SAS_Manual.pdf

Jordan University of Science and Technology

USER’S MANUAL

FOR

SAS-MCT 4.0

A Computer Program for

S

tability

A

nalysis of

S

lopes

Using

M

onte

C

arlo

T

echnique

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

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.

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.

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

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- 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.

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.

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.

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.

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.

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.

F N b c S_m = ′. + ′tanφ

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.

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 =

∑

FH , Smcosα−N′sinα−U.bsinα−khW +Qsinµ+Wwsinβ =0 …(2.1)

0 =

∑

FV , W −Smsinα −N′cosα−Ubcosα +Qcosµ+Wwcosβ =0 ...(2.2)

Solving for N’ ) cos( ) cos( sin cosα − α− + β −α + µ−α = ′ W k W Ub W Q N _{h w} ...(2.3)

Fig. (2.3) Inter-Slice Forces for the Ordinary Method (ZR3 is not equal ZL4).

The Mohr-Coulomb mobilized shear strength, Sm , along the base of each slice is given by;

F N b c

S_m = ′. + ′tanφ′ ...(2.4)

c’ and N′tanφ 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;

M_o =

∑

0

∑

= + + = n i w Q R W W 1 sin ) cos cos ( β µ α

∑

= − + − n i W Q R h W 1 ) cos )( sin sin ( β µ α

∑

∑

= = − + − n i a h n i mR K W R h S 1 1 ) 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 a h W w h n i w R h W k R h Q W Q W W b U W k W Q W b c F 1 1 1 1 ) (cos ) )(cos sin sin ( sin ) cos cos ( tan ] . sin ) cos( ) cos( cos [ sec . α α µ β α µ β φ α α β α µ α α …(2.6)

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),

∑

F_V = 0 ⎥⎦ ⎤ ⎢⎣ ⎡ ₋ ′ _{− + +} = ′ α β µ α cos cos tan 1 Q W ub F b c W m N _W …(2.7) where: F

m_α =cosα+sinαtanφ′ …(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,

∑

M_o =

∑

∑

= = − + − + + n i n i W W Q R W Q R h W W 1 1 ) cos )( sin sin ( sin ) cos cos ( β µ α β µ α 0 )] cos ( [ 1 1 = − + −

∑

∑

= = n i a h n i mR k W R h S α ...(2.9)

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

∑

∑

∑

∑

= = = = − + − + − + + ′ ′ + ′ = n i n i a h W W n i n i R h W k R h Q W Q W W N b c F 1 1 1 1 ) (cos ) )(cos sin sin ( sin ) cos cos ( ) 0 tan sec ( α α µ β α µ β φ α

…(2.10)

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;

∑

F_V = 0

µ β

α

α sin cos cos

cos ) (N′+ub +S_m −W −W_W −Q = … (2.11) α µ β α α cos cos cos sin cos S W W Q ub N′= − − m + + W +

…(2.12) Substituting the value of Sm ,

α µ β α α m Q W ub F b c W N _W ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ′ _{− + +} − =

′ sin cos cos cos ….(2.13)

where

F

m_α =cosα+sinαtanφ′ ….(2.14)

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

∑

F_H = 0

α µ

β

α sin sin cos

sin )

(N′+ub −Wk_h+W_W +Q +S_m

−

= ….(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 i W h H F N b c Q W Wk ub N F 1 1 1 cos tan sin sin sin ) ( α β µ φ α …(2.16)

∑

∑

∑

= = = ′ + − − + ′ ′ + ′ = n i W h n i n i N Q W Wk ub N b c F 1 1 1 sin ) sin sin sin ( cos ) tan sec ( α µ β α α φ α ….(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:

Fcorrected = fo . F

….(2.18)

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

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.

Force Equilibrium

The summing of forces along and normal to the base of the slice are as follows; ) cos( ) cos( sinα+ δ −α − δ −α = _{R R L L} m W Z Z S

+Wk_hcosα −W_Wsin(β −α)−Qsin(µ−α)

….(2.19)

) sin( ) sin( cosα + δ −α − δ −α = ′ W Z_{R R} Z_{L L} N

−Wk_hsinα −ubsecα+W_W cos(β−α)+Qcos(µ−α)

….(2.20)

From Mohr-Coulomb theory

F N b

c

S_m = ′. secα+ ′tanφ′

….(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( tan cos sec . sin ) cos( tan ) sin( ) cos( tan ) sin( α δ φ α δ φ α α α α δ φ α δ α δ φ α δ − − ′ − ′ − ′ − + − − ′ − − − ′ − = R R L R R L L R F W b c FW Z F F Z

) cos( tan ) sin( ] tan ) cos( ) sin( [ ) cos( tan ) sin( cos ) tan tan ( tan sec α δ φ α δ φ α β α β α δ φ α δ α α φ φ α − − ′ − ′ − + − − − − ′ − ′ + + ′ + R R W R R h F F W F F Wk ub ) cos( tan ) sin( ] tan ) cos( ) sin( [ α δ φ α δ φ µ α µ α − − ′ − ′ − + − − R R F F Q ….(2.22)

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,

∑

M_c = 0 µ β δ δ α δ sin sin 2 sin sin 2 ) tan 2 ( cos h b Z b Z b hW hQ Z_{L L L}− + _{L L}+ _{R R} + _W + 0 ) tan 2 ( cos + = − −Wk_hha Z_R δ_R h_R b α ….(2.23)

Solving Eq. (2.23) for hR;

α δ δ δ α δ δ δ δ tan 2 tan 2 2 cos sin tan 2 cos cos cos cos 2 b b b Z Z b Z Z h Z Z h R L R L R L R L L R L R L R = − + + − R R a h R R R R W Z h Wk Z hQ Z hW δ δ µ δ β cos cos sin cos sin − + + ….(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;

) ( . tanδ =λ f x

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).

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;

)

(for all slices

L R δ δ

δ = =

The force equilibrium equation becomes;

) cos( tan ) sin( cos ) tan tan ( tan sec ) cos( tan ) sin( tan cos sec . sin α δ φ α δ α α φ φ α α δ φ α δ φ α α α − − ′ − ′ − + ′ + − − ′ − ′ − ′ − + = F F Wk ub F W b c FW Z Z_{R L} h ) cos( tan ) sin( ] tan ) cos( ) sin( [ α δ φ α δ φ µ α µ α − − ′ − ′ − − − + F F Q ) cos( tan ) sin( ] tan ) cos( ) sin( [ α δ φ α δ φ β α β α − − ′ − ′ − − − + F F W_W ...(2.25)

While the moment equilibrium equation becomes;

α δ δ α tan 2 tan 2 2 tan tan 2 b b b Z Z b Z Z h Z Z h R L R L L R L R = − + + − δ δ µ δ β cos cos sin cos sin R h a R R W Z W k h Z hQ Z hW _{+ −} + ...(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.

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.

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.

W = N_z+S_z … (2.27)

(

)

y z F A C F A U N N

W = cosγ +_⎢⎣⎡ − . tanφ′+ . _⎥⎦⎤sinα … (2.28) Solving Eq. (2.28) for N;

α α φ α m F A U F A C W N y y . tan sin 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_α =cosγ_z +sinαztanφ′ …. (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;

(

)

y x y x y x A α α α α cos cos sin sin 1 . 2 1 2 2 − ∆ ∆ = …. (2.31) 2 1 2 2 1 tan tan 1 cos ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + + = x y z _{α α} 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

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;

(

)

0 tan . . . . . . cos cos . . . = ⎥⎦ ⎤ ⎢⎣ ⎡ + − ′ − + + −

∑

∑

∑

∑

F R A U N R A C d Q ha kW f N x W _{z y} ϕ α γ …. (2.33)

(

)

[

]

∑

∑

∑

∑

+ + − ′ − + = d Q h kW f N x W R A U N R A C F a y z _{. .} cos cos . . . tan . . . . α γ 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.

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,

∑

Fy = 0 0 cos cos ] / . / tan ) . [( sin cos _{z y}− N−U A F+C A F 2 _y +kW _y = N γ α φ α α .... (2.35)

(

)

[

]

∑

∑

∑

+ ′ − + = kW N A U N A C F y z y y α γ α φ α tan cos cos tan . . 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.

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

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 in Measurement procedures Statistical Error in the Mean Random Testing Errors Real Spatial Variation

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 4.0 Reliability index 0.000 0.001 0.010 0.100 1.000 Prob ab ili ty of fail ure

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.

Normal Distribution Generation:

The normal distribution, also known as the Gaussian distribution, has a probability density function given by,

( )

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = 2 2 1 exp 2 1 σ µ π σ x x f_X −∞<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

( )

( )

2 2 1 2 1 s s f_S =

e

− π −∞<x<∞ ….(3.2)

The value of a standard normal variate at a cumulative probability P would be denoted as,

s_p =φ−1

( )

P ….(3.3) where

s

_{= Standard variate = σ}x−µ -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 − = x s ….(3.4) E

( )

x =µ+s×σ ….(3.5)

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:

E

( )

c =µ_c+s×σ_c ...(3.6)

E

( )

φ =µ_φ +s×σ_φ ...(3.7) E

( )

γ =µ_γ +s×σ_γ ...(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

µ

F and standard deviation

σ

F of these factors.

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

F

µ

(

<

)

=φ

( )

−β = p F 1 P_F ...(3.10) or

(

)

_⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − = < = F F F p F P σ µ φ 1 1 ...(3.11) where

µ

F and

σ

F = The mean and the standard deviation of the

safety factor obtained from normal probability paper.

β

= Reliability index.

P

F = Probability of failure.

Log-Normal Distribution Generation:

The density function of the log-normal distribution is:

( )

( )

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − = 2 ln 2 1 exp 2 1 ζ λ π ζ x x x f_X 0≤ x <∞ ...(3.12)

The standard variate (

s)

based on log-normal distribution is:

( )

ζ λ − = x s ln

….(3.13) where

( )

2 2 1 ln µ ζ λ = − ...(3.14)

_⎟⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = 2 2 2 1 ln µ σ ζ

...(3.15)

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 2 1 ln 1 ln 2 1 ln ln µ σ µ σ µ s x ….(3.16)

( )

( )

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + × + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − = 2 2 2 2 1 ln 1 ln 2 1 ln exp µ σ µ σ µ s x E

….(3.17)

E(x) =

expected value based on log-normal distribution.

Based on Log-Normal distribution the reliability index is,

) 1 ln( 1 ) ( ln 2 2 2 2 F F F F µ σ µ σ µ β + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + = ….(3.18)

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:

F = g(x)+e …..(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 i j i j i k j e Var x x C x g x g F Var 1 1 , …..(3.20) where:

[ ]

x_i x_j =

C , Covariance of the parameters xi and xj and

C

[ ]

x_i,x_j =V

[ ]

x_i …..(3.21) if all the xi and xj are uncorrelated for i≠ , then j

[ ]

_∑

[ ]

[ ]

= + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ ≈ k i i i e Var x Var x g F Var 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 order terms. Hence, the approach is called a first-order, second-moment method.

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,

i i i k i k i i i i i i A A b C F α γ φ α γ α sin . . tan . cos . . sec . . 1 1 ′ ′ ′ + ′ =

∑

∑

= = _…(3.23)

The first-order approximation of the variance of the safety factor in eq.(3.23) is; ) ( ) ( ) ( ) ( 2 2 2 γ γ φ φ ⎟⎟_⎠ ′ ⎞ ⎜⎜ ⎝ ⎛ ′ ∂ ∂ + ′ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ′ ∂ ∂ + ′ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ′ ∂ ∂

= Var C F Var F Var

C F F Var ...(3.24) Where

∑

∑

= = ′ = ′ ∂ ∂ k i i i k i i i A b C F 1 1 sin . . sec α γ α ...(3.25)

∑

∑

= = ′ ′ ′ = ′ ∂ ∂ k i i i k i i i A A F 1 1 2 sin . . sec . cos . . α γ φ α γ φ ...(3.25)

[

]

2 1 1 ) sin . . ( ) sin . )( tan . cos . . sec . . ( ) tan sin . )( sin . . (

∑

∑

= = ′ ′ ′ + ′ − ′ ′ = ′ ∂ ∂ k i i i i k i i i i i i i i i i i i A A A b C A A F α γ α φ α γ α φ α α γ γ …(3.26)

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).

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.

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.

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

Circular Non-Circular

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 End Graphs Reliability?

Seismic?

Search for Max. Probability of Failure

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 End Graphs 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

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 End Graphs 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.

Fig. 4.4 Search for Critical General Slip Surface.

Start with New Segment

Yes

Yes

Yes Yes

Prepare Output File

Tables End Graphs 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.

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.

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

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.

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

3

P

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 o P ) C (kN/mP 2 P ) γ (kN/mP 3 P ) 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 20.73 6.71 20.42 6.71 20.42 6.71 20.42 11.58 19.20 11.58 19.20 11.58 19.20 30.80 26.83 30.80 26.83 19.20 22.50 42.07 31.40 62.50 30.18 25.30 23.78 62.50 33.53 31.71 25.00 37.20 25.91 42.68 26.52 62.50 28.35

™ 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).

Fig. (5.3) Objective Tab.

U

Objective TabU: 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.)

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

Method TabU: 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).

Fig. (5.6) Options Tab.

U

Options 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 2-3-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.

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

3 P 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).

Fig. (5.7) Geometry Tab.

U

Geometry 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).

Fig. (5.8) Properties Tab.

U

Properties 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).

Fig. (5.9) Water Table Tab.

U

Water 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).

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).

Fig. (5.11) Stiff Layer Tab.

U

Stiff 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).

Fig. (5.12) Crack Tab.

U

Crack 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).

a- Single Crack.

b- Multi Cracks.

In this example, skip this screen to the reliability tab as shown in Figure (5.14).

Fig. (5.14) Reliability Tab.

U

Reliability TabU: 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).

Fig. (5.15) Earthquake Tab.

U

Earthquake 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).