as Xopt. Once thermal equilibrium is reached, the temperature T is reduced and a

new sequence of moves is made starting from Xopt until thermal equilibrium is reached again.

This process is continued until thermal equilibrium is reached again. This process is continued

until a sufficiently low temperature is reached, at which stage on more improvement in the

objective function can be obtained.

In theory, if simulated annealing algorithm is allowed to run for an infinitely long time with

an arbitrary high T0 and allowing T to tends to 0, the algorithm will find the global minimum

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with a probability of 1.0 In practice, only finite number of run is allowed. With the extensive

experience in simulated annealing by various researchers, it is now clear that unless the

parameters chosen for analysis are unreasonable, convergence to the global minimum can be

achieved in acceptable number of trials. The only exception is the case where T 0 is

unreasonably small so that annealing process cannot process. This parameter is well chosen in

Slope 2000 and is more than enough, hence Slope 2000 will not miss the global minimum for

actual problems.

The details of the mathematics behind the simulated annealing method is too tedious to be

presented here and the reader is advised to consult references 2, 14, 15 and 16 for further

details. There are also many important application of the simulated annealing technique in

various disciplines and interested reader may also consult these references for further detail.

The special features of the simulated annealing method are:

1. The quality of the final solution is not affected by the initial trial, except that more steps

may be required for a poor initial trial.

2.

Due to the discrete nature of the function and constraint evaluations, the convergence or

transition characteristics are not affected by the continuity or differentiability of the

functions.

3.

The algorithm is not confined to convex functions but is suitable for all kinds of

objective function.

4. The evaluation of the global is terminated when the improvement in the solution is less

than a prescribed quantity supplied by the user. That means, the user can control the

precision of the solution.

The first three important features of simulated annealing method are crucial to the location of

general non-circular failure surface and the author’s experience with this method is very

positive. The only limitation to this method is that the number of trials required is usually not

small. With the default values as used in Slope 2000 (precision eps is 0.0001), convergence to

the global minimum is usually achieved within 4000 to 7000 trials for a circular failure

surface with a computer time of about 2 minutes for a Pentium 133 computer. For

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non-circular failure surface, convergence is usually achieved within 12000-23000 trials with a

computer time of about 5 minutes. Since the author’s experience suggest that a precision of

0.0001 is good enough for engineering analysis, a lower limit is hence not adopted by the

author. The reader may use a lower precision eps if necessary.

For the illustration of the accuracy of the simulated annealing method in the location of

critical failure surfaces, the user is advised to read the paper in the appendix.

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3. References

1. Abramson L.W., Lee T.S., Sharma S. and Boyce G.M., Slope Stability and

Stabilization Methods, John Wiley, 1996.

2. Belegundu A.D. and Chandrupatla T.R., Optimization Concepts and Applications in

Engineering, Prentice Hall, 1999.

3.

Bishop A.W., ‘The Use of the Slip Circle in Stability Analysis of Earth Slopes’,

Geotechnique No. 5, 1955.

4. Bromhead E.N., The Stability of Slopes 2/Ed., Blackie Academic & Professional,

London, 1992.

5.

Brunsden D. and Piror D.B., Slope Instability John Wiley & Sons Ltd, 1984.

6. Chen Z.Y. and Morgenstern N.R., ‘Extensions to Generalized method of slices for

Stability Analysis’ Canadian Geotechnical Journal No. 20, pp 104-199, 1983.

7. Chowdhury R.N., Slope Analysis, Elsevier Scientific Publishing Co., New York,

1987.

8.

Espinoza R.D. and Bourdeau P.L., ‘Unified Formulation for Analysis of Slope with

General Slip Surface’, Journal of Geotechnical Engineering, ASCE, Vol. 120 No.7,

pp 1185-1204, 1994.

9. Geotechnical Engineering Office, Geotechnical Manual of Slopes Hong Kong

Government, Hong Kong, 1984.

10.

Graham J., Slope Instability, John Wiley & Sons, 1984.

11.

Hungr O, ‘An Extension of Bishop’s Simplified Method of Slope Stability Analysis to

Three Dimensions’, Geotechnique 37, No.1, pp 113-117, 1987.

12.

Janbu N., Bjerrum, L. and Kjaernsli B., Soil Mechanics Applied to some Engineering

Problems, Norwegian Geotechnical Institute, Publ. No. 16, 1956.

13.

Morgentern N.R. and Price V.E., ‘The Analysis of Stability of General Slip Surface’,

Geotechnique No. 15, pp 79-93, 1965.

14. Otten R.H.J.M. and Ginneken L.P.P.P., The Annealing Algorithm, Kluwer Academic

Publishers, 1989.

15.

Rao S.S., Engineering Optimizatoin, 3/e, John Wiley, 1996.

16.

Sait S.M. and Youssef H., Iterative Computer Algorithms with Applications in

Engineering, IEEE Computer Society, 1999.

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17. Sarma S.K., ‘Stability Analysis of Embankments and Slopes’, Geotechnique 23, No.

3, pp 423-433, 1973.

18. Geo-Slope International, SLOPE/W User’s Guide, Version 3, 1995.

19.

Spencer E., ‘A Method of Analysis of the Stability of Embankments AssumingParallel Inter-Slice Forces’, Geotechnique 17, pp 11-26, 1967.

20. Taylor D.W., Fundamentals of Soil Mechanics, John Wiley, New York, 1984.

21.

Ting Zhao and Donald Ian B., ‘Safety Factor Evaluation Difficulties in Multi-Wedge

Slope Stability Analysis’, International Conference on Computational Methods in

Structural and Geotechnical Engineering, Hong Kong, 1994.

22. Terzaghi K., Theoretical Soil Mechanics John Wiley and Sons, New York, 1943.

23.

Slope 2000 User Guide, 2005.

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4. Appendix A

To assess the applicability of engineers’ experience in performing slope stabilityanalysis, selected cases from projects in Hong Kong are considered. In the following

cases, the same input parameters are used by engineers and Cheng and drastic

differences in the factors of safety are obtained. The differences are due to :

1. limited trials have been considered by engineers due to the use of manual trial and error

approach;

2. ‘failure to converge’ will create further difficulty in the location of critical failure surfaces.

For the results as shown below, the results by engineers are obtained by the use of SLOPE/W

using manual trial and error approaches with limited trials. The more refined results by Cheng are

based on the use of SLOPE 2000 with the same input parameters as used by the engineers.

Re-analysis of some slope designs which are approved by Building

Department/Geotechnical Engineering Office of Hong Kong

Government LPM projects using Morgenstern-Price’s analysis

Case SLOPE 2000 Engineer Difference % Difference (%)

1 1.1956 1.4040 -0.208 -17.4*

2 1.1595 1.4580 -0.299 -25.7*

3 1.5164 1.5800 -0.064 -4.2

4 1.5791 1.6770 -0.098 -6.2

5 1.2220 1.5000 -0.278 -22.7*

6 1.0593 1.4300 -0.371 -35.0*

7 1.4011 1.4200 -0.019 -1.3

8 1.3062 1.7300 -0.424 -32.4*

9 1.1257 1.2200 -0.094 -8.4

10 1.3711 1.4050 -0.034 -2.5

11 1.0665 1.2000 -0.134 -12.5*

12 1.2807 1.4060 -0.125 -9.8*

13 1.1820 1.2000 -0.018 -1.5

14 1.3239 1.4060 -0.082 -6.2

15 1.1798 1.4650 -0.285 -24.2*

16 1.2925 1.4000 -0.108 -8.3

17 1.4210 1.4400 -0.019 -1.3

18 1.0100 1.2100 -0.200 -19.8*

19 1.1586 1.4100 -0.251 -21.7*

20 1.1331 1.4080 -0.275 -24.3*21 1.3430 1.4000 -0.057 -4.2

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22 1.3686 1.4000 -0.031 -2.3

23 1.3056 1.5000 -0.194 -14.9*

24 1.3776 1.4200 -0.042 -3.1

25 1.3700 1.4030 -0.033 -2.4

26 1.4140 1.4780 -0.064 -4.5

27 1.3280 1.4080 -0.080 -6.0

28 1.5930 1.7320 -0.139 -8.7

29 1.1890 1.5100 -0.321 -27.0*

30 1.1080 1.2500 -0.142 -12.8*

31 0.7434 1.2530 -0.5096 -68.5*

32 1.3390 1.4370 -0.098 -7.3

33 1.1870 1.2110 -0.024 -2.0

34 1.2200 1.266 -0.046 -3.8

Average (%)= -12.5

Private jobs using Janbu’s simplified method

Case SLOPE 2000 Engineer Difference % Difference (%)

1 0.933 0.913 0.02 2.15

2 0.75 0.771 -0.021 -2.79

3 1.188 1.209 -0.021 -1.72

4 1.398 1.597 -0.2 -12.45*

5 1.059 1.095 -0.036 -3.33

6 1.612 1.454 0.158 10.86

7 1.15 1.413 -0.263 -18.59*

8 1.284 1.417 -0.133 -9.42

9 1.331 1.446 -0.115 -7.97

10 0.609 0.678 -0.069 -10.24*

11 1.388 1.542 -0.154 -9.97*

12 1.401 1.555 -0.154 -9.92*

13 1.381 1.415 -0.034 -2.4

14 1.356 1.415 -0.059 -4.16

15 1.07 1.125 -0.055 -4.916 1.049 1.06 -0.011 -1.06

17 1.066 1.208 -0.142 -11.78*

18 0.831 0.88 -0.049 -5.55

19 0.788 1.149 -0.361 -31.42*

20 0.763 0.99 -0.227 -22.92*

21 1.377 1.514 -0.137 -9.08

22 1.092 1.41 -0.318 -22.52*

23 0.639 0.684 -0.045 -6.58

24 0.512 0.607 -0.095 -15.73*25 0.637 0.718 -0.081 -11.24*

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26 0.558 0.722 -0.164 -22.73*

27 1.315 1.416 -0.101 -7.13

28 1.16 1.414 -0.254 -17.94*

29 1.169 1.576 -0.407 -25.85*

Average error = 10.5%* cases need further investigation, underline case number means soil nail slopes

It is noticed that :

1. Average error as well as maximum error for analysis using Morgenstern-Price’s analysis is

greater than that by Janbu’s analysis as ‘failure to converge’ by using SLOPE/W will affect

the analysis.

2. In general, the differences between the two set of results are smaller when there is no soil

nail. For soil nail slope, the analysis and the factor of safety is very sensitive to the location

of the critical failure surface and greater differences are found which are expected.

3.

For case 1 in private jobs, some minor input errors by engineers are noted and are corrected

in the study.

Documents

NPGeoCalc Stability SLOPE2000 Theory Manual 1.7