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SLOPE/

WSTUDENT EDITION

Teaching

Guide


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Teaching Guide for the SLOPE/W Student Edition

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CopyrightInformation contained within this document is copyrighted and all rights are reserved by GEO-

SLOPE International Ltd. You may freely reproduce or copy this document in whole or in part,

provided that you include this complete copyright notice and that you do not modify the contents

of this document.

Disclaimer of WarrantyGEO-SLOPE reserves the right to make periodic modifications of this document without

obligation to notify any person of such revision. GEO-SLOPE does not guarantee, warrant, or

make any representation regarding the use of, or the results of, the examples contained in this

document in terms of correctness, accuracy, reliability, currentness, or otherwise.

Teaching Guide for SLOPE/W Student Edition

Copyright 1999

by

GEO-SLOPE International Ltd.

All Rights reserved.


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Table of Contents

Introduction .........................................................................................4The SLOPE/W Student Edition ....................................................................4

Lesson 1 - Ordinary Method of Analysis.....................................................5Lesson 2 Bishops Method of Analysis ................................................... 10

Lesson 3 Janbus Method of Analysis.................................................... 13Lesson 4 Effect of number of slices ..................................................... 15

Lesson 5 Finding the critical slip surface ............................................... 16Lesson 6 Spencers Method ............................................................... 18

Lesson 7 Morgenstern-Price Method..................................................... 20Lesson 8 The Generalized Limit Equilibrium Method.................................. 21

Lesson 9 Non-Circular Slip Surfaces ..................................................... 23Lesson 10 Normal Stress Distributions along a Slip Surface .......................... 24

Concluding Remarks.............................................................................. 24


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IntroductionThis document is directed towards professors who wish to design a slope stability curriculum. It

highlights possible ways that the SLOPE/W Student Edition can be used to teach the

fundamentals of limit equilibrium slope stability analysis. The information is presented here as

10 lessons, any of which you can use in a classroom environment.

The SLOPE/W Student Edition is a software product designed as an aid to learning slopestability analysis. It is an ideal teaching tool for university professors both at the undergraduate

and graduate levels. Using this document as a guide, you can develop a slope stability class that

allows students to solve real slope stability problems in a very short time.

Please note that this document is not intended to present the SLOPE/W interface procedures and

commands; you can use the SLOPE/W online Help for this purpose. If you have never used

SLOPE/W before, it is highly recommended that you complete the detailed tutorial in Chapter 3,

which will provide you with a fairly good understanding of how to use the software. You can also

see Chapters 4, 5 and 6 in the online Help for a detailed SLOPE/W command reference. Since

the online Help is context sensitive, you can highlight any command in the menu and press F1 to

get help on that command. You can also search the entire online Help for specific key words or

use the online Help index.

The data files used for each lesson are provided with this document; they can be found in the

Lessons folder.

The SLOPE/W Student EditionThe SLOPE/W Student Edition is a limited version of the complete full-featured software. It

includes all command names contained in the full-featured edition. Many SLOPE/W commands,

however, are not supported in the Student Edition. Fundamentally, with the Student Edition

you can:

Analyze problems with two different soils plus a bedrock layer

Describe soil using a total unit weight (gamma), cohesion (c) and/or a friction angle (phi)

Specify pore-water pressure conditions with one piezometric line

Examine circular and non-circular slip surfaces

Compute factors of safety using six different methods of slices

The SLOPE/W Student Edition enables students to graphically define and view results on the

screen, allowing them to focus on slope stability fundamentals and not model creation. The

CAD-like graphical user interface makes it possible to define problems on the computer just like

drawing them on paper; the screen becomes the students page and the mouse becomes the

pen. Before analyzing the problem, students can use the Verify command to point out errors

and missing information in the problem definition. Once the problem is analyzed, students can

then view the results in SLOPE/Ws colorful graphical environment, which greatly helps them

interpret, understand and present their solutions.

GEO-SLOPE International, Ltd. provides the SLOPE/W Student Edition free of charge as a

downloadable file from our web site at http://www.geo-slope.com. You can purchase the full-

featured edition of SLOPE/W directly from GEO-SLOPE at an educational discount. For more

information, please visit the GEO-SLOPE web site.


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Lesson 1 Ordinary Method of AnalysisAn introduction to slope stability analysis often starts by evaluating a simple homogeneous slope

using a typical circular slip surface and no pore-water pressure. The potential sliding mass is

divided into slices and a factor of safety is computed on the assumption of no interslice forces.

This is referred to as the Ordinary method in SLOPE/W. Ignoring the interslice forces makes

the factor of safety equation linear (that is, no iterations are required in SLOPE/W).

Consequently, it is relatively simple to compute the factor of safety with a spreadsheet programsuch as Microsoft Excel.

It is highly instructive and educational to calculate the factor of safety manually, slice by slice,

for a few simple examples. Once students have manually done the calculations, they can use

SLOPE/W to verify the results. Going through this exercise also helps to understand the

meaning of the data presented by SLOPE/W.

Consider the simple 1.5H:1V, 6 m high slope shown in Figure 1, with the soil properties listed in

the figure. We are going to analyze a slip surface that intersects the slope toe. The rotation

center is located at coordinate (15, 14), making the slip circle radius 10.198 m. In this case, we

will restrict the slip surface discretization to six slices in order to keep the spreadsheet

calculations to a minimum and still obtain reasonable results.

Figure 1 Simple homogeneous slope analyzed using the Ordinary method

1.476

Unit Weight 18 kN/m3Cohesion 5 kPaPhi 30 degrees

Distance - metres

0 2 4 6 8 10 12 14 16 18 20 22 24

Elevation-

meters

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

In SLOPE/W DEFINE, you can create a perfectly scaled copy of the problem by printing it at a

zoom percentage of 100%. This will result in a scaled printout at 1:100 that students can use to

scale off the mid-height and width of each slice.


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The factor of safety can be computed by completing the following table in a spreadsheet:

Slice B H W W

sin

L N C x L N

tan

1 2.4 2.0 55.1

2 1.8 3.6 37.0

3 1.8 3.4 25.1

4 1.8 2.9 14.3

5 1.8 2.0 4.0

6 1.8 0.7 -6.2

Sum

The meanings of the column headings are as follows:

B = slice width (obtained from the scaled printout)

H = slice mid-height (obtained from the scaled printout)

W = weight

= inclination of slice base

L = sloping length of slice base

C = cohesion

N = normal = W cos

The factor of safety equation is defined as:

F of S = (C L) + ( N tan ) / ( W sin )

You can obtain the summation values in the equation by summing Columns 6, 9 and 10 in the

spreadsheet. This results in a factor of safety equal to 1.47. The SLOPE/W computed factor of

safety using the Ordinary method is 1.476.

You can now verify the forces applied to each slice by using the View Slice Forces command in

SLOPE/W CONTOUR. Once you have selected this command, you can click on any slice and a

free body diagram and force polygon will be displayed. Figure 2 shows the display for Slice 2;

note the absence of interslice forces.


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Figure 2 Free body diagram and force polygon

Slice 2 - Ordinary Method

115.75

43.81

92.439

Included with this diagram is all the information that SLOPE/W has used to compute the factor

of safety. The following list contains the information, including forces, for Slice 2:


Factor of Safety 1.476

Phi Angle 30

C (Strength) 5

C (Force) 11.269

Pore Water Pressure 0

Pore Water Force 0

Pore Air Pressure 0

Pore Air Force 0

Slice Width 1.8

Mid-Height 3.5725

Base Length 2.2539

Base Angle 37.002

Polygon Closure Error 26.622

Anisotropic Strength Modifier 1

Weight 115.75

Base Shear Force 43.81

Base Normal Force 92.439

These SLOPE/W-computed values can be compared with the table values that were used to

compute the factor of safety in the spreadsheet.

At this point, it is important to observe that the force polygon for Slice 2 does not close. It is

particularly bad where the slice base is near horizontal, as shown for Slice 5 in Figure 3. By

ignoring the interslice forces, there is nothing in the analysis to counteract the horizontal


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component of the base shear. Therefore, the slices are not in force equilibrium. This shows that

the Ordinary method can lead to considerable errors in a stability analysis.

Figure 3 Force polygon for Slice 5


63.957

31.079

63.805

You can plot various parameters along the slip surface using the SLOPE/W CONTOUR Draw

Graph command. Figure 4 shows the shear strength and mobilized shear distribution along the

slip surface. Note that the ratio of shear strength to shear mobilized for every slice is a constant

1.47; this ratio is the factor of safety. In other words, the local factor of safety is the same for

each slice and is also the same as the global factor of safety. The graph data in Figure 4 can be

copied to the clipboard and pasted into a spreadsheet to verify that the ratio is a constant.

Figure 4 Shear resistance along the slip surface

Shear Resistance vs. Slice #

Shear

Strength

Shear Mob.

ShearResistance

Slice #

0

10

20

30

40

0 1 2 3 4 5 6

You can extend this simple problem by adding a water table, as shown in Figure 5. It is

relatively simple to do the spreadsheet calculation, even with the added water table. You will

need to insert another column for the pore-water pressure for each slice; the pore-water can then

be included in the shear strength calculation. This example problem will clearly show how

including pore-water pressure in the analysis decreases the resulting factor of safety.


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Figure 5 Introductory problem with water table

1.185


Distance - metres

0 2 4 6 8 10 12 14 16 18 20 22 24

Elevation-meters

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

You could further extend this simple introductory problem by finding the slip surface with the

minimum factor of safety. However, it is probably better to introduce this concept in the next

lesson.

In conclusion, the important points to learn from doing the Ordinary method of analysis are:

1. It is easy to use hand calculations to introduce the basic analysis concept of dividing thepotential sliding mass into slices and then summing the forces on the slices.

2. The Ordinary method only satisfies moment equilibrium.

3. Ignoring the interslice forces means that the individual slices are not in force equilibrium.

4. The failure of the force polygon to close indicates that results from the Ordinary method of

analysis can be in considerable error.


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Lesson 2 Bishops Method of AnalysisThe purpose of Lesson 2 is to show how you can use SLOPE/W to illustrate the effect of including

interslice normal forces in the analysis. Bishops simplified method of analysis will be used,

since it considers the normal forces between the slices but ignores the shear forces. Bishops

method also only satisfies moment equilibrium.

One form of the factor of safety equation for Bishops method is:

+

+

=

sin

tantan1

sec)tan)1((

W

F

rWbc

F

u

This is not the form used in the SLOPE/W formulation, but it illustrates the important point

here thatF(factor of safety) appears on both sides of the equation. This means that the factor of

safety equation is nonlinear and an iterative technique is required to solve forF.

Many textbooks show how to use a spreadsheet table to solve for the Bishop factor of safety,

similar to the approach used in Lesson 1. This technique is also illustrated in the Lam-Whit

example in Chapter 9 of the SLOPE/W online Help. You may want gradate students to do this

as an exercise, but will probably find it inappropriate for undergraduate students.

Figure 6 shows the SLOPE/W results using the Bishop method of analysis for the Lesson 1

problem that included the water table. The first point to notice is the difference in factor of

safety between the two analyses. For the Ordinary method, the factor of safety is 1.185, while

for the Bishop method, the factor of safety is 1.314. This is a significant difference.

SLOPE/W does not display the number of iterations required to reach a solution for Bishops

method. If you are interested, you can view the .FAC output file in a text editor (All SLOPE/W

files are in simple ASCII text format. Chapter 5 of the online Help describes the output file

details). For this simple example, SLOPE/W performed five iterations to get the Bishop factor of

safety.


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Figure 6 Result for Bishop's Method

1.314


Distance - metres

0 2 4 6 8 10 12 14 16 18 20 22 24

Elevation-meters

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

The main reason for the difference in factor of safety between the two methods is that by

including the interslice normal forces, we get much better closure on the force polygons for each

slice. Figure 7 shows the free body diagram and force polygon for Slice 3. Note the normal

forces now acting on the sides of the slice. The force polygon does not close exactly, but it is

much closer than for the Ordinary method. This near closure of the force polygon indicates that

the slice is close to being in force equilibrium.

Figure 7 Free body diagram and force polygon for Bishop's method

Slice 2 - Bishop Method

115.75

48.934

108.1

30.312

51.064


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Another very important point to notice is that the water force acting on the slice base is not

included in the force polygon. The water pressure is used to compute the shear on the slice base,

which is included in the force equilibrium, but the associated water force is not included directly

in the analysis. If the water pressure was not included in the shear force calculation, then the

normal and interslice forces would be different. Therefore, the water pressure is included

indirectly in the analysis through the shear force calculation.

In conclusion, the important points to learn from doing the Bishop method of analysis are:

1. Bishops simplified method of analysis considers the normal forces between the slices but

ignores the shear forces between the slices.

2. Bishops method only satisfies moment equilibrium.

3. Including the interslice normal forces means that Bishops method is close to being in force

equilibrium, as indicated by the force polygon for each slice.

4. The Bishop factor of safety equation is nonlinear, and therefore an iterative technique is

required to solve the equation.


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Lesson 3 Janbus Method of AnalysisSLOPE/W is formulated to solve two factor of safety equations: one with respect to moment

equilibrium and the other with respect horizontal force equilibrium. Remember from Lesson 2

that Bishops method satisfies only moment equilibrium. In the context of SLOPE/W, Janbus

simplified method is identical to Bishops method, except it satisfies only horizontal force

equilibrium. Like Bishops method, Janbus method includes the interslice normal forces but not

shear forces.

Figure 8 shows the SLOPE/W results using the Janbu method of analysis for the Lesson 2

problem. The resulting factor of safety of 1.175 is significantly different from the Bishop value

of 1.314, in spite of the fact that the force polygon closure is quite good, as shown in Figure 9.

The reason for the difference will become clear in a later lesson.

Figure 8 Result for Janbu's method

1.175


Distance - metres

0 2 4 6 8 10 12 14 16 18 20 22 24

Elevation-meters

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14


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Figure 9 Free body diagram and force polygon for Janbu's method

Slice 2 - Janbu Method

115.75

53.213

105.07

30.312

51.064

The important points to learn from doing the Janbu method of analysis are:

1. The Janbu simplified method of analysis is identical to Bishops method, except it satisfies

only horizontal force equilibrium.

2. The Janbu factor of safety can be significantly different from the Bishop value, in spite of the

fact that the force polygon closure is quite good for both methods.


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Lesson 4 Effect of number of slicesIn the previous lessons, we have used only six slices to illustrate some basic concepts and to

compare the SLOPE/W results with hand calculations. In practice, six slices are not enough. If

we re-analyze the problem in Lessons 1-3 using 30 slices (the default number of slices in

SLOPE/W), the factors of safety are as follows:

F of S (6 slices) F of S (30 slices)

Ordinary method 1.185 1.222

Bishops method 1.314 1.328

Janbus method 1.175 1.231

Increasing the number of slices from 6 to 30 does make a difference in the factor of safety.

Increasing the number of slices beyond the default number of 30, however, has very little effect.

You could ask students to try various numbers of slices and plot the factor of safely against the

number of slices as a learning exercise. Generally, SLOPE/W is formulated in such a way that

the results are insensitive to the number of slices, provided that you use at least the defaultnumber of slices as a minimum.

Note that SLOPE/W does not divide the sliding mass into slices with a constant width; the slice

widths will vary. The procedure that SLOPE/W uses to select slice widths is described in the

Theory chapter in the online Help.

The important point to learn from this lesson is:

1. The number of slices used to discretize the potential slip surface can affect the resulting

factor of safety, but once you have a reasonable number of slices, the factor of safety is

insensitive to the number of slices in the SLOPE/W formulation.


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Lesson 5 Finding the critical slip surfaceIn the previous lessons, we have only analyzed one slip surface in order to highlight certain

points regarding slope stability. A more typical need is to examine a wide range of potential slip

surfaces in order to find the one with the minimum factor of safety. This particular slip surface

is known as the critical slip surface. In the SLOPE/W Student Edition, we can do this using the

Grid and Radius slip surface option, as shown in Figure 10. SLOPE/W will analyze 6 potential

circular slip surfaces for each one of the 36 intersection points on the rotation grid, resulting in atotal of 216 trial slip surfaces that will be evaluated.

Figure 10 Problem definition for finding the critical slip surface

1

2

1 2

3 4

5 6

8

9

10

1 1 12

13

14

15 16


Distance - metres

0 2 4 6 8 10 12 14 16 18 20 22 24

Elevation-meters

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Figure 11 shows the critical slip surface that was found. The overall minimum factor of safety is

1.291 (using Bishops method). The minimum factors of safety found at all other Grid center

points have been contoured to assist with the interpretation and presentation of the results.

Note that the minimum value is inside the Grid. This is often used as a guide to indicate that

the minimum factor of safety has been found, and that it does not lie outside the range of

analyzed slip surfaces.


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Figure 11 Critical slip surface search result

1.3

5

1.3

5

1.4

5

1.6

5

1.291

Unit Weight 18 kN/m3Cohesion 5 kPa

Phi 30 degrees

Distance - metres

0 2 4 6 8 10 12 14 16 18 20 22 24

Elevation-meters

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

The position of the critical slip surface is dependent on the soil strength parameters. If the soil

cohesion is zero, the critical slip surface will tend be shallow and parallel to the slope. If Phi ()is zero but cohesion is greater than zero (i.e., the undrained case), the critical slip surface will

tend to be very deep. It is a worthwhile exercise for students to experiment with different

combinations of strength parameters to see the effect on the position of the critical slip surface.

As a broad observation, the position of the critical slip surface will be the most realistic if you use

realistic effective strength parameters.

The important points to learn from this lesson are:

1. The critical slip surface is found by analyzing a wide range of potential slip surfaces and

finding the one with the minimum factor of safety.

2. When the critical slip surface center lies inside the Grid, it is often an indication that the

minimum factor of safety has been found, and that the true minimum does not lie outside the

range of analyzed slip surfaces.

3. The position of the critical slip surface is dependent on the soil strength parameters.


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Lesson 6 Spencers MethodSpencers method considers both normal and shear interslice forces, and satisfies both force and

moment equilibrium. The unique condition in Spencers method is that the ratio of shear to

normal interslice forces is a constant, and is therefore the same for each slice.

SLOPE/W computes one factor of safety with respect to moment equilibrium (Fm) and a second

factor of safety with respect to horizontal force equilibrium (Ff) for various shear-to-normal ratios(this ratio in SLOPE/W is referred to as lambda). The iterative process continues until Fmand Ffare approximately the same. When they are within a specified tolerance, the solution is said to

have converged to the Spencer factor of safety.

Figure 12 shows the SLOPE/W SOLVE window when the Spencers method is used to analyze

the problem from Lesson 5. For Spencers method, the value for Fmis 1.294 and the value for Ffis 1.302. The difference is 0.008, which is within the default allowable tolerance of 0.01. The Fmvalue is actually very close to the Bishop value; the reason for this will be become clear in a later

lesson.

Figure 12 SOLVE window when computing the Spencer factor of safety

If you use the Draw Slip Surfaces command in SLOPE/W CONTOUR, you will see that the value

for lambda is 0.4262, meaning that the interslice shear forces are 0.4262 times the interslice

normal forces. You can check that this is the case by viewing the slice forces or by graphing the

interslice force function, as shown in Figure 13. The specified interslice function is a constant

1.0 for each slice (implied by the Spencer method), and the actual applied function is a constant

value of 0.4262.


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Figure 13 Constant specified and applied interslice force functions

Interslice Force Fn. vs. Distance

Applied Fn.

Specified Fn.

IntersliceForceFn

.

Distance

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 5 10 15

Figure 14 shows a typical free body diagram and force polygon for the Spencer analysis method.Notice that there are now both shear and normal forces on the sides of the slice. On the left side,

the ratio of shear to normal is 12.996/30.496 = 0.426 and on the right side the ratio is

15.135/35.514 = 0.426.

Another important observation is that the force polygon closure is now nearly perfect. Again,

this means that the forces applied on the slice put the slice in near-perfect force equilibrium.

Figure 14 Typical free body and force polygon when using the Spencer method

Slice 8 - Spencer Method

20.075

7.7567

16.929

30.496

12.996

35.514

15.135

The important points to learn about Spencers method are:

1. Spencers method considers both normal and shear interslice forces and satisfies both force

and moment equilibrium.

2. The unique condition in Spencers method is that the ratio of shear to normal interslice forces

is a constant, and is therefore the same for each slice.

3. The force polygon closure for Spencers method is nearly perfect, indicating that the forces

applied on each slice put the slice in near-perfect force equilibrium.


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Lesson 7 Morgenstern-Price MethodThe Spencer method is limited to a constant interslice force function, as described in the previous

lesson. The Morgenstern-Price method is similar to the Spencer method, except it allows you to

specify an interslice force function. The SLOPE/W online Help lists a variety of interslice force

functions that you can choose; in the Student Edition, however, you can only select a constant or

half-sine function.

Figure 15 shows the specified and applied functions resulting from analyzing the Lesson 6

problem using the Morgenstern-Price method. The specified function starts at zero at each end

and peaks at 1.0 near the centre of the slip surface. Since the Lambda value for this analysis is

0.5030, the applied function is 0.5030 times the specified function. In physical terms, this means

that the interslice shear forces are very small relative to the normal forces at the crest and at the

toe of the slip surface. In the middle where the specified function reaches 1.0, the shear to

normal ratio reaches 0.5030. You can check this by viewing the slice forces at the crest, the

middle and the toe of the slip surface.

You will notice that there is virtually no difference in the factors of safety when using the

Spencer method or the Morgenstern-Price method with a half-sine interslice force function. The

reason for this will be explained in the next lesson.

It is worthwhile to note that using the Spencer method is identical to using the Morgenstern-

Price method with a constant interslice force function. Both methods are available so that you

can easily compare them to published literature. Practically, they are the same unless you use a

non-constant interslice force function.

Figure 15 Half-sine specified and applied interslice force functions

Interslice Force Fn. vs. Slice #

Applied Fn.

Specified Fn.

IntersliceForceFn.

Slice #

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25 30

The important points to learn about the Morgenstern-Price method are:

1. The Morgenstern-Price method, like Spencers method, considers both normal and shear

interslice forces and satisfies both force and moment equilibrium. The only difference

between the methods is that the Morgenstern-Price method allows you to specify different

types of interslice force functions.

2. The SLOPE/W Student Edition is limited to two types of interslice force functions: constant

and half-sine.


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Lesson 8 The Generalized Limit Equilibrium MethodThe Generalized Limit Equilibrium (GLE) method embodies the concepts of all other methods.

Using the GLE method is extremely useful for understanding the differences between methods

and for interpreting results.

When you use the GLE method, you must specify a series of lambda values. SLOPE/W computes

Fm and Fffactors of safety for each specified lambda value; you can then plot these factors ofsafety using the Draw Slip Surfaces command in SLOPE/W CONTOUR.

Analyzing the problem in the previous lesson using the GLE method produces a factor of safety

versus lambda graph as shown in Figure 16. Using this graph, you can identify the factors of

safety for several different methods. The Morgenstern Price factor of safety occurs at the point

on the plot where Fm is equal to Ff, since this method satisfies both force and moment

equilibrium. The Bishop method satisfies only moment equilibrium and ignores interslice shear

forces. Since the zero interslice shear condition occurs when lambda is zero, the Bishop factor of

safety therefore lies on the moment equilibrium curve where lambda is equal to zero. The Janbu

simplified method satisfies only force equilibrium and also ignores interslice shear forces; the

Janbu factor of safety is therefore the point on the force equilibrium curve where lambda is equal

to zero.

Figure 16 Factor of safety versus lambda

Factor of Safety vs. Lambda

Moment

Force

Factor

ofSafety

Lambda

1.0

1.1

1.2

1.3

1.4

0.0 0.2 0.4 0.6 0.8

It is important to note the slope of the moment and force equilibrium curves in Figure 16. Themoment curve is essentially flat, while the force curve is at a significant slope. This means that

moment equilibrium is insensitive to interslice shear forces, while force equilibrium is quite

sensitive to interslice shear forces.

Since the moment curve is so flat, the Bishop, Morgenstern-Price, and Spencer factors of safety

are very similar. The Janbu factor of safety is quite different from the rest, however, since it is

based only on force equilibrium. This explains some of the observations made earlier about

factors of safety for the different methods. For more information on this topic, see the Adopting

a Method topic in Chapter 7 of the online Help.


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Generally, moment equilibrium is insensitive to interslice shear forces when the slip surface is

circular and the only applied loading is gravitational force. This may not hold true for

noncircular slip surfaces or when you include highly concentrated line loads such as anchors.

While you cannot apply concentrated loads with the SLOPE/W Student Edition, you should be

aware of it when using the full-featured version in practice.

The important points to learn in this lesson are:

1. The Generalized Limit Equilibrium (GLE) method embodies the concepts of all other

methods. Using the GLE method is extremely useful for understanding the differences

between methods and for interpreting results.

2. The GLE method calculates both moment and force factors of safety for all specified lambda

values.

3. The Morgenstern-Price factor of safety occurs at the lambda value where the moment factor

of safety is equal to the force factor of safety.

4. The Bishop, Morgenstern-Price, and Spencer factors of safety are very similar, since moment

equilibrium is usually insensitive to interslice shear forces. This may not hold true, however,

when you apply highly concentrated line loads or if the slip surface is non-circular.


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Lesson 9 Non-Circular Slip SurfacesThe SLOPE/W Student Edition allows you to analyze non-circular slip surfaces. You can use the

same Grid and Radius technique that have been used in the previous lessons; the only addition

you will need to make is a bedrock layer, which can be added with the KeyIn Soil Properties and

Draw Lines commands. Figure 17 illustrates the shape of the computed slip surface. The slip

surface shape follows the arc of a circle until it intersects the bedrock layer. It then follows the

bedrock surface until it again intersects the slip circle. The soil strength used along the bedrocksurface is the strength of the soil immediately above the bedrock.

Figure 17 Non-circular slip surface

1.373Unit Weight 18 kN/m3Cohesion 5 kPaPhi 30 degrees

Bedrock

Distance - metres

0 2 4 6 8 10 12 14 16 18 20 22 24

Elevation-meters

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

When analyzing non-circular slip surfaces, you will notice that SLOPE/W computes factors of

safety for all specified methods, just like it does for circular slip surfaces. This is true for the

Ordinary and Bishop methods, even though they are traditionally formulated and discussed in

the literature only in terms of circular slip surfaces. The important message here is that the

difference between the methods is notin the shape of the slip surface; the difference is in the

interslice force assumptions and what equilibrium equations each method satisfies. Therefore,

the Bishop method can be applied to non-circular slip surfaces just like the Janbu, Spencer, or

Morgenstern-Price methods.

The important points to learn in this lesson are:

1. Non-circular slip surfaces can be included in the SLOPE/W Student Edition by adding a

bedrock soil layer.

2. You can obtain factors of safety for any analysis method using non-circular slip surfaces.


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Lesson 10 Normal Stress Distributions along a SlipSurfaceA key unknown in limit equilibrium analysis is the normal stress at the base of each slice. In

fact, the normal stress distribution along the slip surface is dependent on the analysis method.

To illustrate these effects, it is useful to plot the normal stresses for the various methods. You

can do this by first graphing the normal stress versus slice number for each method in

CONTOUR. Then, for each method, copy the data to the clipboard and paste it into a

spreadsheet. Once the data is in the spreadsheet, you can delete duplicate columns of slice

numbers and then graph the normal stresses versus slice number for the various methods.

This is a fairly advanced exercise that you will likely only assign at the graduate level.

Concluding RemarksThis document has been designed to illustrate how you can use the SLOPE/W Student Edition to

assist you in teaching the fundamentals of limit equilibrium slope stability analysis. Once you

have gone through these exercises, you will likely develop your own ideas of how to use the

Student Edition in a teaching environment. If your educational institution is involved in

research or advanced studies involving limit equilibrium methods, you will likely want to acquire

the full-featured version of SLOPE/W.

Chapter 8 in the SLOPE/W online Help presents detailed information on limit equilibrium

theory and on the fundamentals used in the SLOPE/W formulation. You may find this

information useful for teaching limit equilibrium fundamentals and for applying the SLOPE/W

Student Edition.

We welcome your feedback on the SLOPE/W Student Edition. Please submit your comments via

email to [email protected].

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