SAS_Manual.pdf

Jordan University of Science and Technology

USER’S MANUAL

FOR

SAS-MCT 4.0

A Computer Program for Stability Analysis of Slopes Using Monte Carlo Technique

By

Abdallah I. Husein Malkawi, Ph.D.

Professor and Dean of Research Geotechnical and Dam Engineering

Email: [email protected]

Waleed Falih Hassan, M.Sc.

Research Associate

Geotechnical Engineering Research Group

Civil Engineering Department Jordan University of Science and Technology

Irbid 22110 – JORDAN

Version 4.0 May, 2003

ii

Copyright © Geotechnical Engineering Research Group 2003 This research was partly conducted at Jordan University of Science and Technology and partly supported by Deanship of Research. The authorship rights are reserved for Jordan University of Science and Technology and the authors.

iii

PREFACE Development of SAS-MCT began in 1997 at Jordan University of Science and Technology as an initiative to develop an easy-to-use computer program for solving complicated slope stability problems involving earth structures such as natural slopes, excavations, dams, or man-made embankments. A new automatic search procedure coupled with a new Monte-Carlo method of both random jumping and random walking types for locating the global critical circular and non-circular slip surface was developed and integrated in the code. This procedure was published in the following well known international Journals;

Husein Malkawi, A.I.; Hassan, W.F and Abdulla, F. (2000) “Uncertainty and reliability analysis applied to slope stability” Structural Safety Journal, 22, 161-187. Husein Malkawi, A.I.; Hassan, W.F. and S.K. Sarma (2001) "A global search method for locating general slip surface using Monte Carlo Techniques", ASCE Geotechnical and Geoenvironmental Journal, August. Husein Malkawi, A.I.; Hassan, W.F. and S.K. Sarma (2001) "An efficient search method for locating circular slip surface using Monte Carlo Technique", Canadian Geotechnical Journal., October. Husein Malkawi, A.I.; Hassan, W.F. and S.K. Sarma (2002) "Closure to discussion of - a global search method for locating general slip surface using Monte Carlo Techniques", by Gautam Bhattacharya, ASCE Geotechnical and Geoenvironmental Journal, December.

Husein Malkawi, A.I.; Hassan, W.F. and S.K. Sarma (2003) "Reply to discussion of - an efficient search method for locating circular slip surface using Monte Carlo Technique", By V.R. Greco, Canadian Geotechnical Journal., February.

The first and second versions of SAS-MCT Software were released in 1999; these versions run under DOS operating systems. Two years later i.e., in 2001, the third version was released SAS-MCT 3.0. The main code was kept in Fortran language whereas the graphics user interface was coded by Dr. Nezar A. Hammouri in Visual Basic. This version runs on PC’s using Windows operating system. Later on, the main code and the supporting graphics were converted into Microsoft® Visual Basic 6.0 by Eng. Mohammad Yamin. This new version SAS-MCT 4.0 runs on PC’s Windows operating systems. Today, there are two versions of the software: SAS-MCT4.0 Standard for ordinary users and SAS-MCT4.0 PRO for researchers and professionals. The SAS-MCT4.0 PRO is more accurate and usually requires more computational time, especially when rigorous methods are used.

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TABLE OF CONTENTS

CHAPTER I INTRODUCTION 1

CHAPTER II DREVATION OF FACTOR OF SAFFTY 3

Introduction 3 Stability Methods used by SAS-MCT Program 7 Ordinary or Fellenius Method 8

Simplified Bishop Method 10 Janbu’s Simplified Method 11 Morgenstern- Price Method 13 Spencer’s Method 16 Three-Dimension Stability Analysis 17 General 17 Bishop method in Three-Dimensions 18 Janbu’s Method in Three-Dimensions 21

CHAPTER III MONTE-CARLO TECHNIAUE TO ESTIMETE UNCERTAINTY 22 Introduction 22

Uncertainty in Soil Properties 22 Safety Factor Distribution 24 Normal Distribution Generation 25 Log-Normal Distribution Generation 27 First-Order Second-Moment Approximation 29

CHAPTER IV SAS-MCT PROGRAM 31

Program Features 31 Program Description and Organization 38

CHAPTER V ILLUSTRATIVE EXAMPLES 40

EXAMPLE 1 40 EXAMPLE 2 70 EXAMPLE 3 95 EXAMPLE 4 104 EXAMPLE 5 134

1

CHAPTER I

Introduction Preamble Stability Analysis Slopes using Monte – Carlo Technique (SAS – MCT)

Version 4.0 is a computer software designed to operate on Windows

operating system. The program is capable of analyzing the stability of man –

made or natural slopes under static and earthquake loading. The program

uses a new developed automatic search procedure coupled with a new

Monte-Carlo method of both random jumping and random walking types for

locating the global critical circular and non-circular slip surface. Calculation of

the factor of safety against instability is performed by one of the following

limiting equilibrium based methods. Ordinary method, Bishop’s simplified

method, Janbu’s method, Spencer’s method, and the generalized limited

equilibrium (GLE) method, a discrete version of Morgenstern – Price method.

The program provides a number of high quality plots. These plots can be

viewed and easily sent to printers. Water can be defined in terms of pore

water pressure ratio (ru) or as a phreatic surface. Total and effective stress

analysis can be performed. Specific circular and non circular slip surface can

be defined and analyzed. Analysis is performed using SI units (kN, m) or

British Units (lbf, ft). Point loads and surcharge loads can be included in the

analysis; inclination of these loads is specified with respect to the vertical axis.

Detailed output files are created to provide the user with extensive information

about the output. In details, the program features the following:

1- Two-Dimensional analysis of slopes assuming circular slip

surface and using either one of the following methods as

specified by the user: Ordinary method, Bishop’s method,

Janbu’s method, Spencer’s method, and the generalized Limited

Equilibrium (GLE) method, which is a discrete version of

Morgenstern-Price method.

2

2- The program can be used to search for critical slip surfaces

based on maximum probability of failure. First order second

moment approximation is used to estimate the reliability index

(β). This option is valid only for Ordinary method of slices.

3- A two-dimensional analysis of slope stability assuming irregular slip

surfaces and using one of the following methods: Janbu’s method,

Spencer’s method, and Morgenstern-Price method (GLE). In this

respect, the program searches for the most critical slip surface by

representing every trial surface by 4,5,6,… to 12 vertices trying to

simulate the shape of the real slip surface. The most critical slip

surface corresponding to these vertices will be also shown graphically.

4- A seismic slope stability analysis using the peseudo-static approach.

The program computes the reduction in the factor of safety due to the

specified acceleration input expressed in percent of ground

acceleration (g).

5- Three-dimensional slope stability analysis using one of the extensions

of Bishop’s or Janbu’s two-dimensional methods.

6- Reliability Analysis is performed using Monte-Carlo simulation

technique. Normal and log normal distribution are considered. The

program generates a large number of different expected soil

parameters and calculates the safety factor for each random set.

These trials are used to construct the distribution of the safety factor.

The corresponding reliability index (β) and the probability of failure Pf

can be obtained. Soil parameters can be assumed to follow either

normal or log-normal distribution.

3

CHAPTER II

Derivation for Factor of Safety

Introduction

The main goal of slope stability analysis is to determine the most

critical failure surface and its associated minimum factor of safety. The factor

of safety is defined as a ratio of resisting force to driving force, both applied

along the failure surface.

The shape of the failure surface may be quite irregular, depending on

the homogeneity of the materials of the slope. If the material is homogeneous,

the most probable critical failure surface will be cylindrical, because a circle

has the least surface area per unit mass; the surface area is being related to

the resisting force and the mass to the driving force. Practically, all stability

analyses of slopes are based on the concept of limiting equilibrium. In most

methods of limiting equilibrium, only the concept of statics is applied. Except

in the simplest cases, most problems in slope stability are statically

indeterminate. As a result, some simplifying assumptions are made in order to

determine a unique factor of safety. Due to the differences in assumptions

variety of methods, which result in different values for the calculated factor of

safety, have been developed. The most popular methods are Fellenius

(1936), Bishop (1955), Janbu (1954, 1973), Morgenstern and Price (1965),

Spencer (1967), and Sarma (1973,1979). Some of these methods satisfy only

overall moment equilibrium, like Fellenius and simplified Bishop methods that

are both applicable only to circular failure surfaces. On the other hand, Janbu,

Morgenstern-price, Spencer, and Sarma methods satisfy both moment and

force equilibrium and are applicable to failure surfaces of any shape. All these

methods use the same principle in the analysis of the slope stability, i.e.,

dividing the failure surface into a number of slices.

4

In the methods of slices, also named “Limit Equilibrium” methods, the

mass of the soil enclosed within the failure surface is divided into (n) discrete

slices that are to be analyzed separately. The analysis of an individual slice is

dependent on the distribution of the normal effective stresses along the failure

surface. Figure (2.1) shows a typical single slice along with the general

system of forces acting upon their prospective locations and lines of action. In

reference to this figure, the “Thrust Line” is defined as that line connecting

points of application of the inter-slice resultant force (Z). Knowing those

variables, the analysis is launched using the force in both directions (x, y), and

moment equilibrium equations.

(a) (b)

Fig. (2.1) (a) A Slope divided into n Slices, and (b) A Typical Slice and

the System of Forces Acting on it.

For such analysis, the number of available equations, (4n), are less

than the problem variables, (6n-1), making the problem statically

indeterminate. Table (2.1) lists the number of variables and equations

provided in such analysis. To overcome this shortage, (2n-1) assumptions

must be made to allow the solution of the stability problems.

5

Table 2.1: Equations and Unknowns Encountered in the Method of Slices.

Equations Condition n Moment equilibrium of individual slice. 2n Single slice force equilibrium in two direction.

n Mohr-Coulomb relationship between shear strength and normal effective stress.

4n Total no. of equations. Unknowns Variable

1 Factor of Safety n Normal force at the base of each slice, N’ n Location of normal force. n Mobilized shear force at the base of each slice, Sm.

n-1 Inter-slice resultant force, Z. n-1 Inclination of inter-slice force. n-1 Location of the inter-slice force (line of thrust). 1 Lambda (λ), where λ is a constant value.

6n-1 Total no. of unknowns

The most common assumption made by almost all-slicing methods is

that the slice normal force is located at the slice base mid-length, reducing the

number of variables to (5n-1) and leaving an additional (n-1) assumptions to

be made.

These assumptions are solely related to the inter-slice resultant forces,

where different assumptions result in different analytic methods. These

include Ordinary Method of Slices (OMS) (Fellenius, 1927,1936), Bishop’s

Simplified and Rigorous methods (1955,1955), Janbu’s Simplified and

generalized methods (1954,1957,1973), Lowe and Karafiath’s (1960), Corps

of Engineer’s (1982), Spencer’s (1967,1973), Morgenstern-Price (1965), and

Sarma’s (1973,1979) methods. Some of these methods discard the existence

of inter-slice forces, try to relate their location to ground and slice base

inclinations, assume constant angle of inclinations, or even define a “portion”

of a function describing the points of action of those inter-slice resultant

forces. Table (2.2) summarizes the static equilibrium conditions of the Limit

Equilibrium based methods, where only Bishop’s Rigorous, Spencer’s,

Morgenstern-Price, and Sarma’s methods are found to fully satisfy all

equilibrium conditions.

6

Table (2.2): Comparison between Different Limit Equilibrium Based Methods.

Force equilibrium Method 1st

Direction*2nd

Direction*

Moment equilibrium

Ordinary or Fellenius. Yes No Yes Bishop’s Simplified. Yes No Yes Janbu’s Simplified. Yes Yes No Corps of Engineers. Yes Yes No Lowe and Karafiath. Yes Yes No Janbu’s Generalized. Yes Yes No Bishop Rigorous. Yes Yes Yes Spencer’s. Yes Yes Yes Sarma’s Yes Yes Yes Morgenstern-Price.(GLE) Yes Yes Yes

* Any of the two orthogonal directions can be selected for the summation of forces.

In the next section, a brief description of the most frequently used

methods in slope stability analysis is presented. The program SAS-MCT

incorporated these methods.

7

STABILITY METHODS USED BY SAS-MCT 4.0 PROGRAM:

In order to calculate the factor of safety against sliding for a slope, it is

important that the geotechnical engineer is familiar with the formulation used

by limit equilibrium methods. The complexity of these procedures ranges from

the simple ordinary method, which is suitable for hand calculations, to the

rigorous methods such as Morgenstern-Price method, which really require the

use of a computer. The complete equations for several popular limit

equilibrium methods, which are used in SAS-MCT program, are presented

next. Figure (2.2) depicts the forces acting on a typical slice.

F = factor of safety. ZL = left inter-slice force. ZR = right inter-slice force. δL = left inter-slice force inclination angle.

Sm = mobilized shear strength.

FNbcSm

φtan. ′+′=

δR = right inter-slice force inclination angle.

U = pore water pressure. hL = height of force ZL. W = weight of slice. hR = height of force ZR. Ww = surface water force. α = Inclination of slice base. Q = external surcharge. β = Inclination of slice top. N’ = effective normal force b = width of the slice. Kh = horizontal seismic coefficient. h = average height of the slice. µ = Angle of inclination of external load.

ha = height to the center of the slice.

Fig. (2.2) Forces Acting on a Typical Slice.

8

Ordinary or Fellenius Method:

The Ordinary method of slices is considered the simplest method of

limiting equilibrium-based approach. The safety factor equation resulting from

this method is linear. This method assumes that for each slice the resultant of

the inter-slice forces is zero (Fellenius, 1936).The method is applicable only to

circular failure surfaces and considers only the overall moment equilibrium,

see Figure (2.3). The indiscriminate change in the direction of the resultant

inter-slice force from one slice to the next results in factor of safety errors that

may be as much as 60% (Whitman and Bailey 1967). This method involves

resolving the forces on each slice normal to the base. Referring to Figure (2.2)

one gets;

0=∑ HF , 0sinsinsin.sincos =++−−′− βµααα whm WQWkbUNS …(2.1)

0=∑ VF , 0coscoscoscossin =++−′−− βµααα wm WQUbNSW ...(2.2)

Solving for N’

)cos()cos(sincos αµαβαα −+−+−−=′ QWUbWkWN wh ...(2.3)

Fig. (2.3) Inter-Slice Forces for the Ordinary Method

(ZR3 is not equal ZL4).

9

The Mohr-Coulomb mobilized shear strength, Sm , along the base of each

slice is given by;

FNbcSm

φ′′+′=

tan. ...(2.4)

c’ and φtanN ′ are the cohesive and frictional shear strength components of

the soil.

The factor of safety is derived from the summation of moment about

the center of rotation;

Mo =∑ 0

∑=

++=n

iw RQWW

1sin)coscos( αµβ

∑=

−+−n

iW hRQW

1)cos)(sinsin( αµβ

∑∑==

−+−n

iah

n

im hRWKRS

11)cos( α ...(2.5)

Where

R = the radius of the circular failure surface.

h = average height of the slice.

ha = vertical height between center of the base slice and the center of

the slice.

If the factor of safety is assumed to be the same for each slice, then

[ ]

∑ ∑ ∑

∑

= = =

=

−+−+−++

′−+−+−++′= n

i

n

i

n

i

ahWw

h

n

iw

Rh

WkRhQWQWW

bUWkWQWbcF

1 1 1

1

)(cos))(cossinsin(sin)coscos(

tan].sin)cos()cos(cos[sec.

ααµβαµβ

φααβαµαα

…(2.6)

10

Simplified Bishop Method:

In simplified Bishop Method, the inter-slice shear forces are neglected, and only the normal forces are used to define the inter-slice forces, Bishop (1955). This assumption implies that there is no friction between any two slices. In this method, overall moment equilibrium and vertical force equilibrium are satisfied. However, for individual slices, neither moment nor horizontal force equilibrium is satisfied. Although equilibrium conditions are not completely satisfied, the method is, nevertheless, a satisfactory procedure and is recommended for most routine work where the failure surface can be approximated by a circle.

The effective normal forces are derived from the summation of forces

in the vertical direction; refer to Figure (2.2),

∑ = 0VF

⎥⎦⎤

⎢⎣⎡ ++−

′−=′ µβα

αcoscostan1 QWub

FbcW

mN W …(2.7)

where: F

m φααα′

+=tansincos …(2.8)

The factor of safety is derived from the summation of moments about a

common point. Summing the overall moment equilibrium of the forces acting

on each slice about the point of rotation,

∑ =oM

∑ ∑= =

−+−++n

i

n

iWW hRQWRQWW

1 1)cos)(sinsin(sin)coscos( αµβαµβ

0)]cos([11

=−+− ∑∑==

n

iah

n

im hRWkRS α ...(2.9)

Substituting the Mohr-Coulomb failure criterion and solving for the factor of

safety gives;

∑ ∑∑

∑

= ==

=

−+−+−++

′′+′

= n

i

n

i

ahWW

n

i

n

i

RhWk

RhQWQWW

NbcF

1 11

1

)(cos))(cossinsin(sin)coscos(

)0tansec(

ααµβαµβ

φα

…(2.10)

11

Janbu’s Simplified Method: Janbu (1954, 1973) suggested the following method. It satisfies vertical

force equilibrium for each slice and the overall horizontal force equilibrium for

the entire slices mass. It is applicable to failure surfaces of any shape. The

normal force is derived from the summation of vertical forces, with the inter-

slice shear forces ignored. Referring to Figure (2.2) one gets;

∑ = 0VF

µβαα coscossincos)( QWWSubN Wm −−−++′= … (2.11)

αµβαα

coscoscossincos QWWSubN Wm +++−−

=′ …(2.12)

Substituting the value of Sm ,

αµβαα

mQWubF

bcWN W⎭⎬⎫

⎩⎨⎧ ++−

′−=′ coscoscossin ….(2.13)

where F

m φααα′

+=tansincos ….(2.14)

The horizontal force equilibrium equation is used to derive the factor of

safety.

∑ = 0HF

αµβα cossinsinsin)( mWh SQWWkubN +++−+′−= ….(2.15)

Substituting the value of Sm and rearranging the equation for overall

horizontal force equilibrium for the sliding mass,

[ ]∑ ∑ ∑= = =

⎥⎦⎤

⎢⎣⎡ ′′+′

+++−+′−=n

i

n

i

n

iWhH F

NbcQWWkubNF1 1 1

costansinsinsin)( αφµβα …(2.16)

If the factor of safety is assumed to be the same for each slice, then

12

∑∑

∑

==

=

′+−−+

′′+′

= n

iWh

n

i

n

i

NQWWkub

NbcF

11

1

sin)sinsinsin(

cos)tansec(

αµβα

αφα ….(2.17)

On the basis of a strictly limited number of such calculations, Janbu

proposed an empirical correction to be applied to the results of calculations

made using his routine method. This is shown in Figure (2.4). The correction

factor is in the nature of an increase in the factor of safety and depends on the

relative depth of the landslide in relation to its length, and on the nature of the

soil properties (cohesion and angle of internal friction). It has a maximum

value of 13% increase in F. The correction should be applied after the routine

procedure has been followed, i.e. the correction is made to the converged

factor of safety, not during the iterative procedure, as follows:

FfF ocorrected .= ….(2.18)

Where fo is taken from the chart in Figure (2.4).

Fig. (2.4) Janbu’s Correction Factor Chart.

13

MORGENSTERN-PRICE METHOD: Morgenstern-Price (1965) introduced a method in which the inter-slice

resultant force angle is assumed to vary according to an arbitrary function,

f(x). However, the general limit equilibrium (GLE) proposed by (Chugh, 1986,

Fredlund, et al. 1981) is adopted, which is a discrete version of Morgenstern –

Price method. It comprises most of the assumptions used in all methods of

slices. This method follows Spencer’s procedure once the assumed function f

(x) is set to a constant value or to any other shape for a discrete version of a

Morgenstern-Price method. Figure (2.5) illustrates some of the functions used

to describe the variation of inter-slice force angle along the slope.

Fig. (2.5) Function Used to Describe the Variation of Inter-Slice Force Angle.

14

Force Equilibrium

The summing of forces along and normal to the base of the slice are as

follows;

)cos()cos(sin αδαδα −−−+= LLRRm ZZWS

)sin()sin(cos αµαβα −−−−+ QWWk Wh ….(2.19)

)sin()sin(cos αδαδα −−−+=′ LLRR ZZWN

)cos()cos(secsin αµαβαα −+−+−− QWubWk Wh ….(2.20)

From Mohr-Coulomb theory

FNbcSm

φα ′′+′=

tansec. ….(2.21)

Substituting Eq. (2.19) into Eq. (2.21); and eliminating N from Eqs.

(2.20) and (2.21) and solving for ZR;

)cos(tan)sin(tancossec.sin

)cos(tan)sin()cos(tan)sin(

αδφαδφααα

αδφαδαδφαδ

−−′−′−′−

+−−′−−−′−

=RR

LRR

LLR F

WbcFWZFFZ

)cos(tan)sin(]tan)cos()sin([

)cos(tan)sin(cos)tantan(tansec

αδφαδφαβαβ

αδφαδααφφα

−−′−′−+−

−−−′−

′++′+

RR

W

RR

hF

FWF

FWkub


αδφαδφµαµα

−−′−′−+−

−RR F

FQ ….(2.22)

15

Moment Equilibrium

Summing moments of forces about the midpoint at the base of a slice

to determine the location of the inter-slice forces, hR; is on the right-hand side

of each slice,

∑ = 0cM

µβδδαδ sinsin2

sinsin2

)tan2

(cos hQhWbZbZbhZ WRRLLLLL ++++−

0)tan2

(cos =+−− αδ bhZhaWk RRRh ….(2.23)

Solving Eq. (2.23) for hR;

αδδδα

δδ

δδ tan

2tan

22cossintan

2coscos

coscos

2bbb

ZZb

ZZh

ZZh

R

L

R

L

R

L

R

LL

R

L

R

LR −++−=

RR

ah

RRRR

WZ

hWkZhQ

ZhW

δδµ

δβ

coscossin

cossin

−++ ….(2.24)

ZL and hL define the boundary conditions for the first slice and ZR and

hR for the last slice. In many cases, these values are zero.

Equations (2.23) and (2.24) provide two equations for solving the unknown

functions ZR,hR, and δ. In order to complete the system of equations, it is

assumed that;

)(.tan xfλδ = In which f(x) is a function of x and λ is a constant. The problem is now

fully specified, λ and F can be determined by solving equations (2.23) and

(2.24) that satisfy the appropriate boundary conditions. The function f(x) can

be assumed to be one of the functions shown in Figure (2.5).

16

Spencer’s method: Spencer’s method (1967,1973) assumes that the angle of inclination of

the inter-slice forces is constant for all slices. It is a special case of the

Morgenstern-Price method. According to Spencer’s assumption;

)( slicesallforLR δδδ ==

The force equilibrium equation becomes;

)cos(tan)sin(cos)tantan(tansec

)cos(tan)sin(tancossec.sin

αδφαδααφφα

αδφαδφααα

−−′−′−+′

+−−′−

′−′−+=

FFWkub

FWbcFWZZ h

LR


αδφαδφµαµα

−−′−′−−−

+F

FQ)cos(tan)sin(

]tan)cos()sin([αδφαδ

φβαβα−−′−

′−−−+

FFWW ...(2.25)

While the moment equilibrium equation becomes;

αδδα tan2

tan22

tantan2

bbbZZb

ZZh

ZZh

R

L

R

LL

R

LR −++−=

δδ

µδβ

coscossin

cossin

R

ha

RR

WZ

WkhZhQ

ZhW

−++ ...(2.26)

ZL and hL define the boundary conditions for the first slice and ZR and hR

for the last slice. In many cases, these values are zero. By using assumed

values for the solution parameters, F and (δ), and considering the known

boundary conditions, ZL and hL, it becomes possible to use Eqs. (2.25) and

(2.26) in a recursive manner, slice by slice, and evaluate ZR and hR for the last

slice. The calculated values of ZR and hR at the boundary are compared with

the given values. An adjustment is made to the assumed values of F and (δ),

and the procedure is repeated.

The iterations are terminated when the calculated values of ZR and hR

are within an acceptable tolerance to the known values of ZR and hR at the

boundary.

17

Three-Dimension Stability Analysis:

General:

Most of the three-dimensional methods developed are simplified

methods and are not rigorous, since they either neglect the inter-column

forces or make assumptions that have not been completely verified.

Hovland (1977) determined the three-dimensional factor of safety for

several example problems. The solutions indicated that the three-dimensional

analysis of slopes give factors of safety that are smaller than the two-

dimensional factors of safety for a certain method. Hutchinson and Sarma

(1985) pointed out that the ratio F3/F2 can approach 1.0, but should not fall

below 1.0, where F3 and F2 are the 3-D and 2-D factors of safety,

respectively. Hunger (1987), indicated that, for all cases, the ratio F3/F2 was

greater than 1.0. Chen and Chameau (1983) found that the ratio F3/F2 might

be less than 1.0 at certain circumstances. Cavounidis (1987) concluded that

the F3/F2 ratios must be equal to or greater than unity and methods that give

F3/F2 ratios below the unity are not accurate.

In the next section, Bishop and Janbu simplified 2-D methods

extended to 3-D methods and are implemented in the program SAS-MCT.

These two methods can be used to calculate the 3-D safety factor for any

specific circular slip surface, where the sliding mass is considered spherical,

and the axis of rotation passes through the center of the slip surface.

18

Bishop Method in Three-Dimensions:

The derivation of the three-dimensional algorithm is based on the two

assumptions proposed by Bishop (1955), namely:

1-The Vertical inter-column shear forces are negligible, Figure (2.6).

2-The vertical force equilibrium of each column and the overall moment

equilibrium of the column assembly are sufficient conditions to determine all

the unknowns. Horizontal force equilibrium conditions in both the longitudinal

(Y) and transverse (X) directions are neglected, similar to the Bishop’s two-

dimensional method.

Hutchison (1981) and Hunger (1987) derive the total normal force

acting at the base of each column using the vertical force equilibrium equation

of a single column as follows (see Figure (2.6));

Fig. (2.6) Forces Acting on a Single Column. The Vertical Inter-Column Shear Forces are not Shown.

19

zz SNW += … (2.27) ( )

yz FAC

FAUNNW αφγ sin.tan.cos ⎥⎦

⎤⎢⎣⎡ +

′−+= … (2.28)

Solving Eq. (2.28) for N;

α

αφα

mF

AUF

ACW

N

yy sintan.sin. ′+−

= …. (2.29)

Where;

W= total weight of the column.

A = the base area of the column.

U = pore pressure at the center of column base.

F

m zz

φαγα′

+=tansincos …. (2.30)

The true base area of the column, A, and the local dip of the sliding

surface at a grid point, (γz), are calculated from geometry;

( )yx

yxyxA

αααα

coscossinsin1

.21

22−∆∆= …. (2.31)

21

22 1tantan1cos

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++=

xyz αα

γ …. (2.32)

Where;

∆x, ∆y = the column width and length.

αx and αy = the inclination of the sliding surface in the direction of the

coordinate axes.

20

The factor of safety is derived from the sum of moments around a

horizontal axis, parallel to the x-axis. The moment equilibrium equation of the

column is written as;

( ) 0tan....

..coscos...

=⎥⎦⎤

⎢⎣⎡ ′−+

−++−

∑

∑ ∑ ∑

FRAUNRAC

dQhakWfNxW yz

ϕ

αγ …. (2.33)

( )[ ]

∑ ∑ ∑∑

++−

′−+=

dQhkWfNxW

RAUNRACF

ay

z ..cos

cos...

tan....

αγ

φ …. (2.34)

Where;

R= moment arm of the resisting force.

x= moment arm of column weight.

f= moment arm of the normal force.

ha= moment arm of horizontal earthquake force acting at the mid point

of each column.

k= % of gravity acceleration.

Q= resultant of the applied point load.

d= moment arm of the resultant of the applied point load.

For a rotational surface, the reference axis is also the axis of rotation

and f is zero in each column.

21

Janbu’s Method in Three-Dimensions: It is also possible to derive the factor of safety from the horizontal force

equilibrium in the direction of motion (y-direction). The normal force is derived

from the vertical force equilibrium as in Bishop, Eq. (2-29), while the safety

factor (F) is calculated from the summation of forces in the y-axis as follow,

∑ = 0Fy

0coscos]/./tan).[(sincos 2 =++−− yyyz kWFACFAUNN ααφαγ .... (2.35)

( )[ ]∑ ∑

∑+

′−+=

kWNAUNAC

Fyz

yy

αγαφα

tancoscostan..cos..

…. (2.36)

This is the three-dimensional equivalent of the Janbu’s simplified

method, presented here without a correction factor (Janbu, et al., 1965). For a

cylindrical slope problem, αx equals zero, and γz equals αy. The above

equations reduce to their well-known two-dimensional forms.

These two methods have been incorporated in SAS-MCT 4.0 program.

The program calculates the 3-D safety factor for the critical circular slip

surface located by the 2-D method.

22

CHAPTER III MONTE-CARLO

Technique to Estimate Uncertainty

Introduction

In SAS-MCT software, Monte-Carlo technique was used to estimate

the uncertainty in slope stability analysis. This technique consists of randomly

generating large numbers of expected soil parameters, shear strength, angle

of internal friction and unit weight of the soil, (c, φ, γ). These parameters are

generated in the range of (± 3 × standard deviation) of each mean value to

establish the distribution of the safety factor and the reliability index (β).

The reliability index expresses uncertainty in the stability analysis and

describes safety of slopes by the number of standard deviations separating

the best estimate of the safety factor F from its defined failure value of 1.0.

This approach was coded in SAS-MCT program. The distribution of the

soil parameters (c, φ, γ) can be assumed either normal or log-normal. The

program generates up to 1000 random trials of different expected soil

parameters and calculates the safety factor for each random set. These trials

are used to construct the distribution of the safety factor, corresponding

reliability index, and probability of failure.

UNCERTAINTY IN SOIL PROPERTIES:

There are many factors that cause the uncertainty in the slope stability

analysis. These include the geological details missed in the exploration

program, the estimation of soil properties, which are very difficult to be

quantified, (i.e. the special variability in the field can’t be reproduced

accurately), fluctuation in pore water pressure, testing errors, and many other

relevant factors as well can be considered.

The uncertainties in soil properties can arise from two sources,

Christian, et al. (1994): 1) Scattering in the data, and 2) Systematic errors in

the estimation of the soil properties. The former consists of inherent spatial

23

variability in the properties and random testing errors in their measurement,

while the latter consists of systematic statistical errors due to the sampling

process and bias in the measurement process itself. Figure (3.1) illustrates

how the uncertainties in soil properties can arise as explained before.

Fig. (3.1) Categories of Uncertainty in Soil Properties. After Christian, et al. (1994).

Uncertainty in soil ti

Systematic Error Data Scatter

Bias inMeasurement

procedures

StatisticalError in the Mean

Random Testing Errors

Real Spatial

Variation

24

SAFETY FACTOR DISTRIBUTION:

When the shape of the probability distribution of the factor of safety is

known, the reliability index can thus be related to the probability of failure,

Figure (3.2). In this program, the Monte-Carlo approach is used to achieve the

distribution of the safety factor resulted from the critical slip surface and its

parameters. This approach consists of randomly generating a large number of

the main variables that contribute to the computation of the safety factor (c,

φ, γ). The generation of these expected values mainly depends on the

distribution of each variable(c, φ, γ). In SAS-MCT program either normal or

log-normal can be used and the standard deviation of each variable(c, φ, γ)

should be input in the data file. The estimation of the uncertainty in the factor

of safety and the corresponding reliability index can be explained as follows;

0.0 1.0 2.0 3.0 4.0Reliability index

0.000

0.001

0.010

0.100

1.000

Prob

abilit

y of

failu

re

Fig. (4.2) Reliability index vs. Probability of failure for normally distributed safety factor.

Fig. (3.2) Reliability Index vs. Probability of Failure For Normally Distributed Safety Factor.

25

Normal Distribution Generation:

The normal distribution, also known as the Gaussian distribution, has

a probability density function given by,

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

−=2

21exp

21

σµ

πσxxfX ∞<<∞− x ….(3.1)

Where µ and σ are the parameters of the distribution N (µ,σ ),

A Gaussian distribution with parameters µ = 0.0 and σ = 1.0 is known

as the standard normal distribution and is denoted appropriately as N

(0.0,1.0). The density function, accordingly, is

( ) ( ) 22

1

21 s

sf eS−

=π

∞<<∞− x ….(3.2)

The value of a standard normal variate at a cumulative probability P

would be denoted as,

( )Psp1−= φ ….(3.3)

where

s = Standard variate = σ

µ−x -3 < S < +3

p = Probability (0.0 - 1.0)

x = Expected value. The procedure starts with generating random numbers between 0.0

and 1.0. These numbers represent the probability. The corresponding

standard variate (s) can be computed through solving the numerical

integration of the standard normal density function using any numerical

method then:

σ

µ−=

xs ….(3.4)

( ) σµ ×+= sxE ….(3.5)

E(x) = Expected value based on normal distribution.

26

Fig. (3.3) Safety Factor on Normal Probability Paper.

The SAS-MCT program generates 50 to 1000 random sets of values of

the soil properties (c, φ, γ) as specified by the user. These values are

generated as follow:

( ) cc scE σµ ×+= ...(3.6)

( ) φφ σµφ ×+= sE ...(3.7)

( ) γγ σµγ ×+= sE ...(3.8)

The corresponding safety factors for each set of values are computed

using any of the methods of slices. These safety factors are then drawn on

normal and log-normal probability paper see Fig. (3.3) and Fig. (3.4)

respectively, to get the mean µF and standard deviation σF of these factors.

Then the reliability index and the probability of failure can be calculated as

follows;

F

Fσ

µβ 1−= ...(3.9)

27

( ) ( )βφ −=<= 1FpPF ...(3.10)

or

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛ −=<=

F

FF FpP

σµφ 11 ...(3.11)

where

µF and σF = The mean and the standard deviation of the

safety factor obtained from normal probability

paper.

β = Reliability index.

PF = Probability of failure.

Log-Normal Distribution Generation:

The density function of the log-normal distribution is:

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

2ln

21exp

21

ζλ

πζx

xxfX ∞<≤ x0 ...(3.12)

The standard variate (s) based on log-normal distribution is: ( )

ζλ−

=xs ln

….(3.13)

where ( ) 2

21ln ζµλ −= ...(3.14)

⎟⎟⎠

⎞⎜⎜⎝

⎛+= 2

22 1ln

µσζ ...(3.15)

28

Fig. (3.4) Safety Factor on Log-Normal Probability Paper.

The expected value can be derived by solving Eq. (3.12) for x,

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+×+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+−= 2

2

2

21ln1ln

21lnln

µσ

µσµ sx ….(3.16)

( ) ( )⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+×+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+−= 2

2

2

21ln1ln

21lnexp

µσ

µσµ sxE ….(3.17)

E(x) = expected value based on log-normal distribution.

Based on Log-Normal distribution the reliability index is,

)1ln(

1)(ln

2

2

2

2

F

F

FF

µσ

µσµ

β

+

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

= ….(3.18)

29

First-Order Second-Moment Approximation

This approach was used by the program SAS-MCT in the search for

maximum probability of failure option. In this approach, the distribution of the

safety factor is not essential and the uncertainty in the safety factor can be

measured by its variance, the performance function (safety factor) can be

described as:

exgF += )( …..(3.19)

Where:

g(x) = the method used in the calculation of the safety factor, which depends

on the geometry and soil properties.

e= modeling error.

Expanding this equation in a Taylor series and trimming to the first

terms yield:

[ ] [ ] [ ]∑∑= =

+∂∂

∂∂

≈k

iji

ji

k

jeVarxxC

xg

xgFVar

1 1, …..(3.20)

where:

[ ] =ji xxC , Covariance of the parameters xi and xj and

[ ] [ ]iji xVxxC =, …..(3.21)

if all the xi and xj are uncorrelated for ji ≠ , then

[ ] [ ] [ ]∑=

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

≈k

ii

ieVarxVar

xgFVar

1

2

…..(3.22)

Making the simplifying assumption that uncertainty in the model error

(e) is uncorrelated with the uncertainties in the model parameters xi and xj.

Equation (3.22) uses the variance (i.e second moment), of the variables, and

it is trimmed to the first-order terms. Hence, the approach is called a first-

order, second-moment method.

30

This approach can be used only with the ordinary method of slices.

Since it is easy to make direct differentiation on the safety factor equation.

Direct differentiation on the other methods is very difficult and complicated.

In order to evaluate the reliability index and the corresponding

probability of failure for each slip surface, the solution of the partial derivatives

of Eq.(3.22) is necessary. According to the ordinary method of slices, for a

slope, the safety factor is defined as,

iii

k

i

k

iiiiii

A

AbCF

αγ

φαγα

sin..

tan.cos..sec..

1

1

′

′′+′

=

∑

∑

=

= …(3.23)

The first-order approximation of the variance of the safety factor in

eq.(3.23) is;

)()()()(222

γγ

φφ

′⎟⎟⎠

⎞⎜⎜⎝

⎛′∂

∂+′⎟⎟

⎠

⎞⎜⎜⎝

⎛′∂

∂+′⎟

⎠⎞

⎜⎝⎛

′∂∂

= VarFVarFCVarCFFVar ...(3.24)

Where

∑

∑

=

=

′=

′∂∂

k

iii

k

iii

A

b

CF

1

1

sin..

sec

αγ

α ...(3.25)

∑

∑

=

=

′

′′

=′∂

∂k

iii

k

iii

A

AF

1

1

2

sin..

sec.cos..

αγ

φαγ

φ ...(3.25)

[ ]

2

1

1

)sin..(

)sin.)(tan.cos..sec..()tansin.)(sin..(

∑

∑

=

=

′

′′+′−′′=

′∂∂

k

iiii

k

iiiiiiiiiiii

A

AAbCAAF

αγ

αφαγαφααγ

γ…(3.26)

31

CHAPTER IV

SAS-MCT PROGRAM

Program Features:

Stability Analysis of Slopes using Monte-Carlo Technique (SAS-MCT4.0)

is a computer software programmed in Microsoft® Visual Basic 6.0 capable of

analyzing stability of man-made or natural slopes under static and earthquake

loading. The most critical slip surface and its corresponding safety factor are

evaluated using Monte-Carlo technique and methods of slices. The main frame

of the program is shown in Fig. (4.1). The program aims at solving the following

problems in slope stability analysis:

1- Two-dimensional analysis of any slope stability configuration

assuming circular slip surface, and using either one of the following

methods- as specified by the user:

a- Ordinary method,

b- Bishop’s method,

c- Janbu’s method,

d- Spencer’s method, and e- Morgenstern-Price method (GLE).

The program aims at evaluating the global minimum safety factor using

Monte Carlo technique. In this respect, the program supports a user friendly

interface wizard for file preparation in which the geometry of slopes, layers,

and the properties of materials are encoded. The program also shows the

critical surface searching routine graphically, and locates the most critical slip

surface. See Fig. (4.2).

32

2- The program can be used to search for the critical circular slip

surface as in point (10 above, but based on maximum probability of failure.

First-order second-moment approximation is used to estimate the probability

of failure (pf). This option is valid for ordinary method. See Fig. (4.3).

3- Two-dimensional analysis of slope stability assuming irregular slip

surface using one of the following methods;

a- Janbu’s method,

b- Spencer’s method, and

c- Morgenstern-Price method (GLE).

In this respect, the program searches for the most critical slip surface by

representing every trial surface by 4,5,6, …. to 12 vertices trying to simulate the

shape of the real slip surface. The most critical slip surface corresponding to

these vertices will also be shown graphically. See Fig. (4.4).

4- The program also calculates the safety factor for a specified circular

and non-circular slip surfaces.

5- Seismic slope stability analysis using the Pseudo-static method. The

program computes the reduction in the safety factor due to a specified

acceleration input- expressed in percent of ground acceleration (g).

6- Three-dimensional slope stability analysis using one of the

extensions of Bishop’s or Janbu’s two-dimensional methods. The most critical

slip surface will also be shown in three-dimensional view.

33

7- SAS-MCT program user may perform Reliability Analysis. In this

part, the program generates a large number of different expected soil

parameters assuming either normal or log-normal distribution and calculates

the safety factor for each random set. These trails are used to construct the

distribution of the safety factor and the corresponding reliability index (β) and

the probability of failure Pf.

8- Stability analysis can be conducted using either total or effective

parameters.

9- Stability analysis can be conducted using both SI and British units.

34

Fig. 4.1 Main Frame of SAS-MCT Program.

Generation of N Different Values of Soil Properties c, φ, and γ to estimate the Uncertainty in the calculated Safety Factor.

Circular Non-Circular

Start

Prepare Input File

Specified Slip Surface Search for Min. Factor of Safety


Input Xc,Yc,R Input Vertices See Figure(4.2 and 4.4) See Figure (4.3)

Ordinary Bishop Janbu Spencer Morg-Price

Calculate Factor of Safety

No

Yes

Input Seismic Coefficient

Prepare Output File

No

Yes

Generate New C,Phi,Gama

Prepare Normal Probability Paper

Tables Graphs End

Reliability?

Seismic?

Search for Max. Probability of Failure


35

Fig. 4.2 Search for Critical Circular Slip Surface Based on Min. Safety Factor.

Yes

Yes

Yes

Yes

Loop Until Tolerable Difference between the Values of Safety Factor in the Iterative Procedure. SW=Search Width. Prepare Output File

Tables Graphs End

Start

Search for Critical Slip Surface Based on Min. Safety Factor

Ordinary Bishop Janbu Spencer Morg-Price

Generates the First Slip Surface, Points A, B

Calculate the Safety Factor for this Slip Surface, Fmin = F1

Random Walking Random Jumping & Walking

Random Jumping

Move Point A

Calculate the Safety Factor F

F>Fmin.

Decrease Width of the Search

SW>dmin

Move Point B

Generate a New Slip Surface

i = i + 1

i>2000

Calculate the Safety Factor Fi

Fi>Fmin

Jumping or Jumping &

Walking F=Fmin

Increase Search Width

Fmin.=Fi

Random Jumping Only

Random Jumping and Walking

36

Fig. 4.3 Search for Critical Circular Slip Surface Based on Max. Probability of

Failure.

Yes

Yes

Yes

Yes

Loop Until Tolerable Difference between the Values of Safety Factor in the Iterative Procedure. SW=Search Width. Prepare Output File

Tables Graphs End

Start

Search for Critical Circular Slip Surface Based on Max. Probability of Failure

Generates the First Slip Surface, Points A, B

Calculate the Probability of Failure for this Slip Surface, PfMax. = Pf1

Random Walking Random Jumping & Walking

Random Jumping

Move Point A

Calculate the Safety Factor F

Pf>Max

Decrease Width of the Search

SW>dmin

Move Point B

Generate a New Slip Surface

i = i + 1

i>2000

Calculate the Prob. of Failure

Pf<Max

Jumping or Jumping &

Walking PfMax=Pf

Increase Search Width

PfMax=Pf

Random Jumping Only

Random Jumping and Walking

Ordinary Method (Only)

Note: The Probability of Failure are Calculated Based on First Order Approximation.

37

Fig. 4.4 Search for Critical General Slip Surface.

Start with New Segment

Yes

Yes

Yes

Yes

Prepare Output File

Tables Graphs End

Start

Search for Critical General Slip Surface

Generate the First Slip Surface with Four Vertices n=4

Calculate the Safety Factor for this Slip Surface, Fmin=F1

F>Fmin

Janbu Spencer Morg-Price

i = 1

Rotate Segment i

Calculate the Safety Factor, F

Decrease the Angle of Rotation

D.R>drmin

Increase the Number of Vertices

i = i + 1

nVertices>12

F=Fmin Increase the

Angle of Rotation

nSegment>n-1

D.R = Degree of Rotation. drMin. = Min. Degree of Rotation.

n = Number of Vertices.

i = Number of Segments.

38

Program Description and Organization:

The program is composed of different classes and modules written in

Microsoft® Visual Basic 6.0. The following are the main subroutines and

functions that are used in the program:

1- Subroutine ORDINARY is used to determine the factor of safety by

Ordinary method, i.e., Fellenius method of slices.

2- Subroutine BISHOP is used to determine the factor of safety by simplified

Bishop’s method of slices.

3- Subroutine JANBU is used to determine the factor of safety using Janbu’s

simplified method of stability analysis.

4- Subroutine SPENCER is used to determine the safety factor via Spencer’s

method of stability analysis.

5- Subroutine MORG is used to determine the factor of safety using

Morganstern-Price method of stability analysis.

6- Subroutine 3DBISH is used to determine the three-dimensional safety

factor for the most critical slip surface observed in two-dimensional analysis

using the extension of Bishop’s two-dimensional analysis.

7- Subroutine 3DJANBU is used to determine the three-dimensional safety

factor for the most critical slip surface observed in two-dimensional analysis

using the extension of Janbu’s two-dimensional analysis.

8- Subroutine JANNON is used to calculate the safety factor for non-circular

slip surface using Janbu’s simplified method of stability.

39

9- Subroutine SPENNON is used to calculate the safety factor for non-

circular slip surface using Spencer’s simplified method of stability for non-

circular slip surface.

10- Subroutine DEVIDER is used to increase the number of vertices of the

critical slip surface by putting a new one in the center of the largest

distance between two adjacent vertices. The subroutine checks that the

new vertex satisfies the boundary condition.

11- Function RAN1 is used to generate random numbers extracted from a

uniformly distributed population in the range [0,1].

12- Function FO is used to calculate the correction factor of Janbu’s

simplified method. This function depends on the relative depth of the

landslide in relation to its length and on the nature of the soil properties.

13- Subroutine NORMAL is used to generate (n) random sets of the

parameters (c, φ, and γ) and to calculate the safety factor for every set of

values, where n varies from 50 to 1000.

14- Subroutine AREANOR is used to calculate the standard normal variate

(gasdev) for the (n) cumulative probabilities generated by Subroutine

NORMAL.

15- Subroutine ZNOR numerically integrates the standard normal

distribution equation using Least Square method.

16- Subroutine PAPER is used to construct the normal probability paper.

17- Subroutine GAWS is used to fit the data in the normal probability

paper.

40

CHAPTER V

Illustrative Examples

Example 1

Description:

This example is taken from Stable5M manual and is analyzed using

Janbu’s method. The safety factor obtained using PC Stable5M is 1.371.

Figure (5.1) shows the cross section and Table (5.1) shows the summary

information for this example. The physical properties of soil layers and the

coordinates of the geometry are shown in Tables (5.2 & 5.3) respectively.

Two cases will be conducted for this example. In Case (I) the slope will

be analyzed assuming circular slip surface, while case (II) assumes a

noncircular slip surface.

Fig. (5.1) Slope Geometry.

41

Table (5.1) Summary Information. Item Case I Case II

Objective of the Search

Find Minimum Safety Factor

Method of Search Janbu’s Method Slip Surface Circular Non-Circular Type of Analysis 3-Dimension 2-Dimension Type of Search Random Jumping --- Stress Analysis Type Effective Stress Search Limits (3.5-62.5) m No. of Layers 3 Layers No. of Slices 50 Slices Unit Weight of Water 9.81 kN/mP

3P

Existence of Water Table

Water Table

Storage Type Not Filled Existence of Stiff Layer Stiff Layer (Bearing Stratum) Existence of Cracks No Cracks Reliability Analysis Not Required Seismic Data Static Analysis External Loads No External Loads Table (5.2) Soil Properties.

Layer No. Description φ (P

oP) C

(kN/m P

2P)

γ (kN/m P

3P)

1 Layer 1 0 0.00 18.30 2 Layer 2 14 23.94 18.30 3 Layer 3 14 23.94 19.52

Table (5.3) Geometry. Layer 1 Layer 2 Layer 3

X Y X Y X Y 0.00 20.73 0.00 20.73 0.00 20.736.71 20.42 6.71 20.42 6.71 20.42

11.58 19.20 11.58 19.20 11.58 19.2030.80 26.83 30.80 26.83 19.20 22.5042.07 31.40 62.50 30.18 25.30 23.7862.50 33.53 31.71 25.00

37.20 25.91 42.68 26.52 62.50 28.35

42

File Preparation: Open SAS-MCT program, on the File menu click “New”, the

browser will appear that requests you to specify the path of the file to

be created. Input wizard screen will appear as shown in Figure (5.2).

Fig. (5.2) Input Wizard.

UGeneral TabU: Contains general information about the project. Feel free to

fill in this information (Optional Information). Then, click Next button to move

into objective tab as shown in Figure (5.3).

43

Fig. (5.3) Objective Tab.

UObjective Tab U: In this tab the user will be prompted to specify the

objective of the search and the slip surface mechanism. The objective of

the search will be one the following three options:

1. Minimum Safety Factor: This option can be used when

minimum safety factor is required.

2. Maximum Probability of Failure: First-order second moment

approximation is used to estimate the reliability index (β).

(Note: Maximum Probability of Failure can be used only with

the Ordinary method.)

44

3. Safety Factor for a Specified Slip Surface: This option can be

used when the user wants to calculate the safety factor for a

slope with a predefined slip surface.

For the slip surface mechanism, the user can either use a circular or a

noncircular slip surface. (Note: When calculating the safety factor for a

specified slip surface, the user will be prompted to enter the radius, x-

coordinates, and y-coordinates for the circular slip surfaces. Otherwise,

the x & y coordinates will be defined for the noncircular slip surface.)

In this example, the aim is to find the minimum safety factor with

circular slip surface (Case I) and the minimum noncircular slip surface

(Case II). Click Circular Slip Surface, then, click next button to move into

method tab as shown in Figure (5.4).

Fig. (5.4) Method Tab.

U

45

Method Tab U: Five common methods of stability analysis can be used

for the simulation process, these are:

1. Ordinary Method.

2. Bishop’s Method.

3. Janbu’s Method.

4. Spencer’s Method.

5. Morgenstern-Price’s Method.

When using Morgenstern-Price’s method, the user has to specify the

Morgenstern-Price function option that will be used, see Figure (5.5) and

these options are as follows:

1. Constant Function.

2. Half-Sine Function.

3. Clipped-Sine Function.

4. User-Specified Function.

Note: Ordinary and Bishop’s methods are not valid for noncircular slip

surface analysis.

In this example, use Janbu’s method, then click Next button to move

into options tab as shown in Figure (5.6).

46

Fig. (5.5) Types of Functions Used for Morgenstern-Price’s Method.

47

Fig. (5.6) Options Tab.

UOptions TabU: In this tab the user must specify the desired options such

as:

1. Type of Analysis: 2-dimensional analysis or 3-dimensional

analysis (Note: 3-dimensional analysis is only valid for

circular slip surfaces), in this example case I select 3-

dimensional analysis, and for case II select 2-dimensional

analysis.

2. Type of Search: (only for circular slip surface) The user might

choose one of the three types of search: random jumping,

random walking, or both random jumping and walking. Select

the random jumping option.

48

3. Slices, Layers, and unit weight of water: The user must

specify the number of slices for the slope to be divided into.

The user can choose from 5 to 50 numbers of slices. Also,

the number of soil layers forming the slope must be entered.

Select 50 slices for case I & II. For SI and BS units take the

unit weight of water to be equal to 9.81 kN/mP

3P and 62.4 lb/ftP

3 P,

respectively.

4. Stress Type: The user may use either total stress analysis or

effective stress analysis. Select effective stress for case I &

II.

5. Search Limit: The user should pay some attention for the

search limit, because the critical slip surface will be bounded

by these values. Also, the user must notice that the minimum

value of x should be at least greater than or equal to

minimum x-coordinate of the slope. The maximum value of x

should be less than or equal to maximum x-coordinate. Enter

x-minimum = 3.5m, x-maximum = 62.5m.

6. Both SI and BS units are available; the user must select the

units for the analysis.

Then, click the Next button to move into the geometry tab as shown in Figure

(5.7).

49

Fig. (5.7) Geometry Tab.

UGeometry TabU: In the geometry tab the user enters the x & y

coordinates that define each layer in the slope, noting that layer 1 will be the

upper most one and layer 2 will be directly below it and so on.

In this example, enter the coordinates of the slope geometry as in

Table (5.3). Then, click the Next button to move into the properties tab as

shown in Figure (5.8).

50

Fig. (5.8) Properties Tab.

UProperties TabU: In the properties tab screen, the user will be prompted

to enter the physical properties of the different soil layers (description-

optional, friction angle, cohesion, and unit weight).

In this example, enter these values as in Table (5.2). Then, click the

Next button to move into water table tab as shown in Figure (5.9).

51

Fig. (5.9) Water Table Tab.

UWater Table TabU: In this tab the user must specify whether there is a

water table or not simply by clicking on the check box beside the Existence of

Water Table field. The user may skip this screen if there is no water table.

Otherwise, the user should specify whether it is the water table or the pore

water pressure ratio option. Also, should specify the storage type whether

filled with water or not. When the storage is filled with water, the user will be

prompted to enter the x-intersection of water level with the topography, see

Figure (5.10).

52

Fig. (5.10) Storage Filled with Water.

In this example, no water storage is chosen and water table option is used.

Enter the coordinates of the layer that define the water table as shown in

Figure (5.10). Then, click the Next button to move into Stiff layer tab as shown

in Figure (5.11).

53

Fig. (5.11) Stiff Layer Tab.

UStiff Layer TabU: In this tab the user must specify whether there is a stiff

layer or not simply by clicking on the check box beside the Existence of Stiff

Layer field. The user may skip this screen if there is no stiff layer. Otherwise,

the user should enter the coordinates that define the stiff layer.

In our example, a stiff layer exists and is defined by the coordinates as

shown in Figure (5.11). Then, click the Next button to move into the crack tab

as shown in Figure (5.12).

54

Fig. (5.12) Crack Tab.

UCrack TabU: In this tab, the user is asked to specify whether there is a

cracked layer or not. The user may skip this screen if there is no cracked

layer. Otherwise, the user will be prompted to enter the x-position of crack and

its depth, see Figure (5.13a), and whether it is filled with water or not. It is

worthy to say that if there is a cracked layer (layer with multi cracks), see

Figure (5.13b), the user may skip this screen and consider this layer as a

layer with a friction angle and cohesion equal to zero (i.e.: φ=0, c=0).

55

a- Single Crack.

b- Multi Cracks.

Fig. (5.13) Presence of Cracks.

56

In this example, skip this screen to the reliability tab as shown in Figure

(5.14).

Fig. (5.14) Reliability Tab.

UReliability Tab U: The user may use this screen when reliability analysis

is required, and the number of generation data must be entered. Also, the

standard deviation for the friction angle, cohesion, and unit weight, for each

soil layer, must be supplied.

In this example, reliability analysis is not required, so skip this screen

by clicking the Next button to move into the earthquake tab as shown in

Figure (5.15).

57

Fig. (5.15) Earthquake Tab.

UEarthquake TabU: In the earthquake tab, the user is asked to specify

the seismic data by choosing either static or earthquake analysis. When

earthquake analysis is required, seismic coefficient as a percent of gravity

acceleration should be entered.

In our example, static analysis is conducted by clicking on static

option. Then click the Next button and move into the last tab in the input

wizard, the loading tab, as shown in Figure (5.16).

58

Fig. (5.16) Loading Tab.

ULoading Tab U: Within this tab the user can simulate the external loading

acting upon the slope by choosing either line load or distributed load (see

Figures 5.17 a & b).

In this example, there is no external loading action. Click the Ok button

to finish the file preparation and a message will appear asking the user to

save the current file. Click Yes.

59

a- Line Load Option.

b- Distributed Load Option.

Fig. (5.17) Presence of External Loading.

60

After completing the file preparation, click on the View menu >> Graphs

>> Slope Geometry to see the slope. After that, it is time for slope stability

processing. This can simply occur by clicking on Process menu >> Start or

(press F5). A screen will appear showing that the analysis process is in

progress (see Figure 5.18).

Fig. (5.18) SAS-MCT Running Screen.

61

Viewing Output After the running process is completed, the user can simply view the

output graphs and tables by clicking on the View menu >> Graphs or Tables

>> …

UCase I (Circular Slip Surface): The following Figures (5.19 to 5.23) show the output graphs capability of the

SAS-MCT 4.0 for circular slip surface.

Fig. (5.19) Critical Circular Slip Surface (FS=1.255).

Fig. (5.20) Converging Circular Slip Surfaces.

62

Fig. (5.21) All Tested Circular Slip Surfaces.

Fig. (5.22) 3-Dimensional Circular Slip Surface (FS=1.791).

63

Fig. (5.23) Safety Factor Converging.

UCase II (Non-Circular Slip Surface):U

Similarly, for non-circular slip surfaces the following Figures (5.24 to 5.32)

demonstrate that the software is a powerful and an efficient tool for analyzing

earth slop problems.

Fig. (5.24) Critical Non-Circular Slip Surface (4 Vertices, FS=1.275).

64


Fig. (5. 26) Critical Non-Circular Slip Surface (6 Vertices, FS=1.256).

65



66



67



68

Output Reports Another facility of the SAS-MCT program is printing reports for the input and

the output of the problem. It can be simply used by clicking on the Print

Report button on the tool bar menu or Ctrl+P as a shortcut; the print report

screen will appear as shown in Figure (5.33). It is clear that there are three

sections in the print report screen, Input File, Tables, and Graphs.

The second section is the Tables section that enables the user to print

prepared tables for the calculation processes such as the reliability

calculations, converging iterations, see Figure (5.34), and all iterations done

by the program, and so on. Finally, at the graphs’ section, the user may print

multiple graphs at a time.

The user may print general information for the analyzed slope problem that

can be written in the general section of the input wizard or the first tab. Also

the SAS-MCT program will prepare a summary sheet for the analyzed slope,

see Figure (5.35). (Note: The user may view the summary sheet by clicking

on View menu >> Summary Information or Ctrl+F1 as a shortcut).

Fig. (5.33) Print Report Screen.

69

Fig. (5.34) Converging Iterations (Example 1 Case I).

Fig. (5.35) Summary Information Screen (Example1 Case II).

70

Example 2 This example is taken from Greco, V.R. (1996) “Efficient Monte Carlo

Technique for Locating Critical Slip Surface”, Journal of Geotechnical

Engineering, Vol. 122, No.7, July. Slope geometry is shown in Fig. (5.36).

Physical properties of soil layers and the coordinates of the geometry are

shown in Tables (5.4 & 5.5).

Fig. (5.36) Slope geometry for example 2.

Table (5.4) Geotechnical Parameters for Example 2.

Layer No. c’(kN/mP

2P) φ’( P

οP) γ(kN/mP

3P)

1 49 29 20.38

2 0.0 30 17.64

3 7.48 20 20.38

4 0.0 30 17.64

71

Table (5.5) Cross Section Geometry.

Layer 1 Layer 2 Layer 3 Layer 4

X y x y x y x y

0.0 40.0 0.0 40.0 0.0 35.0 0.0 26.0

60.0 40.0 260.0 40.0 260.0 35.0 260.0 26.0

180.0 100.0

200.0 100.0

260.0 70.0

Several authors analyzed this example and Greco, (1996) summarized their

results along with his results and they are given in Table (5.6) below. This

example is further analyzed using SAS-MCT4.0 program. Different types of

analysis with different cases of loading are used. The results of the analyzed

slope are presented herein after.

Table (5.6) Summary Results using Different Search Methods. Method Factor of Safety

BFGS

DFP

Powell

Simplex

1.423

1.453

1.402

1.405

Pattern Search

Monte Carlo

1.400

1.401

Range of values of safety factors for slip surfaces Method

4 vertices 7 vertices 13 vertices

Pattern Search

Monte Carlo

1.438-1.775

1.437-1.625

1.406-1.421

1.407-1.431

1.4-1.406

1.4-1.413

72

Output for Example No. 2, Case I Circular Slip Surface

82

Output for Example No. 2, Case II Non-Circular Slip Surface with Earthquake Loading

87

Output for Example No. 2, Case III Specified Slip Surface, Line of Thrust

95

Example 3

Karameh dam is situated in the Dead Sea Rift and it is built on deep

soft compressible deposits. Fig. (5.37) presents the cross section. Tables (5.7

and 5.8) show the slope geometry and the soil parameters for the dam.

Fig. (5.37) Cross Section of Karameh Dam.

Table (5.7) Cross Section Geometry.

Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 Layer 6

x Y X y x y x Y x y x Y

0.0 57.0 0.0 57.0 0.0 57.0 0.0 40.0 0.0 39.0 0.0 22.0

83.0 60.0 83.0 60.0 57.0 60.0 140.0 40.0 140.0 39.0 230.0 22.0

143.0 80.0 143.0 80.0 77.0 50.0 145.0 80.0 145.0 34.0

157.0 80.0 157.0 80.0 141.0 50.0 157.0 80.0 157.0 34.0

217.0 59.0 167.0 77.0 145.0 80.0 162.0 39.0 162.0 39.0

230.0 59.0 173.0 42.0 157.0 80.0 230.0 39.0 230.0 39.0

230.0 42.0 162.0 39.0

230.0 39.0

96

Table (5.8) Soil Properties for Karameh Dam Project.

No. Description γ (kN/mP

3P) c’ (kN/mP

2P) φ’

1 Zone 7B Downstream 18.2 80.0 0.0 2 Filter/ Gravel 21.0 0.0 38.0 3 Zone 7B Upstream 18.2 100.0 0.0 4 Core 18.0 40.0 0.0 5 Lower Laminated 16.9 95.0 0.0 6 Lower Clay 18.3 95.0 0.0

This example was analyzed using SAS-MCT 4.0. The results are

shown below.

97

Output for Example No. 3 Circular Slip Surface

104

Example 4 Example 4 is a homogeneous slope with the presence of a weak thin layer.

This example was originally used by Fredlund and Krahn (1977). Fig. (5.38)

shows the cross section of the analyzed slope. Tables (5.9 and 5.10) show

the coordinates of slope and properties of the soil layers, respectively.

Fig. (5.38) Shows the Cross Section for Example 4.

This example is analyzed using SAS-MCT 4.0. Several loading conditions

were considered, such as line load, uniformly distributed load, crack option

etc. Several illustrative plots are enclosed.

105

Table (5.9) Cross Section Geometry for Example 4.

Layer 1 Layer 2 Layer 3 Water Table

x Y x Y x y x y

0.0 6.0 0.0 5.5 0.0 5.0 0.0 6.0

6.0 6.0 48.0 5.5 48.0 5.0 6.0 6.0

30.0 18.0 48.0 12.0

48.0 18.0

Table (5.10) Geotechnical Soil Properties for Example 4.

Layer No. γ (kN/mP

3P) c’ (kN/mP

2P) φ’

1 18.8 29.0 20.0 2 18.8 0.0 7.0 3 18.8 29.0 20.0

106

Output for Example No. 4, Case I Circular Slip Surface

113

Output for Example No. 4, Case II Non-Circular Slip Surface

118

Output for Example No. 4, Case III Circular Slip Surface with Line Load

123

Output for Example No. 4, Case IV Non-Circular Slip Surface, Existence of Crack

128

Output for Example No. 4, Case V Non-Circular Slip Surface, Distributed Load

134

Example 5

This landslide occurred in 1952 during the construction of Congress Street in

Chicago, the north side. Fig. (5.39) shows the cross section of the Congress

street project and Tables (5.11 and 5.12) present the statistical values for the

shear strength parameters and slope coordinates, respectively.

Fig. (5.39) Shows the Cross-Section for Example 5.

135

Table (5.11) Statistical Values for the Undrained Strengths at the Congress

Street Project.

Material Type Mean value

(µ)

Standard

Deviation (σ)

Coefficient of

variation (%)

γ (kN/mP

3P) 12.28 n/a n/a

φBu B ( P

oP) 30.0 0.0 0.0

Sand and

miscellaneous

fill (1) CBu B (kN/mP

2P) 0.0 0.0 0.0

γ (kN/mP

3P) 20.75 n/a n/a

φBu B ( P

oP) 0.0 0.0 0.0

Stiff gritty blue

clay (upper

layer) (2) CBu B (kN/mP

2P) 50.77 25.89 51.0

γ (kN/mP

3P) 20.11 n/a n/a

φBu B ( P

oP) 0.0 0.0 0.0

Medium gritty

blue clay

(middle clay)

(3) CBu B (kN/mP

2P) 29.69 7.72 26.0

γ (kN/mP

3P) 20.11 n/a n/a

φBu B ( P

oP) 0.0 0.0 0.0

Medium gritty

blue clay

(middle layer)

(4) CBu B (kN/mP

2P) 37.36 11.95 32.0

Table (5.12) Slope Coordinates.

Layer 1 Layer 2 Layer 3 Layer 4 Water

Table

x y x y X Y x y x y

0.0 2.8 23.5 13.411 15.9 9.144 7.16 3.05 25.0 14.6

7.16 2.8 40.0 13.411 40.0 9.14 40.0 3.05 40.0 14.6

19.2 11.277

21.03 11.277

28.3 16.760

40.0 16.760

136

Output for Example No. 5, Case I Search Based on Minimum Factor of Safety

(Janbu Method)

146

Example No. 5, Case Ia

Search Based on Minimum Factor of Safety

(Ordinary Method)

153

Output for Example No. 5, Case II Search Based on Maximum Probability of Failure

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