BG Manual 07

IER

BGSLOPE 6.16

c'

P O LY G

ONAL

COMP OSITE F I X E D PLAN E

Fs=1.422

1.0

f(x)

by Dr Milan Maksimoviæ

f(x)

Fs=1.415

optional NENVE

1.0

Fs=1.417

1.0

f(x)

Slope Stability for PC

IRCL E

2006

C

BEZ

B G S L O P E 6.16 (ver. 2006) SLOPE STABILITY SOFTWARE FOR PERSONAL COMPUTERS By Milan Maksimović ABSTRACT The package for IBM-PC and compatibles performs 2D limit equilibrium analyses, of non-homogeneous sections of soil and rock slopes with arbitrary distribution of pore water pressures, external loading and inertia forces due to earthquake acceleration in any direction, by the methods of slices, considering arbitrary and circular slip surfaces. The shearing strength can be defined with the standard linear Coulomb or the nonlinear failure envelope in terms of effective stresses. The undrained shear strength can vary linearly with depth in each material zone. Circular slip surfaces are analyzed by the extended Bishop's method. An automatic search option, using the method of the steepest descent, can be selected for determination of the critical slip circle as well as the option using the grid of centers with the range of passing points which may automatically define more than 3000 slip circles in the single run. Arbitrary slip surfaces are analyzed by the General method that satisfies all equilibrium conditions. Five arbitrary shape types of slip surfaces can be generated as polygonal, circular, composite, fixed plane shape and Bezier curves. The fully interactive and fast search procedure for the critical slip surface of the arbitrary shape is used. The back analyses of the failed slopes and landslides can be performed in a single run. Programs are menu driven; variety of branching in the data preparation stage and computation can be carried out in an interactive work, including graphic presentation of cross sections, slip surfaces, line of thrust, etc., on monitor and hard copy graphic. Program exports HP-GL (.plt) files for graphic presentation. Running of the programs is made as simple as possible without any need for the programming knowledge. User, the geotechnical engineer, can see what he/she is doing, in each step of the analysis using all the user-friendly features of the package. The geometry of the cross section can be described with up to 100 lines, up to 25 material zones, up to 10 surcharge loads and up to 20 line loads in any direction. Arbitrary slip surfaces can be defined with up to 100 points, either automatically generated (Bezier or composite slip surfaces), or entered explicitly (polygonal slip surface). The number of slices is initially specified by a user up to 100, (default is 40), though the software can handle up to 150 automatically generated slices.

I

BGSLOPE 6.16 SLOPE STABILITY SOFTWARE FOR PERSONAL COMPUTERS LIST OF CONTENTS Page 1. INTRODUCTION

1

2. GETTING STARTED

3

3. GENERAL FEATURES OF THE PACKAGE

9

4. SOIL and ROCK STRENGTH and THE FACTOR OF SAFETY

12

5. METHODS OF THE ANALYSIS

22

5.1 PROGRAM BE.EXE

23

5.2 PROGRAM GE.EXE

29

6. PREPARATION OF INPUT DATA

32

6.1 MAIN DATA

33

7. USING BE.EXE

46

7.1 RESULTS FROM BE.EXE

67

8. USING GE.EXE

71

8.1 RESULTS FROM GE.EXE

91

9. MORE ON MENU OPTIONS - CHANGES

96

10. EXAMPLES OF STABILITY COMPUTATIONS

104

11. PROGRAM BGP.EXE

129

11.1 MENU

129

11.2 NOTES ON GRAPHIC POST-PROCESSING

134

11.3 EXAMPLES

134

Addendum No. 1 HANDLING TENSION CRACK

149

REFERENCES

152

INDEX

155

II

1. INTRODUCTION

BGSLOPE 6.16 SLOPE STABILITY SOFTWARE FOR PERSONAL COMPUTERS 1. INTRODUCTION BGSLOPE is a software package written for PC and compatibles for the general solution of slope stability problems by a two-dimensional limiting equilibrium methods. The programs were originally developed for the DOS operating system, although accessible from the WINDOWS environment. Programs can handle arbitrary non-homogeneous cross sections, external loading including seismic accelerations and the nonlinear failure envelope, in addition to the conventional Coulomb failure law for soils, rock mass and rock discontinuities. The application is rather wide and covers the problems of stability of embankments, cuts, landslides, earth dams, mine-tailings dams, soil and rock slopes in open pit mining, deep excavations, retaining structures and bearing capacity problems in nonhomogeneous and/or inclined soil. The development of this package started in l968 for main frame computers and that phase of application and the development has practically ended in 1979. The early version of the package for PC (Ver. 1.0) is briefly described by author (Maksimović, 1988). Software was continuously upgraded, and the present version BGSLOPE 6.16 is the results of development completed in year 2006, after 37 years of the development and application in hundreds of different projects worldwide. Some improvements were suggested by users and other envisaged and introduced by writer during intensive use in many projects. The software package BGSLOPE 6.16 consists of the setup program, three main programs and five supporting files: (0) BGSETUP.EXE Setup program performs initialization of five supporting files and sets parameters for your hardware configuration. You can start any of the main programs from here after saving configuration. You will not need this program again unless you change your hardware configuration, if at all. Main programs are: (1) BE.EXE (Bishop Extended method), uses CIRCULAR slip surfaces and handles the grid or automatic search option, or interactive search for finding the critical slip circle. (2) GE.EXE (GEneral method) is used for the analysis of ARBITRARY slip surfaces which can be generated as POLYGONAL, BEZIER CURVES, FIXED PLANE, COMPOSITE, and CIRCULAR, using the method developed by author, (3) BGP.EXE exports HP-GL files for graphic presentation of results by other programs which can handle *.plt files and produce graphs using printer for presentation of results obtained by programs (1) and (2). It can produce text files with final results transferred or saved by BE.EXE and GE.EXE.

1

1. INTRODUCTION

In this text two programs (1) BE.EXE and (2) GE.EXE, which actually perform computations, are called stability programs. You can shift from one to another main program transferring main data without exiting to DOS or WINDOWS as shown in Fig. 1.1, which indicates the possible shifting from one program to another. That means that the whole package works as a single larger program. Running of the package can be started from any of the four programs i.e. BGSETUP.EXE, BE.EXE, GE.EXE or BGP.EXE. In most practical cases in the routine analyses you will usually start from BE.EXE or GE.EXE.

Fig. 1.1 Switches and flow of data and results An optional software package NENVE may be included. This package handles results of soil shear strength tests (direct shear or triaxial). It is used for derivation of parameters of the nonlinear failure envelope of hyperbolic type proposed by writer (Maksimović, 1988, 1996-a,b). Appropriate curve fitting techniques are programmed and graphic presentation of results is included. Besides, the optional NENVE package contains a database of collected results published in international literature and options to convert some other forms of the nonlinear failure envelopes like Hoek-Brown, Barton, and power type strength envelopes. Processing of data is interactive and enhanced to a level of final presentation of graphs on the screen and hard copy can be obtained using HP-GL graphic format. Some examples of the nonlinear failure envelopes are presented in Section 4 of this Manual. Note: In this manual it is not possible to give examples for all the imaginable combinations that might occur in the practical application. Intelligent user will note, after certain short practice, that the use of these programs is rather simple and that programs work efficiently if the physical and the soil mechanics aspects of the problem are properly defined. This manual is not a textbook. In the case of difficulty, user is advised to examine given examples in more detail, and/or to consult some references on the subject.

2

2. GETTING STARTED

2. GETTING STARTED You can install the BGSLOPE package on either a hard disk or floppy-disk system. The graphic card VGA is required in order to use graphics. Working from hard disk is highly recommended. Create a new directory (say BGSLOPE) on your hard disk and copy all the files from directory BGSLOPE in this directory on the hard disk. Initially, BGSLOPE is configured by assuming that you have an IBM-VGA color card and the appropriate color monitor. Only if these assumptions are not valid for your system configuration, you should run BGSETUP which will display the following menu: 1 GRAPHIC MODE - CURRENT is VGA-COLOR (640x480) 30 ROWS 2 ORIGIN ( X0 = 320 Y0= 240 ) 3 HEADINGS ARE 1: (Left blank in this Manual) 2: Manual-examples 3: (Left blank in this Manual) Cap. start is 4: Fig. 4 VIEW-CHECK IF GRAPHIC MODE IS ADEQUATE 5 SAVE CONFIGURATION 6 R U N BE.EXE... (Bishop method) 7 R U N GE.EXE... (General method) 8 R U N BGP.EXE... (*.PLT & Printing ) 9 QUI T CHOICE ? 1 GRAPHIC MODE.... The graphic mode, best available for your system and the software package, (640 x 480 pixels) will be selected automatically. The package is delivered with the blue background color. You can select some other background color (black, green, cyan, red, magenta or brown) from the sub-menu offered after you have initiated your graphic mode. 2 ORIGIN... Initially, the origin for the graphic presentation of cross sections and slip surfaces is placed automatically in the central point of the screen, as shown in Fig. 2.1 and the scale is taken as 10 (pixels horizontal size per meter). The initial position of the origin can be easily changed by using this ORIGIN option. Note that the origin of the screen is in the top left corner of the visible area of the screen. The sub-menu offers the following choices: SELECT THE POSITION OF ORIGIN 1 CENTER – DEFAULT 2 DOWN – CENTER 3 DOWN – LEFT 4 MID-HEIGHT – LEFT 5 OTHER

(See Fig.2.1, p.5) (See Fig.2.2, p.5) (See Fig.2.3, p.6) (See Fig.2.4, p.6)

CHOICE ?

3

2. GETTING STARTED

The conventional position of the origin using option 3 (DOWN - LEFT) is shown in Fig. 2.3. You may select any OTHER position of the origin, but the position of the origin must remain within the visible area of the screen. Program will check your entry for the new ORIGIN and will force its position within the permissible area if necessary. During the initial stage of the application of the package you may accept the default position, because the position of the origin as well as the scale can be easily changed later in any of the other main programs to suit the need in any particular circumstances using MOVE/ ZOOM ... option available in all three programs. 3 HEADINGS... Option permits the entry of three headings which will appear in the top right corner of the graph produced by the program BGP.EXE and the beginning of the figure caption (Cap. start) as the fourth. In this Manual, the first heading is left empty, the second is Manual-examples, the third is empty, and the Caption start is Fig.. For the beginning you may wish to leave all headings as they are, or enter empty strings, or enter your personal name, etc. If you press 3 and , the following sub-menu will appear: HEADINGS ARE 1: (Left blank, could be yours or company name) 2: Manual-examples(should be replaced or left blank) 3: (Left blank in this Manual) Cap. start is 4: Fig. OK (Y/N) ? If you enter Y or y or just press , that would mean YES, it is OK, confirming that you wish that these four labels on your drawings produced by BGP.EXE. If you enter N or n, that would mean NO, it is NOT OK for these four labels and the possibility to change any of these, one by one, is offered in this form: CHANGE No. (1/ 2/ 3/ 4) : ? Press the number of the heading which you wish to enter or change and , type the corresponding new heading and . Again you will have on the screen: OK (Y/N) ? The new loop starts until you make all the changes which you wish answering to respond to OK (Y/N) ? with Y or y. At any rate, these 4 headings can be also entered or changed in program BGP.EXE, (Section 11) in some later stage during the use of the program for solving particular tasks.

4

2. GETTING STARTED

Fig. 2.1 The default position of origin

Fig. 2.2 Down - centre position of the origin 5

2. GETTING STARTED

Fig. 2.3 Down - left position of the origin

Fig. 2.4 Mid-height - left position of the origin 6

2. GETTING STARTED

4 VIEW-CHECK... Performs a simple check of graphics. The screen should show a graph similar to one of those shown in Fig. 2.1, to Fig. 2.4. The geometry of simple embankment should be shown with multiple colored lines if you are using color monitor. If OK, the system of coordinate axes should appear on the screen and the local screen coordinates will be shown in the top left as values X0 and Y0. Entered headings previously described in 3. (if any) will appear in the top right corner of the screen and Cap. start will be shown in the bottom left. 5 SAVE... Save your configuration. This procedure will generate five supporting files with the names INI.* with different extensions, as follows: INI.BG,

INI.BGP,

INI.PLT,

INI.SS and INI.TXT

If you had run BGSETUP from your floppy, you can copy all INI.* files in the newly opened directory of your hard disk, or floppies. You will probably not need to run BGSETUP again on your system, 6. RUN BE.EXE... Starts the program based on Bishop Extended method using last options previously saved in INI.* files and the INITIAL MENU (BE) of BE.EXE is offered for further action. It is unlikely that you will use this option frequently, as you will usually start running the stability programs directly from WINDOWS (or rarely from DOS). 7. RUN GE.EXE... Starts the program based on General Extended method using last options previously saved in INI.* files and the INITIAL MENU (GE) of GE.EXE is offered for further work. It is unlikely that you will use this option frequently, as you will usually start running the stability programs directly from WINDOWS (or rarely from DOS). 8. RUN BGP.EXE… Starts the program BGP.EXE which handles results (printing and graphics) from both stability programs using last options previously saved in INI.* files. It is unlikely that you will use this option very often, as you will usually start running this program directly from WINDOWS (or rarely from DOS). 9. QUIT - exit to DOS or WINDOWS

You must place the whole BGSLOPE package in one directory. Be sure that you have the appropriate five supporting INI.* files in the same directory or in the same floppy.

7

2. GETTING STARTED

EXAMPLES of MAIN input DATA used in this Manual and enclosed with the software, are stored in 8 files as follows: 1. 2. 3. 4. 5. 6. 7. 8.

EX-1.BG EX-2.BG EX-3A.BG EX-3B.BG EX-4A.BG EX-4B.BG CULIIN.BG PRANDTL.BG

Simple slope, Linear envelope Simple slope, Nonlinear envelope Simple slope, Tension crack empty Simple slope, Tension crack + water Riverside, Linear envelopes Riverside, Nonlinear envelopes SOFT CLAY, Cu increases with depth. Bearing capacity to be used only with PRANDTL.SS

MAIN DATA from #1 to #7 can be used by both stability programs (BE.EXE and GE.EXE), while the data set #8 is for GE.EXE only in conjunction with the slip surface of the same name (#6 below). EXAMPLES of ARBITRARY SLIP SURFACES handled with GE.EXE used in this Manual and enclosed with the software, are: 1. 2. 3. 4. 5. 6.

P-1.SS Z-1.SS F-1.SS M-1.SS C-1.SS PRANDTL.SS

Polygonal Bezier type Fixed plane type coMposite type Circular Polygonal to be used only with PRANDTL.BG

SLIP SURFACES from #1 to #5 can be used with MAIN DATA #1 to #6. Novice user is encouraged to run some of the examples given here in order to initialize some familiarity with the software, even before further reading the Manual in detail. Author hopes that the user will not need to look at the Manual frequently. The effort is made to provide immediate user friendly interface on the screen.

8

3. GENERAL FEATURES

3. GENERAL FEATURES OF THE PACKAGE Running of the programs is made as simple as possible. You can concentrate on the geotechnical aspects of the problem, without any need for the use of the programming knowledge. You will be able to see what are you doing in each step of the analysis. All programs are menu driven. Menus are displayed in a logical sequence and the optimum selection of options is offered in each stage of the analysis. Stability programs (BE.EXE and GE.EXE) offer basically three main menus, namely: INITIAL..., WORKING... and CHANGES.... Flow between menus in stability programs is shown in Fig. 3.1. The INITIAL MENUs are described in Section 5, the WORKING MENU options for the program BE.EXE are shown in Section 7, for the program GE.EXE in Section 8, and MENU CHANGES in Section 9. Each menu contains several options as well as branching to other submenus and to other two programs. Most of the menu options are self-explanatory.

Fig. 3.1 Flow between menus of stability programs and the possibility to switch to BGP which handles export of HP-GL - .plt files Choices offered in both stability programs can be briefly summarized as follows: -INITIAL MENU permits entering the data via keyboard or loading from the saved file, display the cross-section and slip surface(s) on the screen, initiation of the computation, saving input data and exiting from the program. VIEW GRAPH from this menu displays the geometry of the cross section and slip surfaces before computation. VIEW GRAPH will show SOIL LIMITS in different colors for different materials. Handy to check if material zoning is adequately described. ZOOM/ MOVE... option permits you to change the scale of the graph and to change the location of the origin of the coordinate axes. -WORKING MENU controls the branching after the initial computation stage has been completed. One of the important options here is INTERACTIVE SEARCH for Fs,min. which is placed in an interactive window (Fig. 7.8 to Fig. 7.14, p. 53-57, and

9

3. GENERAL FEATURES

Fig. 8.14 to Fig. 8.17, p. 85-88. You can perform search for the critical slip surface by varying the parameters of the failure line and see the result immediately. Besides, you can try, for example, to select the new initial circle for the automatic search in BE.EXE or check the solution for some different choice of the f(x) function in GE.EXE. VIEW GRAPH option from this menu displays the results of computation, showing, for example, grid of centers with the critical slip circle, (first example shown in Fig. 7.7, p.51, or the path of the finite difference cross with smallest safety factor in program BE.EXE, as shown in the first example of this kind in Fig. 7.2, p. 47. The graph in GE.EXE will show the result, the cross section with the slip surface analyzed and line of thrust of interslice forces for used f(x), as shown in Fig. 8.8 to Fig. 8.13, p.79-82, and Fig. 10.23 to Fig. 10.33, p.120-125. SAVE .BGP FILE (*.PLT & Printing) is used to save file for further processing, like exporting graphic HP-GL files (written in HewlettPackard Graphic Language) and text files for printing, which is handled by the program BGP.EXE. Direct transfer of data and results to BGP.EXE is also possible using SWITCH… option via temporary file which is transparent to all programs and named TEMP.BGT. From this menu you can SWITCH and transfer MAIN DATA as well as some slip surfaces to any of the other two programs (see Fig.1.1, p. 2). -CHANGES is the menu that permits data editing or variation of parameters. It is rather handy in data input stage as well. If you enter some wrong input, you do not have to restart the typing procedure from the beginning, but just continue as nothing wrong is done, and after completing the entering, call this menu and correct typing error(s) and check the effects on the graphic display before initiating the computation. ZOOM/ MOVE option permits you to change the scale of the graph and to change the origin of the coordinate axes in order to adjust the graph according to changes performed. SHIFTING THE CROSS SECTION shifts the section with respect to the system of axes; combined with options ZOOM and/or MOVE provides the full control of the screen graphics during the work, or at some stage when the output of the screen graph to the file or to the printer via .PLT file is asked for. VIEW GRAPH will show SOIL LIMITS in different colors for different materials. Handy to check if material zoning is adequately described after you had made some changes. Options offered by this menu are described in more detail in Section 9. Note that all the changes on the input data are performed in the computer memory only and changes are not made permanent until you explicitly save the file on your hard disk or diskette. Menu functions in BGP.EXE, all placed in a single menu and shown in Section 11, can be summarized as follows: It reads files with results produced by stability programs and handles printing output of results. It can export HP-GL files which can be used for producing graphics on printer or processed with other programs which can handle HP-GL graphic format. This program has all the graphic control features as in both stability programs described previously.

10

3. GENERAL FEATURES

GENERAL NOTES: SAVING and LOADING files. Whenever you choose any of these options, files with relevant extensions residing in a directory will be shown. When typing the file name of up to 8 characters for SAVING or EXPORT, never type extension, as it will be automatically added to the entered file name. Files for LOADING are easily selected using arrow keys and typing of the file name for loading is completely avoided. Five file types are defined by extensions as follows: *.BG file with main input data which describe the geometry of the cross-section, soil parameters and loading. It can be formed, saved or loaded by the both stability programs, BE.EXE and GE.EXE. *.SS file with slip surface of arbitrary shape. It can be generated, saved and loaded by GE.EXE only, though the file describing circular slip surface can be formed from BE.EXE. Once defined, the slip surface can be saved from BGP.EXE as well. *.BGP file with input data and results of computation can be formed by any of the two stability programs. Such a file can be loaded by BGP.EXE program for final presentation of results. It is also convenient to store some intermediate results in this form and use them for further analyses as all the MAIN DATA and RESULTS can be transferred to stability programs for further processing. *.PLT is a HP-GL file formed and saved by BGP.EXE which is used for graphical presentation of results on plotters or printers with commercially available programs which handle HP-GL (.PLT) graphic format. *.TXT text file which contains saved printable input data with results of computation. The number of data files with the same extension i.e. *.BG, *.SS, *.BGP , *.TXT and *.PLT) is limited to 80 in the directory. You will be warned if this limit is approached and advised to move some files to other directory or delete files which are not needed. In order to avoid the possible problem which may arise during or after extensive analyses, it is advisable to save (or move) some files occasionally to some other directory if their number approaches close to 80 (say 70-75). This limitation of 80 files might be more critical for *.BGP files. Pay attention in advance.

11

4. SOIL and ROCK STRENGTH

4. SOIL or ROCK STRENGTH AND THE FACTOR OF SAFETY Methods for the computation of the index of relative stability called the factor of safety are oriented toward the calculation of the value:

Fs = where

τf τm

(1)

τf is the actual available shearing strength of the soil τm is the average shear stress on the hypothetical failure surface mobilized to maintain the body of soil in equilibrium.

The value of Fs is the chief unknown in a limit equilibrium analysis. The shear strength of soil or the rock τf is traditionally described as the Coulomb's linear relationship between the effective normal stress on the failure plane. Linear parameters are cohesion c ′ and the angle of the shearing resistance φ ′ , parameters being constant and independent of the stress level, or:

τ f = c ′ + σ n′ tan φ ′

(2)

Total stress analysis for saturated clays may be performed by using the values for cu , (which can vary linearly with the depth), and φ u = 0 for the Mohr-Coulomb strength parameters. Straight line is considered as an approximation, which is conventionally used for all types of soil. The linear Coulomb failure envelope could be considered, more or less, only as the reasonable approximation for loose sands and normally consolidated clays.

Fig. 4.1 Real failure envelope with c’=0 and the linear approximation Most soil types and rocks exhibit curved failure envelopes (Fig. 4.1). The cohesion term in noncemented soils is the consequence of the extrapolation of the line to the zero stress level. The angle of the shearing resistance decreases with the increase of normal effective stress level. Early work on this topic is reported by Maksimović (1978), who presented several forms of nonlinear failure envelopes. Charles (1982), Charles & Soares (1984) and 12

4. SOIL and ROCK STRENGTH

Costa Filho & Thomaz (1984) used power type expression, which has significant limitation due to the fact that it is valid in a limited stress range and parameters depend on units. The power type failure envelope does not have an asymptote and has vertical tangent in the origin, if cohesion is zero, and parameters have no physical meaning. New expression, (Maksimović 1988, 1989-a,b,c, 1992, 1993, 1995, 1996-a,b), very suitable for description of the nonlinear failure envelope in terms of effective stresses for most soil and rock types, describes the secant angle of the shearing resistance as the function of the normal effective stress on the plane of failure in the form:

φ ′ = φ B′ +

∆φ ′ 1 + σ n′ p N

(3)

and the soil or the rock shearing strength becomes:   ∆φ ′ τ f = c′ + σ n′ tan φ B′ + 1 + σ n′ / pN  

    

(4 )

where c′ is cohesion, ( c ′ > 0 for cemented soils and rock only and c′ = 0 for most soils and rock discontinuities) ′ φ B is the basic angle of friction, (usually equal to the friction angle at constant volume). ∆φ ′ is the maximum angle difference, (contribution of dilatancy and/or particle reorientation) is the mean angle stress, the value of the normal effective stress for which the secant pN angle of the shearing resistance equals to the mean value between the initial value φ ′B + ∆φ ′ and basic φ ′B i.e., for the stress level at which φ ′ = φ ′B + ∆φ ′/2 .(It basically reflects the deformability of the material). The geometrical description of these parameters and functions of the hyperbolic failure criterion is shown in Fig. 4.2. The envelope in Fig. 4.2-a has a tangent in the origin inclined at angle φ ′0 which is the sum of the basic angle φ ′B and the maximum angle difference ∆φ ′ . In a semi-logarithmic plot (Fig. 4.2-c) the point M corresponding to pN is a point of central symmetry, while φ ′0 and φ ′B are the left and the right asymptotes. Both expressions, (3) and (4), are dimensionally consistent. Values of the angles can be taken in degrees or radians and the description of the normal effective stress σ ′ must be in the same units as the median angle pressure pN. The optional software package NENVE can be used to convert the variety of the proposed nonlinear failure envelopes for soils, rock mass and rock discontinuities to the more general but simple nonlinear failure envelope of the hyperbolic type briefly described here. 13

4. SOIL and ROCK STRENGTH

Fig. 4.2 Parameters of the non-linear failure envelope The hyperbolic expression described here reduces to the conventional Coulomb straight-line envelope for three combinations of parameters, i.e. when ∆φ ′ = 0 then φ ′ = φ ′B or when p N = 0 then φ ′ = φ ′B and when p N = ∞ then φ ′ = φ ′B + ∆φ ′ . In this sense the conventional Coulomb linear failure envelope is simply just one special case of the described nonlinear failure envelope for geotechnical materials. The first case listed above is also handy in practical application of this software.

14

4. SOIL and ROCK STRENGTH

This nonlinear failure law applies to the peak shearing strength of coarse-grained materials (rockfill, gravel and sand), dense fine-grained soils (clays) and for the residual shearing strength of clays and clay minerals, rock discontinuity and rock mass, or almost to all types of soils and rocks with better accuracy then the conventional linear failure law. The dimensionally consistent expression is valid from the zero stress level to practical infinity. Some examples are shown here, and for more, user may consult papers listed in literature or examine examples shown here. For the first example, (Maksimović, 1989), which is shown in Fig. 4.3, and Fig. 4.4, parameters in eq. (2) are derived by considering the relationship between the angle of the shearing resistance and the normal stress based on data for the compacted London clay (Atkinson & Farrar, 1985). In the conventional range of testing stresses (150-300 kPa) the failure envelope can be described by a linear relationship defined by the angle of the shearing resistance (Fig. 4.4-a). However, in the low stress range (5-25 kPa) the envelope is curved and can be described by a power type expression, as shown in the enlarged detail in Fig. 4.3.

Fig. 4.3 Detail of the envelope from Fig. 4.4-a in the low stress range Four points for mentioned values of normal stresses are selected for the derivation of the proposed parameters as shown in Fig. 4.4. It can be seen (Fig. 4.4-a) that in the stress range from 100 kPa to 400 kPa the failure envelope is very close to the straight line, but at the lower stresses, (Fig. 4.3) the linear extrapolation towards the zero normal stress gives the cohesion value and the unsafe, higher values of the shearing strength. Plots shown in Fig. 4.4b and Fig. 4.4-c indicate that the description of the shearing strength from combined linear and the power type expressions can be substituted by a single expression of the proposed hyperbolic type with rather high accuracy. The implication of the NON-LINEARITY of the failure envelope for compacted clays is discussed by Charles (1982), Charles & Soares (1984), Day & Axten (1989), Day (1994,1996), Day & Maksimović (1994) and others.

15

4. SOIL and ROCK STRENGTH

Fig. 4.4 Parameters of the nonlinear failure envelope for compacted clay

16

4. SOIL and ROCK STRENGTH

Fig. 4.5 Residual shearing strength

17

4. SOIL and ROCK STRENGTH

The angle of the residual shearing resistance very often depends on the magnitude of normal stress acting on the failure plane, as shown by Skempton & Petley (1967), Bishop (1971), Lupini, Skinner & Vaughan (1981), Chandler (1984), Skempton (1985), Lambe (1985), Maksimović (1989, 1995), Stark & Eid (1994) and others. The curvature of the envelope of the residual shearing strength can be attributed to the different degree of orientation of platy particles with increasing parallelism to the failure plane as the normal stress on the failure plane increases. The residual failure envelope can be described using the expression of the same type, and adding the subscript r to parameters to designate that it describes the shearing strength after large shearing displacements. An example based on data reported by Stark & Eid (1994) and discussed by Maksimović (1995) is shown in Fig. 4.5. The curvature of the envelope does not seem very significant in the graph of the envelope, but the variation of the angle, in a relative sense is very pronounced. The shape of the actual envelope shown in Fig. 4.5-a is not very clear unless it is shown as the variation of the angle of the shearing resistance with stress level in linear (Fig. 4.5-b) and in semi-logarithmic plot (Fig. 4.5-c). It can be seen that the curve fits the data with remarkable accuracy. The NON-LINEARITY of the coarse grained material like sand, gravel and rockfill is the consequence of the variable dilatancy that decreases with the rise of the normal stress level and grain crushing. The peak shearing strength envelopes of coarse-grained materials used for construction of large dams are shown in Fig. 4.6 and Fig. 4.7. Strength-density relationship for sands in terms of parameters of the nonlinear failure envelope is described by Maksimović (1993), as an alternative to the logarithmic description described by Bolton (1986).

Fig. 4.6 Failure envelope for the sand-gravel mixture

18

4. SOIL and ROCK STRENGTH

Fig. 4.7 Crushed basalt rockfill, Dmax = 76.2 mm, Cu=11.6. It is worth noting that the same form of the failure law applies to rock discontinuities and to jointed rock mass.

Fig. 4.8 Failure envelope of the rock discontinuity The most popular failure criterion for rock discontinuities of the logarithmic type proposed by Barton and Choubey (1977) can be easily converted to the failure criterion of the proposed hyperbolic type (Maksimović, 1996) without using the software simply by taking the basic angle unchanged, taking pN=JCS/10 and ∆φ=2 JRC. The results of the direct shear test on sandstone discontinuity are evaluated. Six data points based on results reported by Barla, Forlati and Zaninetti (1985) are used and a very good approximation obtained, as shown in Fig. 4.8. This failure envelope in terms of Barton and Choubey parameters for the shear strength along the rock discontinuity would correspond to JRC=28.82/2=14.41 and JCS=10 x 2277 kPa =22.77 MPa. 19

4. SOIL and ROCK STRENGTH

The nonlinear failure criterion of the power type for the jointed rock mass by Hoek & Bray (1981) as well as recent version Hoek (1995) can be also converted to the hyperbolic form of equation 3 using “NENVE” package. The main principle for the conversion of Hoek (1995) strength envelope for the rock mass into the envelope of the hyperbolic type, is to compute c’ as the shearing strength for the zero value of the normal and select three non-zero normal stresses to compute unknown parameters. In this way it is achieved that the hyperbolic failure envelope and the Hoek (1995) envelope have 4 identical points: at zero normal stress, this is the cohesion term, and at three different non-zero normal stresses. The difference between the two envelopes (Hoek 1995 and hyperbolic) is negligible for all practical purposes within the stress range considered. The maximum difference in terms of the angle of shear resistance for the normal stresses between points used for conversion is of the order of (+/-) 0.1-0.2 degrees only. An example of such conversion is shown in Fig. 4.9. It can be seen that the approximation within rather wide stress range is very good and in this case, the non-zero cohesion term is included in the equation 4. The main advantage of this approach is that the Hoek (1995) failure criterion, which is essentially defined in terms of principal stresses σ1 and σ3, is expressed in terms of shear strength τf and the normal effective stress on the failure plane σn . The non-linearity of the failure envelope is maintained.

Fig. 4.9 Failure envelope of the rock mass - conversion of Hoek 1995 criterion. Slope stability software package described in this manual can handle both the conventional linear Coulomb's failure law, (eq. 2, Fig. 4.1), as well as the described nonlinear failure envelope of hyperbolic type, (eq. 4) shown in Fig. 4.2 through Fig. 4.9 as a very versatile one. 20

4. SOIL and ROCK STRENGTH

The nonlinear envelope and its variations, as well as the methods of derivation of the parameters from tests or conversion from a number of other proposed forms of the nonlinear failure envelope are described in more detail separately. The optional package "NENVE" is used for derivation of parameters from the shearing test data and presentation of the failure envelopes on suitable plots. Package contains a database with a number of results of test and derived parameters for various geotechnical materials. All data on the shape of the curved failure envelopes indicate that the curvature of the failure envelope is most pronounced in the lower stress range. In general, the significance of the stress level can be expressed in terms of the Stress Level Ratio (SLR) for the curvature of the failure envelope (Maksimović 1996) using the described parameters. Any failure envelope of the proposed type can be separated approximately into two or three segments, in terms of SLR (Stress Level Ratio), as defined and shown in Fig.4.10. High curvature is in the stress range SLR2.

21

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5. METHODS OF THE ANALYSES Of the limiting equilibrium models available, those which divide the mass above the assumed shear surface into slices, are most ideally suited to being programmed to solve general slope stability problems. It is widely accepted that methods, which satisfy all equilibrium conditions, give essentially the same results. Bishop (1955) simplified method, limited to circular shear surfaces, gives virtually the same results as methods that satisfy complete equilibrium. The Morgenstern and Price (1965) method for the slip surfaces of arbitrary shape requires numerical integration over each slice, which is a lengthy procedure. The General method developed by writer (Maksimović 1970, 1979, 1988, see also Bromhead 1986) can be considered as an equivalent to the Morgenstern & Price method, but significantly simpler. For the discussion on different methods you may consult Bromhead (1986, 1992, 1999) and/or Fredlund (1984). This software package is based on the following two methods: 1. Bishop's method, for circular slip surfaces, and 2. General method developed by writer, for slip surfaces of arbitrary shape. Arbitrary slip surfaces can be generated as polygonal, Bezier curves, fixed plane, composite and circular arcs as shown in Fig. 5.7, page 30. As mentioned earlier, the methods implemented in this software package are extended to handle external loading, earthquake accelerations and the nonlinear strength envelope. Due to the nature of the methods, all computations for the safety factor are iterative. In the Bishop Method there are two nested iterative loops and in the General method, three. Tolerances adopted for the convergence are very small. All the computations are carried out in a DOUBLE PRECISION mode, involving sign and 15 digits.

22

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5.1 Program BE.EXE Large number of circular slip surfaces are usually treated by the application of the Bishop (1955) Extended method, which is extended here by the introduction of the nonlinear failure envelope and modified to include external line loads, distributed (surcharge) loads and inertia forces due to earthquake loading in any direction. At the very beginning of the run, only options 1, 3, 5, 10, 11 and 12 are offered, though the complete set of options on the screen in the INITIAL MENU are as follows: CHOICE I N I T I A L 1 2 3 4 5 6 7 8 9 10 11 12

MENU

( BE )

ENTER MAIN DATA (kbd) ENTER SLIP-CIRCLES LOAD MAIN DATA PRINT MAIN DATA (Screen) VIEW GRAPH ZOOM/ MOVE/ AXES/ MIRROR SAVE MAIN DATA PRINTING OPTION IS: (Blank) COMPUTE! W O R K I N G M E N U. . . C H A N G E S. . . QUIT

CHOICE No.? 1 ENTER MAIN DATA (kbd) is an option described in detail in Section 6. 2 ENTER SLIP-CIRCLES initiates the computation aiming for the critical slip-circle. The practical problem is to find the minimum of the function FS=FS(x,y,R) with three variables, i.e., two center coordinates of the circle and the radius. The time for execution of the search for the minimum of the function of three variables (3D search) will be of the order of a seconds. The computation is for an order of magnitude shorter and significantly safer, if the problem is, at least initially, reduced to the search for the minimum of the function FS(x,y,R) of two variables only (2D search), where "r" is a parameter treated in each search as the constant selected by the user. Two options are made available initially in this menu. First, shown in Fig. 5.1-a, is that each family of slip circles has one common PASSING POINT (an exit point P in the zone of the toe, for example). The second option is that the family of slip-circles is tangential to certain horizontal plane defined with elevation YP, as shown in Fig. 5.1-b, which in some cases could be the lower boundary of some weak zone or only a trial level not related to any particular feature of the cross section under consideration. However, 3D search can be performed either in an interactive way dealing with only successive slip-circles at the time, or by massive computation with up to 3025 clip circles in a single run starting from WORKING MENU. For details see Section 7.

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5. METHODS

(a)

( b)

Fig. 5.1 The single passing point and the single passing horizontal tangent (point P). For the search of the critical slip-circle four options are built in the program: i ii iii iv

Automatic search (steepest descent) Grid of centers Local grid Interactive search

The routine for the Automatic search (i) is based on the method of steepest descent. This method is very valuable during the practical work, though the Grid option (ii) is common in most computer programs of this kind. These two options can be initiated from the INITIAL MENU. Options (iii) and (iv) can be initiated from WORKING MENU (BE).

Fig. 5.2 Path of steepest descent

Fig. 5.3 Spider - 5 trial centers

i. Automatic search (steepest descent). To reach the solution it is necessary to start from some reasonable and arbitrary center T (Fig. 5.2) and arrive to the center C, which is such that FS(x,y,R)=FS,min. The paths from point T to C will follow the trajectory of steepest gradient (Fig. 5.4). As the method, (Maksimović, 1986), requires calculation of the first and the second partial derivatives of FS(x,y,R), the local finite difference grid, consisting of five points, 24

5. METHODS

forming "the finite difference cross," which is in the program referred to as SPIDER, as shown in Fig. 5.3, is defined with the central point T, as point number 1, placed initially by the user. Factors of safety for five trial slip-circle centers, and the direction of steepest descent are automatically computed. By using Taylor's expansion to the polynomial of second degree, estimated length and the direction of the step is calculated, central point of the finite difference cross moved to the new point and the procedure automatically repeated until the calculated step becomes some small tolerable value. In the program this tolerance is taken as 0.2 of the opening "a" of the grid, (SPIDER'S LEG), as at the stationary point first partial derivatives vanish.

Fig. 5.4 An example of the Automatic search option Practical experience show that the value of "a", can be taken as about 1/10 to 1/20 of the slope height. Then the error in the location of the critical circle center becomes not more than 1-2% of the slope height, being within the zone surrounded by the smallest circle with the center in the central point 1 (Fig. 5.3) of the “spider”. To avoid excessive long jumps from one position of the point 1 to the next trial location, it is limited in the program to the length of the diagonal or R = 2a 2 , as indicated with a large dotted circle shown in Fig. 5.3, p.24. An example of this option is outlined in Fig. 5.4 showing the initial center, path, and the critical circle. The number of steps is limited to 30 and in usual circumstances the minimum will be reached for the fixed tangential level or one common point for all slip circles. In the case that the initially assumed center is far away from the critical one, or if step "spiders leg" is too short, after making 30 steps and the criterion for the minimum Fs has not been reached after 30 steps, you are going to get suggestion on the screen. Usual restriction on the practical significance of the automatic search is the consequence of continuity and unimodality of the function FS(x,y,R). It might have more than one minimum, and the procedure will reveal the minimum along the path of the steepest descent from the chosen starting point T. To check if some another minimum exists, you can try another starting point, (Fig. 7.1 to Fig. 7.4, p.4748), or use the GRID OF CENTERS, (Fig. 5.5 and Fig. 7.5 to Fig. 7.7, p.50-51). 25

5. METHODS

ii. Grid of centers. The Grid option, shown in Fig. 5.5, as an alternative to Automatic search from the first option in MAIN MENU, offers a possibility to generate the grid containing up to 150 centers. It is convenient to use this option in any stage of computation, usually to prove previously estimated critical circle based on Automatic search and show that in rather wide area of possible centers the estimated critical is the only one, or to indicate if some other local minimum might be considered.

Fig. 5.5 Grid option. Application of Local grid of smaller size could be next. iii. Local grid. The LOCAL GRID option (No. 5, pages 57-58) in the WORKING MENU, is used after some computation is performed and some smallest safety factor found for a particular circle or set of circles and the particular passing point. This option, when chosen, will automatically generate the quadrilateral grid of the prescribed SIZE with the last critical center placed in the center of the local grid, as shown in Fig. 5.6. You can chose variable size and shape of the local grid in order to find the accurate position of the critical circle. The LOCAL GRID density is fixed to 11, meaning that 11 x 11 =121 centers will be generated within the given size of the local grid (see Fig. 5.6, Fig. 7.15 through Fig. 7.22). Besides, you can select up to 25 passing points within the prescribed range of positions for the same grid and generate 3025 slip circles for a 3 D search in a single run. The LOCAL GRID proportions (H/S), pages 57-58, can be altered in menu CHANGES (option 11, page 65) and than tilted left or right, as explained in Section 7, page 57. iv. Interactive search option (No. 2, p. 53-57) INTERACTIVE SEARCH for Fs,min., in the WORKING MENU (see Section 7) deals with a single slip circle in such a manner that the changes of the value of Fs due to changes of the coordinates of the circle center and/or the passing point P (see Fig. 5.1) performed by using only arrow and +/- keys, can be followed in an interactive window (Fig. 7.8 to Fig. 7.14, p.54-57). 26

5. METHODS

Fig. 5.6 Local grid placed after Automatic search. A set of up to 25 tangents or passing points can be prescribed in a single run. Options (iii) and (iv) are not offered in the INITIAL MENU shown in page 23, but they are offered in the WORKING MENU which will be described later in Section 7. Some other possibilities are also described in WORKING MENU (BE). In some cases, particularly when the size of the spiders leg “a” is small, the function FS(x,y,R) might be rather irregular in the vicinity of local minimum, and the procedure might fail to converge to the minimum, oscillating in the vicinity of that point. In such a case you might find the use of the LOCAL GRID option rather helpful. This LOCAL GRID option with a large number of slip-circles might be used after Automatic search or the Grid had been executed, to investigate the area in the zone of the slip-circle with the smallest factor of safety obtained in the previous computation. The size of the grid and the spacing of passing points can be arbitrary small making the accuracy of the critical center coordinates very high with high density of slip-circles. Examples of the LOCAL GRID are shown in Fig. 5.6, Fig. 7.16 through Fig. 7.22, Fig. 10.12 and Fig.10.13. The combined procedures of Automatic search, the predefined, usually larger Grid and the LOCAL GRID option as well as the INTERACTIVE SEARCH for Fs,min., when used sensibly, will provide the satisfactory answer for a critical slip circle and the smallest FS to an engineer with reasonable geotechnical training and experience. For more on the slipcircle analyses see Section 7. Other options from the MAIN MENU (BE) shown in page 23 are rather selfexplanatory, and will be described here for completeness. 3 LOAD MAIN DATA When this item is selected, a list is obtained with all the available *.BG files in the directory. Move the highlighted bar to the name of the file you wish to load and press the key. The screen shall display the cross section of the slope. 27

5. METHODS

4 PRINT MAIN DATA (Screen) or (File) Input data will be printed depending on the active option 8 PRINTING OPTION. In the case that the active option is Blank, main data will be printed on the screen. 5 VIEW GRAPH Self-explanatory. Graph on the screen will show the definitions of slip surfaces and the cross section, before computation. The soil limit lines will be shown in different colors; one color for one material. Useful for rough checking whether the zoning of the cross section is properly described. 6 ZOOM/ MOVE/ AXES/ MIRROR Option is explained in Section 6, page 38. 7 SAVE MAIN DATA When this item is selected, a list is obtained with all the existing *.BG files in the directory. Type the name of the data file you wish to save and press the key. Do not type extension. If you enter the name of the already existing data file, it will be overwritten by the new one. Pay attention, there is no warning message. 8 PRINTING OPTION IS: (Blank) or (Screen) or (File) permits you to control printing of input data and results of computation. PRINTING OPTION might be: (Blank) is default option meaning that the whole process of computation will not be shown on the screen when COMPUTE ! option is chosen. It will affect printing of the results of computation only. The message COMPUTING...PLEASE WAIT... appears and only the final result, (Fs) will be shown on the screen without details, and THAT WILL SIGNIFICANTLY SHORTEN THE TIME OF EXECUTION. (Screen) means that main data or intermediate phases of computation, when requested by this option from the menu, will be displayed on the screen. (File) option implies that, input data and results will be saved in the *.TXT file and later if needed, sent to the printer or simply examined on the screen. Note that this option, even if selected, will be deactivated during computation. 9 C O M P U T E ! Initiates computation for the defined slip circles. When computation for this set is finished, the graph showing results will appear, and the further control of the program is transferred to WORKING MENU (BE) shown in page 52. 10 W O R K I N G M E N U. . .Shows and offers WORKING MENU (BE)… 11 C H A N G E S. . . Shows and offers menu CHANGES (BE), page 65. 12 Q U I T You will be asked to confirm that you are quitting the program. Be sure that you have saved the files before you quit the program.

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5.2 Program GE.EXE General slip surfaces, including the circular ones, are analyzed by the method that satisfies all equilibrium conditions. For more details and the critical review, you may consult Bromhead (1986, 1992, 1999) and Maksimovic (1979, 1988). The method is analogous to the Morgenstern and Price (1965) method as it uses the distribution function of inclinations of interslice forces f(x), though computationally simpler. At the beginning, the screen shows the following INITIAL MENU: CHOICE I N I T I A L 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

MENU

( GE )

ENTER MAIN DATA (kbd) ENTER SLIP-SURFACE (kbd) LOAD MAIN DATA LOAD SLIP-SURFACE PRINT MAIN DATA (Screen) PRINT SLIP-SURFACE (Screen) VIEW GRAPH ZOOM/ MOVE/ AXES/ MIRROR COMPUTE! PRINTING OPTION IS: (Blank) SAVE MAIN DATA SAVE SLIP-SURFACE W O R K I N G M E N U. . . C H A N G E S. . . QUIT

CHOICE No.? 1 ENTER MAIN DATA (kbd) is an option described in detail in Section 6, page 33. 2 ENTER SLIP-SURFACE initiates the definition of the slip surface. The description of the slip surface of arbitrary shape can be entered in five different ways. The Arbitrary slip surfaces shown in Fig. 5.7 can be described as: Polygonal slip surface defined by point coordinates, Bezier curves generated by control points linked to Bezier polygon, Fixed plane type defines the fixed central plane and polygonal shapes on sides Composite defined by a circular arc and an intersecting fixed straight line, Circular shape which is treated as any arbitrary slip surface. You may decide which one to use in a process of the continuous application, depending on circumstances in each particular problem. For details see Section 8, page 71. 29

5. METHODS

Fig. 5.7 Five shapes of slip surfaces handled by GE.EXE To simplify the definition of the function f(x) with minimum number of parameters, six types are envisaged, and presented later in more detail, (see Fig 8.6 and pages 77-78). The final result, the factor of safety, is rather insensitive to the choice of f(x), if the solution is physically admissible. Examples of the application of different f(x) are shown in sequence of figures from Fig. 8.8 to Fig. 8.15 and from Fig. 10.18 to Fig. 10.28 and in some other figures in Section 10. This program GE.EXE can be also used for computation of the critical horizontal acceleration that gives FS=1.0 for a given slip surface. Rigid body displacements due to the square acceleration pulse with a value larger than critical can be easily calculated in addition to the conventional pseudo-static seismic analysis (option 7 in pages 83-84). For more on the analyses using arbitrary slip surfaces see Section 8 and Section 10. 3 LOAD MAIN DATA When this item is selected, a list is obtained with all the available *.BG files in the directory. Using arrow keys, move the highlighted bar to the name of the file you wish to load and press the key. The screen should display the cross section. 4 LOAD SLIP SURFACE When this item is selected, a list is obtained with all the available *.SS files in the directory. Using arrow keys, move the highlighted bar to the name of the file you wish to load and press the key. The screen should display the cross section of the slope including the slip surface. 5 PRINT MAIN DATA (Screen) or (File) Input data will be either shown on the screen or saved in the *.TXT file depending on the active option 10 PRINTING OPTION. In the case that the active option is Blank, main data will be shown on the screen.

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5. METHODS

6 PRINT SLIP SURFACE (Screen) or (File) Data describing the slip surface will be either shown on the screen or saved in the .TXT file, depending on the active option 10 PRINTING OPTION. In the case that the active option is Blank, slip surface data will be shown on the screen just Screen option was active. 7 VIEW GRAPH Self-explanatory. Graph will show the definition of the slip surface and the cross section, before computation. The soil limit lines will be shown in different colors; one color for one material beneath the line. Useful for rough checking whether the zoning of the cross section is properly described. 8 ZOOM/ MOVE/ AXES/ MIRROR Option is explained in Section 6, page 38. 9 C O M P U T E ! Initiates computation for the defined slip surface. When computation for the slip surface is finished, the graph showing results will appear, and the further control of the program is transferred to WORKING MENU (GE), page 83. 10 PRINTING OPTION IS: (Blank) or (Screen) or (File) permits you to control printing of input data and results of computation. Initially, as the program is started, this PRINTING OPTION might be: (Blank) meaning that the whole process of computation will not be shown on the screen when COMPUTE ! option is chosen. If used, option will skip printing details of the results of iterative computation. Only the final result, (Fs) will be shown on the graphic screen without details. (Screen) means that main data or intermediate phases of computation, when requested from the menu, will be displayed on the screen. (File) option implies that, input data and/or results will be saved in the .txt file which can be later sent to the printer or examined. This option, is automatically made inactive during iterative computation in order not to produce huge redundant files. 11 SAVE MAIN DATA When this item is selected, a list is obtained with all the existing *.BG files in the directory. Type the name of the data file you wish to save and press the key. Do not type extension. If you enter the name of the already existing data file, it will be overwritten by the new one. Pay attention, there is no warning. 12 SAVE SLIP-SURFACE When this item is selected, a list is obtained with all the existing *.SS files in the directory. Type the name of the slip surface file you wish to save and press the key. Do not type extension. If you enter the name of the already existing data file, it will be overwritten by the new one. Pay attention, there is no warning message. 13 W O R K I N G M E N U. . . Shows and offers WORKING MENU (GE)…(see p. 83) 14 C H A N G E S. . . Shows and offers menu CHANGES (GE)…(see p.90) 15 Q U I T You will be asked to confirm that you are quitting the program.

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6. INPUT DATA

6. PREPARATION OF INPUT DATA Input data for both stability programs consist of the two separate sets as follows. 1. MAIN DATA must be given first for definition of the geometry of the cross section, distribution of materials, material properties, pore water pressures, external loading, seismic coefficients, etc. This set of data is identical for both stability programs, BE.EXE and GE.EXE. MAIN DATA can be saved or read from files by both stability programs, regardless of the program used for entering and saving of the data. File name extension is .BG Preparation of this set of input data is described in the next Section 6.1. 2. SLIP-SURFACE(s) is the second set and it depends on the program used (BE.EXE or GE.EXE). Slip surfaces of circular shape for BE.EXE (Bishop Extended method) are described in Section 7. Slip surfaces for GE.EXE (General Method) of arbitrary or circular shape are described in Section 8. The slip circle analysis by the Bishop Extended method and BE.EXE is usually done first. After completion of the slip circle analysis by BE.EXE, you can read the same main data file by GE.EXE and check some non-circular slip surfaces, or compare the results computed by the extended Bishop's method with the result obtained by the General Method, which satisfies all equilibrium conditions, using SWITCH to… option. 6.1 MAIN DATA Trial slip-surfaces may be used for either the right or left face of the slope. No restrictions are placed on the direction in which the slope faces. See Addendum 1, page 149. Positive direction of "x" is horizontal with coordinates increasing to the right hand side. The positive direction of "y" is upward. The cross-section to be analysed is drawn in a convenient scale, basically as shown in Fig. 6.1 (simple slope, p.39), Fig.6.2 (slope with vertical crack, p.39) and Fig. 6.3 (moderate size problem, p.41). It is recommended to use A-3 size millimeter paper or similar to draw moderate size and/or complex sections, and to follow the principles shown in Fig. 6.2. Draw your x & y axes. Writer's preference is to place "x" in such way that all or most points have positive values of "y" and that major part of the cross section is on the (+)ve side of "x" for the right face slope.

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6. INPUT DATA

Smallest example for preparation of input data is given in Fig. 6.1, p.39 and Fig. 6.2, p.39. A moderate sized example for preparation of input data is given in Fig. 6.3, p.41. Boundaries of the cross section, limits of soil zones and piezometric lines are approximated by a straight line segments. * End points of line segments are numbered from 1 to "N", * Zone boundary lines and piezometric lines are numbered from 1 to "L", and * Soil zones are numbered from 1 to "M". If the shear strength of some material zone(s) has to be described with the nonlinear failure envelope, these material zones must be either numbered first, or all materials should be defined as they had the nonlinear failure envelope. Linear envelope can be described with nonlinear parameters by taking ∆φ=0 and with an nonzero value for parameter pN The sequence of numbering of points and lines in the section is arbitrary. Vertical lines are not needed for computational description of the cross section. If vertical line is prescribed, with the material definition in the range from 0 to the number of defined materials “M”, stability programs will warn you that this is not permitted. If, in spite of that, you start computation without making corrections, an error will be reported. However, if you want to have some vertical lines or any line of arbitrary orientation on your graph, you can prescribe to a line the material number outside the predefined range (0 to M). The negative material numbers are appropriate for such a purpose as they will show as discontinuous lines on the graphs on the screen and on HPGL plots, but will be ignored in computations. Such lines are called dummy lines. ALL VALUES ENTERED IN THE PROGRAM MUST BE IN "SI" UNITS, in kN, m, (kilo-Newton, meter). The unit weight of water is fixed value in the stability programs γ w = 9,807 kN/m3.

MAIN DATA are entered after the first option is chosen in the INITIAL MENU of each stability program. The entering is performed in a "question and answer bases", or simply by following the screen instructions. (i) TITLE : is an alpha-numeric label giving the name of the problem. title. The suggested length of the title is not more than 40 characters, preferably less, if you wish to have a neat appearance on the screen. If you just press without any title, the title will be automatically entered as "Untitled". (ii) COMNT : is the additional alpha-numeric comment to the problem with the suggested length as in (i). Make your comment as brief as possible. If you just press without any comment, this label will remain empty.

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(iii) NUMBER OF POINTS (max.100) for the definition of the geometry of the cross section (N). Entering 0 (zero) or pressing only will return you to the INITIAL MENU. This comes handy if you initiated the procedure by not wishing to do so. (iv) NUMBER OF LINES (max.100) which describe the boundaries of the cross section, internal zoning and piezometric lines (L). Entering 0 (zero) or pressing only will return you to the INITIAL MENU. (v) NUMBER of MATERIALS (max.30) with different properties in the cross section, (M). Entering 0 (zero) or pressing only will return you to the INITIAL MENU. (vi) NUMBER OF MATERIALS WITH NONLINEAR ENVELOPES is the number of zones for which the soil shearing strength is to be described in the proposed unconventional manner (MN). Note that MN=0 Pore water pressure in the base of the slice will be computed by taking that r u = PP (pore pressure ratio), and the value of the pore water pressure will be computed from u = r u xW / b where W is the weight and b is width of the slice. 2). PP0, check Addendum 1, for details. If you have some other surcharge loads, beside due to the crack, the surcharge due to the crack must be entered as the first surcharge load. Surcharge loads are shown with dark red on color screen. The height of the strip representing the surcharge is

36

6. INPUT DATA

controlled by a value of the LOAD SCALE for which the default value 20 is used meaning that 20 kPa corresponds to 1 m. This LOAD SCALE for graphical presentation of the surcharge can be altered in BGP.EXE, page 132. However, this scale for the surcharge load due to the presence of the tension crack is governed by the description of the crack and not by this value of the LOAD SCALE that is applicable to all other surcharges. (xv) SEISMIC COEFFICIENTS kx, ky are two values expressing the components of inertia forces in terms of fraction of gravity. Most frequently, only horizontal acceleration kx is used, and the second value, ky is taken as zero. Note that the positive horizontal acceleration induces horizontal forces in the horizontal +x direction. If the slope is "left oriented", the negative value of kx shall normally cause the drop in the value of Fs and vice versa. Positive vertical inertia forces will act in (+)ve direction of "y" axis. Program will automatically use the saturated unit weight for soils below the TAILWATER LEVEL (YW) for computation of inertia forces. * * * Entry (xv), described above, is supposed to be the last entry in the input stage and the graph will automatically appear on the screen. The scale and the position of the cross section on the screen might need some adjustments, depending on the size of the section and position of the section with respect to origin. The option offered in the INITIAL MENU of BE.EXE is 6 ZOOM/ MOVE/ AXES/ MIRROR and in the second stability program GE.EXE is option 8 ZOOM/ MOVE/ AXES/ MIRROR. This set of options are also offered in several other menus. Available options are explained in the next page. The window showing the available options will appear as, for example, shown in Fig. 6.5, p.43. * * * As mentioned before, this set of main input data from (i) to (xv) can be entered in an identical manner using any of the two slope stability programs (BE.EXE or GE.EXE). After saving this set of data in a file, the same file can be loaded by any of the two slope stability programs. The data file formed by these programs will have an extension .BG added automatically to the users defined name when saved on the disk. Three examples of input data are shown in the following pages 39 to 42 and included as files within this BGSLOPE 6.16 package. EXAMPLES of MAIN input DATA used in this Manual are stored in 8 files. File names are given in page 8 of this Manual. It is advisable for a novice user to load and/or print and/or examine these examples in some detail before attempting to use the package for solving his/her particular slope stability problem. After performing adjustments shown as an example in Fig. 6.5 and 6.6, you are ready to enter slip surfaces, depending on the method selected, as described in Sections 7 & 8. *** 37

6. INPUT DATA

ZOOM/ MOVE/ AXES/ MIRROR… option is offered in several menus in this package. Its purpose is to control the graph on the screen as well as in graphic (HP-GL) files. In any ZOOM/ MOVE/ AXES/ MIRROR window (Fig. 6.5 & Fig. 6.6, pages 43, 44) you can perform several adjustments using: ZOOM controls the scale of the cross section. The scale is initially set automatically to 10, meaning that 1 meter of length corresponds to 10 pixels on the screen. It can be altered in by pressing + (plus on numeric key pad) to increase the size of the section, or (minus on numeric key pad) to decrease the size. Scale increment (+) or decrement (-) is 1.02. The number in the left lower corner of the screen indicates the length of divisions on coordinate axes or on the scale bar. Cross-section loaded from the file will have the scale valid when the file was saved. MOVE controls the position of the origin and the cross section on the screen. MOVE using arrows is performed in 5 pixel increments in both stability programs. ARROWS TO MOVE Use arrow keys to move the origin in desired direction. Cross-section loaded from the file will have the origin valid when the file was saved. letter A in Axes display is used as a toggle for showing or removing coordinate axes or the scale bar in the lower left corner of the screen. This option might give a neater appearance of the graph on the screen and on the drawing produced from the HP-GL, .PLT file exported by BGP.EXE. Cross-section loaded from the file will keep the consequences of this option when the file was saved. For example, compare windows shown in Fig. 8.14 (with axes, p.85) to Fig. 8.15 (no axes, p.86) or Fig.10.13 to Fig. 10.14, p.113. letter R in miRror can be used to produce the mirror image, right slope becomes left and vice versa. (option rarely used).The sign of "x" of point coordinates is changed. Origin is moved in a new position - a mirror image of the point with respect to vertical line of the symmetry of the screen. This practically means that the whole graph has rotated for 180 with respect to the vertical center line of the screen. If you use the option twice in sequence, you will return to the initial orientation of the face of the slope. An example is shown in Section 9, Fig. 9.6 and Fig. 9.7, p.103. letter C, in show Circles is an option offered in BE.EXE only. It works as a toggle. By pressing C you can make faster moves or zooms on the screen by not drawing slip circles in each step of the movement or in each change of scale. Press C again to see circles.

38

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Fig. 6.1 Simple slope. Section, coordinate system, points and lines. INPUT DATA for SIMPLE SLOPE shown in Fig. 6.1 above for the linear envelope. BGSLOPE 6.16 Licensed: ***** Copyright: Dr M. Maksimovic Example 1 Simple slope Linear envelope NUMBER OF POINTS 4 NUMBER OF MATERIALS 1 TAILWATER LEVEL -80 COORDINATES: Point x 1 -10.000 2 12.000 3 36.000 4 60.000

NUMBER OF LINES NUMBER OF SLICES

NUMBER OF LINE LOADS

3 40

SOIL LIM. & PIEZ. LINES: Line T1 T2 MT 1 1 2 1 2 2 3 1 3 3 4 1

y 18.000 18.000 9.000 9.000

SOIL PARAMETERS: Mat. Gamma c 1 20.00 25.00

01-01-2006

PhiB 16.00

PP 0.40

0

NUMBER OF DISTRIBUTED LOADS COEFF. OF SEISMICITY kx, ky

0 0.0000

0.000

In the Example 2, only the table with the soil parameters differs and looks like: SOIL PARAMETERS: Mat. Gamma c 1 20.00 0.00

PhiB 16.30

39

PP 0.40

DPhi 48.10

pN 28.20

6. INPUT DATA

Fig. 6.2 Simple slope with tension crack. INPUT DATA for slope shown in Fig. 6.2 with tension crack + water. BGSLOPE 6.16 Licensed: ***** Copyright: Dr M. Maksimovic

01-01-2006

Example 3-B Tension crack+water Linear envelope NUMBER OF POINTS 5 NUMBER OF LINES 4 NUMBER OF MATERIALS 1 TAILWATER LEVEL -80 NUMBER OF SLICES 40 COORDINATES: Point x 1 11.990 2 -20.000 3 36.000 4 60.000 5 12.000

SOIL LIM. & PIEZ. LINES: Line T1 T2 MT 1 1 2 1 2 1 5 1 3 5 3 1 4 3 4 1

y 15.000 15.000 9.000 9.000 18.000

SOIL PARAMETERS: Mat. Gamma c 1 20.00 25.00

PhiB 16.00

TENSION CRACK DEPTH Hc= 3 NUMBER OF LINE LOADS 1 I XQ YQ 1 0.01 16.00 NUMBER OF DISTRIBUTED LOADS No. from to LOADING 1 1 2 60.00 COEFF. OF SEISMICITY kx, ky

PP 0.40 TOP y = 18 QX 45.00

With water QY 0.00

1

0.0000

0.0000

Note that the horizontal force due to water pressure in the crack is entered (edited) to be QX=45 kN/m for the unit weight of water 10.0 kN/m3 though the program would calculate it as QX=44.13 kN/m for the unit weight of water 9.807 kN/m3.

40

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Fig. 6.3 Geometry of cross sections for Examples 5 & 6.

41

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INPUT DATA for the slope section shown in Fig. 6.3 with nonlinear envelopes. BGSLOPE 6.16 Licensed: ****** Copyright: Dr M. Maksimovic Example 4-B Riverside Nonlinear envelopes NUMBER OF POINTS 17 NUMBER OF MATERIALS 7 TAILWATER LEVEL 10 COORDINATES: Point x 1 26.800 2 12.000 3 10.000 4 2.000 5 0.000 6 -4.500 7 29.400 8 -10.500 9 -10.500 10 -10.500 11 35.000 12 55.000 13 55.000 14 -10.000 15 40.000 16 -10.000 17 55.000 SOIL Mat. 1 2 3 4 5 6 7

NUMBER OF LINES 18 WITH NON-LINEAR ENVELOPES NUMBER OF SLICES 40 SOIL LIM. & PIEZ. LINES: Line T1 T2 MT 1 2 3 1 2 3 4 1 3 4 5 1 4 5 6 2 5 5 2 2 6 2 1 2 7 8 6 3 8 6 1 3 9 9 1 4 10 1 7 4 11 10 7 5 12 7 11 5 13 11 12 5 14 9 1 0 15 1 7 0 16 7 13 0 17 14 15 6 18 16 17 7

y 10.700 17.000 18.000 18.000 17.000 15.000 10.000 16.000 13.700 10.000 9.000 7.000 10.000 8.000 5.000 5.000 5.000

PARAMETERS: Gamma c 20.00 0.00 20.00 0.00 18.50 0.00 18.50 0.00 8.70 0.00 8.40 0.00 9.50 0.00

NUMBER OF LINE LOADS

01-01-2006

PhiB 32.00 16.30 20.00 20.00 20.00 17.00 35.00

PP 0.00 0.00 0.00 -1.00 -1.00 -1.00 -1.00

DPhi 12.00 48.10 25.00 25.00 25.00 15.00

pN 400.00 28.20 80.00 80.00 80.00 30.00

0

NUMBER OF DISTRIBUTED LOADS COEFF. OF SEISMICITY kx, ky

42

0 0.0000

0.0000

6

6. INPUT DATA

Fig 6.4 Screen view of the cross section-start from default origin and scale

Fig 6.5 ZOOM/ MOVE... Start adjusting scale and origin to the whole section

43

6. INPUT DATA

Fig 6.6 ZOOM/ MOVE...Finished adjustments of scale of the whole cross section

Fig 6.7 SREEN VIEW. Finished adjustments of scale of the whole cross section

44

6. INPUT DATA

Fig 6.8 ZOOM/ MOVE… Adjusted scale and origin to start some work.

Fig 6.9 SCREEN VIEW. Adjusted scale and origin from Fig. 6.8.

45

7. BE.EXE

7. USING BE.EXE After the MAIN DATA are entered or read from the file, you still have the INITIAL MENU, and you should enter some slip-circles by selecting the second option in the menu: 2 ENTER SLIP-CIRCLES option from the INITIAL MENU (same option is offered as 1 from WORKING MENU, which you might choose at some later stage of the computation). You can use one of the options by answering to the request from the screen: DEFINE the PROCEDURE - Automatic search or Grid (A/G) ? If you just press , or any other character except A, or G, the program will return you to the INITIAL MENU. If you enter G or A, the graph with the cross section will automatically appear on the screen and you will be asked to enter details. Two possible choices are: A - means Automatic search for the critical slip-circle. The graph of the cross section will appear on the screen as shown in Fig. 7.1 and Fig. 7.3. Program asks for your initial guess: INITIAL CENTER Xc, Yc ? After that you should enter the size of the opening of the finite difference grid "a", described earlier in Figure 5.3 as the SPIDER, by answering the question: SPIDER'S LEG (def.=1) a = ? The suggested value of a is 1/10 to 1/20 of the slope height approximately. If you just press , the program will assume the default value of 1.0 m. Next question from the screen asks for the Passing PoinT coordinates (briefly PPT): PPT XP,YP ? If XP is different from 0 (zero), program will take that the pair of given coordinates (separated by ","), define the point through which all the slip circles in this group must pass, as shown in Fig. 7.1 and Fig. 7.2. If XP is entered as 0 (zero) this value is accepted as the signal that the value YP entered afterwards, defines the elevation "y" of the horizontal tangent for slip-circles in this set, as shown in Fig. 7.3 and Fig. 7.4.

46

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Fig. 7.1 Starting Automatic search, circles passing through the toe.

Fig. 7.2 Graph after Compute with start shown in Fig. 7.1 47

7. BE.EXE

Fig. 7.3 Starting Automatic search, tangential circles.

Fig. 7.4. Graph after Compute with start shown in Fig. 7.3 48

7. BE.EXE

G - meaning GRID OF CENTERS. First example is shown in Fig. 7.5. The Grid will be defined with coordinates for the CENTRAL POINT, the HEIGHT of the GRID and the WIDTH of the GRID. The additional information required is the TILT SHIFT value, which, if entered as zero generates vertical grid, if (+)ve the grid will be tilted to the left and when (-)ve, tilted to the right. The value of the TILT SHIFT is the distance for which the lower corner points of the grid are shifted from the position, which would otherwise define the vertical and orthogonal grid. The upper corners of the grid boundary will be shifted in the opposite direction for the same distance. Note that slightly inclined grid, which is dependent on the TILT SHIFT value, might have certain advantages as the contours of safety factors could be smoother when compared to the vertical grid. After that answer the question: NUMBER OF NODES NX, NY. See Fig. 7.5. NX is the number of nodes in "x" direction and NY is the number of nodes in "y" direction. The product NX * NY is limited to max. 150 meaning that up to 150 circles can be considered within this option. If the product NX x NY is equal to zero, control will be switched to the INITIAL MENU. You can generate centers of the circle along a single horizontal line if you enter the HEIGHT of the GRID as zero, or along any inclined or vertical line if you choose WIDTH of the GRID as zero. If both WIDTH and HEIGHT are entered as zero, one single circle will be defined, provided that you enter 1 (number one) for NX or NY (depending which value is required on the screen). PPT XP,YP Passing PoinT(s) coordinates shall be entered next in an identical manner as in the Automatic search option in page 46. After completing the entering of data for one of the selected options described above, the graph will reappear on the screen, and you will be able to see if entered data describe what you wanted. Graph will show the cross-section and defined slip-circles, as shown in Fig. 7.6, page 50.

Note: You should pay attention that any slip-circle should not consist of two or more sliding bodies, passing through the air between them. Program will not recognize that, giving erroneous results or an error message, and you should take care that this does not happen. If that happens, you should use dummy lines (with negative MT pointers) which will appear on the graph, but will be ignored in computation.

After you have examined any of the graphs, just press any key to continue, and the INITIAL MENU will be offered. Press 9 to COMPUTE.

49

7. BE.EXE

7.5 Entering GRID. Tangential circles.

Fig. 7.6 Circles defined by GRID shown in Fig. 7.5 50

7. BE.EXE

If the graph has shown what seems to be the acceptable definition of your problem, you can choose 9 COMPUTE ! option, (which is No.9 in MAIN MENU of both stability programs), to obtain results for this set of slip-circles. If PRINTING OPTION IS (Blank), the default option at the start of the run, only the final result (Fs) will be shown. While computing, the program will show the message: COMPUTING ... PLEASE WAIT ... Above this message on the screen you will be able to follow the progress of computation. In the case that you have used the Grid option the value of each computed FS will appear. In the case of the Automatic search only the safety factor of the each current step will appear. While solving smaller problems and when using fast machines you will probably not be able to read the numbers. Immediately after completion of computation the VIEW GRAPH from the WORKING MENU will be automatically called and you will be able to see graph with results, path to a critical center as shown in Fig. 7.2 and Fig. 7.4, or the grid with contours, as shown in and Fig 7.7. The next set of the convenient options are numbered from 2 to 5 in the WORKING MENU, page 52. These options can be used only if you have some initially estimated critical circle from some previous runs like the ones shown in Fig.7.2 or Fig.7.4 or Fig.7.7.

Fig. 7.7. Graph after Compute with start shown in Fig. 7.5 and Fig.7.6. 51

7. BE.EXE

If PRINTING OPTION IS (Screen), the table with results will appear on the screen during computation, but it should be used rarely as the computation is fast and you will not be able to follow what is on the screen. If PRINTING OPTION IS (File), just before starting computation the printing option is automatically reverted to (Blank) during computation, though you can save .TXT file with results afterwards using option 8 in WORKING MENU. In last two cases (Screen) and/or (File), after the completion of computation, you should press any key, the control will be switched to the VIEW GRAPH with the position of the critical circle in this set, as shown for example, in Fig. 7.2, Fig. 7.4 and Fig. 7.7. Press any key again to reach the WORKING MENU with an outline of results in the top left corner of the screen. WORKING MENU in BE.EXE looks as follows: CHOICE W O R K I N G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

MENU

( BE )

ENTER SLIP-CIRCLES INTERACTIVE SEARCH for Fs,min NEW INITIAL CENTER NEW PASSING POINT(s) LOCAL GRID 11 x 11 H/S = 1.0 (Note: H/S can be changed) VIEW GRAPH CONTOUR INTERVALS CI=0.02 (Note: CI can be changed) REPEAT PRINTING RESULTS (Screen) or (File) PRINTING OPTION IS: (Blank) or (Screen) or (File) SWITCH to GE. . . (General method) I N I T I A L M E N U. . . C H A N G E S. . . SAVE .BGP FILE (HP-GL & Printing ) SAVE CRITICAL CIRCLE FOR GE. . . SWITCH to BGP... (HP-GL & Printing ) ZOOM/ MOVE/ AXES/ MIRROR QUIT

CHOICE No.? Depending of results obtained and your estimate if the results obtained represent the smallest safety factor for the considered cross section, you may choose other possibilities which include further search for the critical circle by selecting options numbered from 1 to 5: 1 ENTER SLIP-CIRCLES (same as option 2 in INITIAL MENU). This option is used normally only in some initial stages as only the center of circles can be varied but the PASSING POINT is fixed, as shown in examples from Fig. 7.1 to Fig.7.7. Note that the PASSING POINT(s) can be varied using options 2, 4 and 5.

52

7. BE.EXE

2 INTERACTIVE SEARCH for Fs,min can be selected and the initial slip circle for the interactive search will be the one from the previous run. The search is performed in an interactive window like the one shown in Fig. 7.8 to Fig. 7.13 in pages 54-57. The graphical content on the screen and the procedure for the interactive search will depend on the type of the current PPT (PASSING POINT) in the previous run. For example, if you select this option after the result shown in Fig.7.2, and press C (for Compute) and the window will be like the one shown in Fig. 7.8 and you can move the PPT in the horizontal direction. However, if the PPT is defined with XP=0 implying that YP defines the tangent for all slip circles in the set, point defining the horizontal tangent can be moved in vertical direction only, as shown in Fig. 7.13. In any case, you can now either move the PPT using +/- keys (what would be the reasonable thing to do as the slip circle is critical for the previous passing point) or move the center (changing the center coordinates of the slip-circle) using arrow keys ← ↑ → ↓ . The amount of the movement will depend on the value of Step which is initially set to 1.0 (meters). In an example shown in Fig. 7.8, in the previous run, the PPT was specified with non-zero value of XP, it defines the point through which all slip circles must pass. In this program it is taken that such a point can be moved only horizontally, changing XP value while the value of YP remains unchanged, as shown in Fig. 7.8 and Fig. 7.9. Radius of the circle is computed automatically and shown in the window in Fig. 7.9 or Fig. 7.11. After performing changes you can immediately Compute! hitting letter C or number 9 after each move. The result of computation appears in the bottom right corner indicating the type of the change in FS which can be either No change, (white color), or Increase (red color) or Decrease (green color). Green color encourages you perform the change of the selected variable in same direction until you get Increase (red color) in order to obtain minimum. It works like the traffic signal. The interactive iterative procedure is based on the concept that you change one variable (center or passing point) while the safety factor is decreasing, and to skip to the other variable until the Fs is further decreasing. When the center and the passing point arrive to such a position that any change causes the Increase of Fs, the Step is reduced, and the procedure repeated. From the practical experience, the Step should be reduced to 1/3 to 1/5 of the previous value for the next set of the interactive iterations. To change Step in an iterative window press S and enter the new desired value. You can iterate to any desired accuracy or until all 5 visible significant digits stop changing. Note that when you press SPACE TO EXIT program will compute automatically the safety factor for the current slip circle and show the VIEW GRAPH window from WORKING MENU.

53

7. BE.EXE

Fig. 7.8 Interactive window for slip-circle for non-zero XP.

Fig. 7.9 Point P moved and the new circle from the same center is defined. 54

7. BE.EXE

Fig. 7.10 New Fs is computed and decreasing.

Fig. 7.11 Center moved and the new circle defined for the same PASSING POINT P. 55

7. BE.EXE

Fig. 7.12 Interactive window for slip-circle for zero XP (YP tangent).

Fig. 7.13 Point P moved and the new circle from the same center is defined. 56

7. BE.EXE

Fig. 7.14 New Fs is computed and decreasing. 3 NEW INITIAL CENTER can be selected for a new run under A (Automatic search) option, but the PASSING POINT will remain the same as in the previous run. 4 NEW PASSING POINT(s) can be entered, but the previously existing grid of centers if the currently active option was G (Grid) which will remain the same. The previous center will define the initial circle if the active option was A (Automatic search). 5 LOCAL GRID option is the most powerful feature for the search of the critical slip-circle because you can compute up to 11 x 11 x 25 = 3025 slip-circles and their safety factors Fs in a single run. The central point of the grid is automatically placed in the center of the estimated critical circle from any previous run. This option generates a grid of 11 x 11 =121 center nodes, as shown in Fig. 7.15, and allows you to enter two limits within which the program can insert up to max. 25 equally spaced PASSING POINTS. When selected, you will be asked to enter the SIZE OF GRID, and TILT (LLL/LL/L/ V / R/RR/RRR), meaning Left-Left-Left, Left, Vertical, Right or Right-Right, etc. The default, if you only press , is that you have selected V for the Vertical grid. Definitions for these entries are shown in Fig. 7.15 for the V, R & RR cases only, and L and LL options just imply tilting of the grid in the opposite direction. If you press only for the SIZE OF THE GRID, control will return back to the WORKING MENU. The height H of the grid will be automatically taken as H/S Ratio multiplied by the SIZE. Note that the default value of the H/S is initially, by default, taken as 1.0 (top of the Fig. 7.15) but this value can be changed, usually increased, to suit better your needs, (bottom of Fig. 7.15). See also option 11 in CHANGES (BE), page 65. 57

7. BE.EXE

Fig. 7.15 Local grid. Density is 11 centers per row and column (121 centers). When this option 5 Local Grid is selected, the last critical slip-circle will be shown, for example, see Fig. 7.16 and Fig. 7.19, where you should enter what is required. First is the SIZE OF GRID, and TILT. After that you will be informed about the last valid PASSING POINT co-ordinates (shown as LAST PPT XP,YP). Next entry is the number of PASSING POINTS (No. of PPTs … on the screen) which you wish to have in the next run, with max. 25, and the default number (if you only press ) is 11. The value 11 is taken as a default because in this case the passing points or tangents will be spaced at 1/10 of the interval defined by point co-ordinates of PASSING POINTS, (FIRST PPT P1 x,y …and the SECOND PPT P2… on the screen for brevity). In the case that for the P1 you enter x not equal to zero, and some y, for P2 you will have to enter x, y values (see Fig. 7.16). To minimise the input data for definition of these limiting passing points, in the case that you enter x=0 for P1 and some y value, only y value will be required for P2 (see Fig. 7.19), as you have selected the tangent at various levels y. After you have defined P2, the graph will show the set of limiting slipcircles drawn from the corner points and the central point of the grid through each of 58

7. BE.EXE

passing point. For example, in Fig. 6.16, only 5 passing points have been selected and 25 limiting slip circles are shown in Fig. 7.17, though 121 x 5=605 slip-circles are defined for computation in this case. In the second example, which starts from Fig. 7.19, only 7 passing points (tangents) have been prescribed; Fig. 7.20 shows 5x7=35 slipcircles, though 121 x 7 = 847 slip-circles are defined for computation. After you have seen the limiting slip-circles (Fig.7.17 and Fig.7.20), press space bar and the INITIAL MENU will show, allowing you to select option 9 for COMPUTE ! In the centre of the screen the message COMPUTING ... PLEASE WAIT ... will appear and on the top of this message you might be able to follow the progress of computation. As soon as the computation is finished, the screen will show results, for example like those shown in Fig. 7.18 and Fig. 7.21. The smallest Fs from these two trials are practically the same (Fs=1.6737 in Fig. 7.18 and Fs=1.6738 in Fig. 7.21). You might further refine your search by selecting some smaller SIZE OF GRID and define closer points P1 to P2 in such away that the last critical passing point is between new P1 and P2, and to increase the number of passing points between P1 and P2. Besides that, you can change H/S Ratio, (see option 11 in page 65), usually to some value larger than 1, say 1.6, for example and use this same option to obtain results for final presentation. The result for the currently considered example might look something like shown in Fig. 7.22.

Fig. 7.16 LOCAL GRID applied after computation with results in Fig. 7.2

59

7. BE.EXE

Fig. 7.17 Limiting circles for LOCAL GRID defined in Fig. 7.16

Fig. 7.18 Results for LOCAL GRID and circles defined in Fig. 7.16 and 7.17. 60

7. BE.EXE

Fig. 7.19 LOCAL GRID applied after computation with results in Fig. 7.4

Fig. 7.20 Limiting circles for LOCAL GRID defined in Fig. 7.19 61

7. BE.EXE

Fig. 7.21 Results for LOCAL GRID and circles defined in Fig. 7.19 and 7.20.

Fig. 7.22 Results for LOCAL GRID after some refinements. 62

7. BE.EXE

6 VIEW GRAPH will show cross section with results, the critical slip-circle, the path to a critical center or the grid with contours and passing point(s). 7 CONTOUR INTERVALS are set to 0.02 by default and are relevant to the Grid and the Local grid options only. If contours are to dense, the value should be increased and vice versa. The value of the contour interval is shown as CI=… in screen graphs as well as in graphs produced using HP-GL format. 8 REPEAT PRINTING RESULTS option is in the WORKING MENU. If in the previous run the GRID option was used, then the whole table of results can be revealed (pages 69, 70). In the case of Automatic search, only the last set of five circles can be seen on the screen or saved to the .TXT file. (Step 5 in page 68). In the Local grid case, only the smallest Fs for each center can be seen on the screen or saved to the .TXT file. The table of results is very similar to the table of results from the Grid option. Note: Printing of final results on your printer can be performed in two ways: I. After completion of computation after activating (File, see next option 9) and asking for REPEAT PRINTING RESULTS (Screen) or (File). II. From the program BGP.EXE if you have saved the *.BGP file with results of computation by any of the two stability programs. 9 PRINTING OPTION IS: (Blank) or (Screen) or (File) permits you to control printing of input data and results of computation. Initially, as the program is started, this PRINTING OPTION will be: (Blank) meaning that the whole process of computation will not be shown on the screen when COMPUTE ! option is chosen. If used, (Blank) will affect printing of the results of computation only. The message COMPUTING...PLEASE WAIT... appears and only the final result, (Fs) will be shown on the screen without details. (Screen) means that output of data as well as intermediate phases of computation, when requested from the menu, will be displayed on the screen. (File) option implies that, input data and/or results will be saved in *.TXT file. 10 SWITCH TO GE...(General Method) will save MAIN DATA to a temporary file (TEMP.TMP), start running GE.EXE which reads this temporary file. MAIN DATA and the critical circle are transferred and the computation for the circular slip surface using the General method will be carried automatically. The function f(x) SHAPE No. 2 is set by default, see Fig. 7.23, next page.

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7. BE.EXE

Fig. 7.23 Result of SWITCH TO GE...-Critical circle analyzed by General method 11 I N I T I A L M E N U . . . Brings INITIAL MENU (BE), (Section 5, page 23) 12 C H A N G E S . . . Brings menu CHANGES (BE) (See next page) 13 SAVE .BGP FILE (HP-GL & Printing) is offered in WORKING MENU of both stability programs. This option is very handy for saving files with MAIN DATA and results for later processing, plotting and/or printing using BGP.EXE. You have to give the name of the file only, and the extension will be automatically set to .BGP. 14 SAVE CRITICAL CIRCLE FOR GE . . .will form and save the file with circular slip surface with an extension .SS, which can be later loaded and analyzed by a General method. The default function f(x) No.2 is set automatically. 15 SWITCH to BGP... (HP-GL & Printing ) may be used alternatively to option 13. You may select this option and transfer MAIN DATA and results of computation for printing and/or produce graphic files. Note that in this case the file with results (TEMP.TMP) will be temporary only. You can return from BGP.EXE back to BE.EXE with results of computations and continue the analyses after, for example, saving graphics in HP-GL, *.PLT format. 16 ZOOM/ MOVE/ AXES/ MIRROR option is described previously in Section 6, page 38. It is placed in this menu to facilitate the adjustments after performed computations, for

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example, if the center of the critical circle, or the grid with contours is not within the current visible screen area. 17 Q U I T . . .You will be asked to confirm that you are leaving BE.EXE. **** Menu CHANGES (BE) contains 10 options (1 to 10) for MAIN DATA editing which are the same for both stability programs and described in more detail in Section 9, and specific options for slip-circle analyses, as well as options for the control of graphics. These options are: 11 12 13 14 15 16 17

LOCAL GRID Ratio H/S = 1.00 (Note: H/S can be changed) ZOOM/ MOVE/ AXES/ MIRROR ( X0=320 Y0= 240 ) SHIFT SECTION and/or CHANGE SCALE RMX=10.00 VIEW GRAPH (with axes) or (without axes) INITIAL MENU... WORKING MENU... COMPUTE !

These options can be described as follows: 11 LOCAL GRID Ratio H/S effective in WORKING MENU in LOCAL GRID… option can be changed. H/S Ratio can be selected in the range 0.2 to 5.0. If you only press without selecting the Ratio H/S, the default value of 1.0 will apply. 12 ZOOM/ MOVE/ AXES/ MIRROR option is described previously in Section 6, page 38. It is placed in this menu to facilitate the adjustments after some changes in geometry are performed. 13 SHIFT SECTION and/or CHANGE SCALE... can be used to relocate the geometry of the cross section by shifting it for some increment in x and y directions with dx and dy values with respect to origin. Handy if you change your mind about the relative position of the cross section with respect to origin after you have calculated all the coordinates for input in main data. Moreover, the scale can be changed by explicitly entering the appropriate value. An example is given in Section 9, page 102, 103, Fig. 9.4, Fig. 9.5 and Fig. 9.6. 14 VIEW GRAPH will show the cross section with lines in different colors. Namely, each line describing the upper boundary of the same material zone will be of the same color. It is used to check whether material zoning is described properly. 15 & 16 bring you respective MENU(s) 17 COMPUTE ! offers a possibility to restart computation immediately after changes have been completed, without resorting to the INITIAL MENU.

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*** Program will give correct results if the problem is properly defined. The Bishop's expression for FS is solved by iteration. The number of iterations is limited to 100 by the program. If the convergence is not reached, the program will give you the appropriate message. The tolerance ∆Fs for the acceptance of the result is 0.00000001, the accuracy much higher then required in practice. It is well known fact that in some cases the Bishop method might give dubious results if tension stresses appear on base of some slices. This can happen at the top part of the slope which has a cohesion but the tension crack is not introduced, or in the case of the high friction angle, upward inclined base of the slice and low safety factor. If that happens, an asterisk * will follow the Fs result on the screen and in the printed output as a warning signal. If that happens, it is advisable to rerun the same circle using the General method (GE.EXE). In the case that the shearing strength is defined with the nonlinear failure envelope, there is an additional iteration cycle on normal stresses on the base of each slice. The tolerance in this case depends on the currently active option TOLERANCES described in Section 8, page 89. The Default tolerance for normal stresses acting on the base of each slice is ∆σ’=0.1% and the Small is ∆σ’=0.01%. The corresponding tolerances for the normal stresses on the base of the slice are used in General Method as well, in GE.EXE, as shown in page 89. As the amount of input data for definition of the grid of centers, automatic search or single slip-circle is rather small, saving this set of data in a separate file, without results of computation, is not envisaged by the program. However, the complete set of input data, slip circles and results can be saved in .BGP type files, and recovered by BE.EXE using SWITCH TO BE…(Bishop Method) option in BGP.EXE program, as described later in Section 11, p.132.

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7.1 RESULTS FROM BE.EXE All input data and results of computation when running stability programs can be seen on the screen and/or printed to text files. As an alternative, results of computation can be viewed and/or printed to text files using BGP.EXE. Table of results (pages 68-70) contains heading with following meaning: No. Number of the slip circle. If Automatic search is used, numbers are from 1 to 5 in each step for each position of the spider. Examples of this kind for cases shown in Fig. 7.1 to Fig. 7.4 and in page 68, top. In the case that the Grid option is used, like in the case shown in Fig. 7.5 to Fig. 7.7, or in the case of the LOCAL GRID, the numbers follow the sequence from 1 to NX*NY. Examples of this kind are given in pages 68 and 70. Xc and Yc are coordinates of the center of the slip-circle and R is Radius SLICES Number of slices for a particular circle. Note that this number is only approximately equal to the prescribed NUMBER OF SLICES, as explained earlier. Fs Factor of safety accurate to four significant digits. In the case that tension stresses on one or more base of a slice is encountered, result is marked with * (asterisk) as a warning only, as shown in all the cases in this Section, starting from Fig. 7.2 and in pages 68-70. The main reason is the existence of the cohesion term, which causes this to happen locally and inevitably if such a shear strength description is used for the upper part of the slope. The influence of this tension can be checked by running the same case using GE.EXE (see Example 10.1 in page 104). Most frequently, the influence is not very significant. This physical inadmissibility can be avoided be either introducing the tension crack at the top of the slope as in Examples 3-A and 3-B (p. 110), or using the nonlinear failure envelope with smaller or zero cohesion, (see Example No.2, p.107). The asterisk sign * does not appear if all bases of slices are in compression (no tension). At the end of the table program writes the position of the critical slip-circle with smallest factor of safety from the group considered. The table described above will appear on the screen if PRINTING OPTION is (Screen), and/ or printed to the .TXT file using PRINTING OPTION is (File). 7 REPEAT PRINTING RESULTS option is in the WORKING MENU. In the case of Automatic search, only the last set of five circles can be printed. If the Grid option was used in the run, then the whole table of results for each slipcircle center can be revealed. If the LOCAL GRID option was used in the run, with several Passing PoinTs, then the printed table shows the result for each slip-circle with the smallest Fs in each center.

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Table of results for the case shown in Fig. 6.1 (p.39) and Fig. 7.2 (p.47). Example 1 Simple slope Linear envelope BISHOP-EXTENDED SLOPE STABILITY ANALYSIS PASSING POINT 36.000 9.000 AUTOMATIC SEARCH FOR CRITICAL CENTER INITIAL CENTER Xc, Yc 20.00 35.00 SPIDER's LEG 0.80 RESULTS OF COMPUTATION No. Xc Yc R SLICES Fs 1 25.925 28.272 21.747 40 1.684 2 26.725 28.272 21.388 40 1.694 3 25.125 28.272 22.129 40 1.689 4 25.925 27.472 21.041 40 1.684 5 25.925 29.072 22.459 40 1.686 TOTAL NUMBER OF STEPS : 5 Fsmin 1.6837 CENTER Xc, Yc 25.925 28.272 R= 21.747 CIRCLE THROUGH THE POINT 36.000 9.000

* * * * * *

Table of results for the case shown in Fig. 6.1 (p.39) and Fig. 7.7 (p.51). Example 1 Simple slope Linear envelope BISHOP-EXTENDED SLOPE STABILITY ANALYSIS GRID CENTRAL POINT x, y 28.00 27.00 HEIGHT of the GRID 15.00 WIDTH 10.00 SHIFT FOR TILT -2.00 NX, NY 10 15 PASSING POINT 0.000 5.000 RESULTS OF COMPUTATION No. Xc Yc R SLICES Fs 1 21.000 19.500 14.500 40 2.093 2 22.111 19.500 14.500 40 2.004 3 23.222 19.500 14.500 40 1.933 4 24.333 19.500 14.500 40 1.884 5 25.444 19.500 14.500 40 1.860 6 26.556 19.500 14.500 40 1.866 7 27.667 19.500 14.500 40 1.905 8 28.778 19.500 14.500 40 1.967 9 29.889 19.500 14.500 40 2.056 10 31.000 19.500 14.500 40 2.175

* * * * * * * * * *

…omitted for brevity… 51 52 53 54 55 56 57 58

22.429 23.540 24.651 25.762 26.873 27.984 29.095 30.206

24.857 24.857 24.857 24.857 24.857 24.857 24.857 24.857 68

19.857 19.857 19.857 19.857 19.857 19.857 19.857 19.857

40 40 40 39 40 39 40 40

1.818 1.751 1.707 1.689 1.692 1.718 1.770 1.855

* * * * * * * *

7. BE.EXE

59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88

31.317 32.429 22.714 23.825 24.937 26.048 27.159 28.270 29.381 30.492 31.603 32.714 23.000 24.111 25.222 26.333 27.444 28.556 29.667 30.778 31.889 33.000 23.286 24.397 25.508 26.619 27.730 28.841 29.952 31.063

24.857 24.857 25.929 25.929 25.929 25.929 25.929 25.929 25.929 25.929 25.929 25.929 27.000 27.000 27.000 27.000 27.000 27.000 27.000 27.000 27.000 27.000 28.071 28.071 28.071 28.071 28.071 28.071 28.071 28.071

19.857 19.857 20.928 20.928 20.928 20.928 20.928 20.928 20.928 20.928 20.928 20.928 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 22.000 23.071 23.071 23.071 23.071 23.071 23.071 23.071 23.071

40 40 40 41 40 40 39 40 40 40 40 40 40 40 39 40 40 39 40 40 40 40 40 40 40 39 40 40 39 40

1.971 2.111 1.788 1.726 1.694 1.681 1.687 1.714 1.766 1.848 1.967 2.111 1.763 1.714 1.686 1.677 1.686 1.714 1.766 1.847 1.965 2.113 1.745 1.706 1.683 1.677 1.688 1.718 1.770 1.849

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

40 40 40 40 40 39 39 39 40 40 40 40 39

1.901 2.005 2.150 1.737 1.721 1.717 1.726 1.748 1.785 1.839 1.916 2.021 2.162

* * * * * * * * * * * * *

…omitted for brevity… 138 139 140 141 142 143 144 145 146 147 148 149 150

32.492 33.603 34.714 25.000 26.111 27.222 28.333 29.444 30.556 31.667 32.778 33.889 35.000

33.429 33.429 33.429 34.500 34.500 34.500 34.500 34.500 34.500 34.500 34.500 34.500 34.500

28.428 28.428 28.428 29.500 29.500 29.500 29.500 29.500 29.500 29.500 29.500 29.500 29.500

CRITICAL CIRCLE No. 74 CENTER Xc, Yc 26.333 27.000 CIRCLE TANGENTIAL TO THE LEVEL

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Table of results for the LOCAL GRID shown in Fig. 7.18 (p.60). Example 1 Simple slope Linear envelope BISHOP-EXTENDED SLOPE STABILITY ANALYSIS GRID CENTRAL POINT x, y 26.33 27.00 HEIGHT of the GRID 6.00 WIDTH 6.00 SHIFT FOR TILT 0.00 NX, NY 11 11 5 PASSING POINTS From 36.000 9.000 to 41.000 9.000 PPT 1 36.000 9.000 PPT 2 37.250 9.000 PPT 3 38.500 9.000 PPT 4 39.750 9.000 PPT 5 41.000 9.000 RESULTS OF COMPUTATION No. Xc Yc 1 23.333 24.000 2 23.933 24.000 3 24.533 24.000

R SLICES Fs 19.633 40 1.765 19.251 40 1.738 18.881 40 1.719

* * *

…omitted for brevity… 78 79 80 81 82 83 84 85 86 87

23.333 23.933 24.533 25.133 25.733 26.333 26.933 27.533 28.133 28.733

28.200 28.200 28.200 28.200 28.200 28.200 28.200 28.200 28.200 28.200

23.002 22.677 22.363 22.704 22.389 22.730 22.415 22.757 23.112 22.784

40 40 40 41 41 40 39 40 40 39

1.742 1.719 1.701 1.686 1.677 1.674 1.676 1.683 1.696 1.715

* * * * * * * * * *

40 39 40 40 40 40 40 40

1.723 1.704 1.678 1.680 1.685 1.695 1.712 1.740

* * * * * * * *

…omitted for brevity… 101 102 116 117 118 119 120 121

23.933 24.533 26.333 26.933 27.533 28.133 28.733 29.333

29.400 29.400 30.000 30.000 30.000 30.000 30.000 30.000

23.701 24.039 24.270 23.975 24.295 24.628 24.320 24.023

CRITICAL CIRCLE No. 83 Fsmin = 1.6738 * CENTER Xc, Yc 26.333 28.200 R= 22.730 CIRCLE THROUGH THE POINT 38.500 9.000

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8. USING GE.EXE After the MAIN DATA are entered or read from the file, the slip surface can be entered either from INITIAL MENU (p.29) from the keyboard (option 2), or loaded from the file (option 4), if you have formed it and saved previously. 2 ENTER SLIP-SURFACE (kbd) is option which is used when you wish to define the slip surface from the keyboard. TITLE FOR THE SLIP-SURFACE is entered next. Suggested length of title is up to 20 characters, preferably less. If you just press the TITLE: “Untitled” will be assigned automatically. After that the following five choices will be offered: ENTER SLIP-SURFACE TYPE P B F M C SHAPE ?

Polygonal Bezier Fixed plane coMposite Circular

Five possibilities shown in Fig. 8.1 are available, which can be chosen by the entering your choice either as P, or B or F or M or C. Small letters will work equally well. If you enter any other letter, or just press , control will return to the INITIAL MENU. Handy in the case that you wish to cancel the process of defining the slip surface.

Fig. 8.1 Five shapes of arbitrary slip surfaces handled by GE.EXE Entering of data for any type of the slip surface can be checked as each entry appears simultaneously on the screen showing the slope cross section.

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P means that you will have to enter x, y co-ordinates that define the Polygonal slip-surface as shown in Fig. 8.2. Note that x coordinates of these points will be automatically arranged in ascending order of "x" and the sequence of entering is arbitrary. You will be able to see on the screen as each point is entered initially. The number of points that define the polygonal slip surface is limited by the software to max. 100. However, there is an advantage of using only up to 9 (nine) points, because in such a case you will be able to use the interactive search for finding the critical slip surface of the polygonal shape, Fig. 8.14, p.85) . That is why in the first line in Fig.8.2 you see the input reminder …(max. 9 -100). It is important that both end points of the polygonal slip surface are placed "in the air" or above the external boundary of the slope in order to ensure the intersection of the slip surface with the slope contour lines. For f(x) SHAPE No. see pages 77-78.

Fig.8.2 Entering polygonal slip surface Some results of computation for this slip surface are shown in Fig. 8.9. The interactive changes performed on the shape this polygonal slip surface are shown in Fig. 8.14.

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B is the signal that using the Bezier curve will generate the arbitrary slip surface. Control point coordinates are entered in sequence that corresponds to a manner in which lines are connected, as shown in Fig. 8.3. It is important that both end points of the Bezier polygon are placed "in the air" or above the external boundary of the slope in order to ensure the intersection of the slip surface with the slope contour lines. The number of Bezier control points is in the range from min. 3 to max. 7. The number of points that approximate the continuous curve is 40 by default and the max. number of points for which the coordinates can be are computed along the Bezier curve is 100. This number can be changed later in an interactive window, Fig.8.15, p.86. For f(x) SHAPE No. see pages 77-78.

Fig.8.3 Entering Bezier type slip surface Results of computation for this slip surface are shown in Fig. 8.10, p.81. Since this initial trial slip surface does not pass along the weak layer, the window for changes is shown first in Fig.8.15 and explained in page 86. It is writer’s opinion that the smooth Bezier curve is the rather versatile one and you are encouraged to try to use it frequently. Some tests on the f(x) distributions are shown in Section 10, Fig. 10.25 to Fig. 10.28.

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F means that you intend to enter a FIXED PLANE slip surface. This type of slip surface is basically polygonal, with a possibility to define a fixed plane for a portion of the slip surface. Before entering the definition of this type of slip surface, you have to define the coordinates of two neighboring points which define the line of the fixed plane, usually within the weak material zone or weak layer. In an example shown in Fig. 8.4 it is a plane just bellow the line 14-15 in an example shown in Fig. 6.3 (p.41). The values of y for selected x=15 and x=25 are calculated and to ensure that the line is within zone 6, just slightly bellow its upper boundary, the vertical distance of 0.1 meters is subtracted from the calculated y value. The corresponding y values are 6.4 and 5.8. Note that you have to enter points in increasing x sequence. Points 1 and 2 are entered to the left and 5 and 6 to the right from the Point No.3 & Point No.4, which define the fixed plane. You will be able to see on the screen as each point is entered initially. The number of points that define the polygonal slip surface is limited by the software to max. 100. However, there is an advantage of using only up to 9 (nine) points, because in such a case you will be able to use the interactive search for finding the critical slip surface of the polygonal shape, see Fig. 8.16, p. 87. It is important that both end points of the polygonal slip surface are placed "in the air" or above the external boundary of the slope in order to ensure the intersection of the slip surface with the slope contour lines. For f(x) SHAPE No. see pages 77-78.

Fig.8.4 Entering Fixed plane type slip surface

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M means that you intend to enter a coMposite slip surface. Composite slip surface is defined by a circle center coordinates Xc, Yc, PASSING POINT coordinates XP, YP, (the distance between these two points define the initial radius R), and two pairs of coordinates (Point No.1 & Point No.2) which define the line or the fixed plane intersected by a circle, as shown here in Fig.8.5. The number of points which define the composite slip surface is 50 by default. You can change this value later in an interactive window (Fig. 88, p.88) to make the circular arcs smoother by increasing the number of points up to 100. This M type slip surface is similar to F type. M type slip surface contains circular arcs above the fixed plane. The F type slip surface has polygonal segments above the fixed plane. For f(x) SHAPE No. see pages 77-78.

Fig. 8.5 Entering coMposite slip surface Results of computation for this slip surface and the default f(x) are shown in Fig. 8.12, p.82. How to performed interactive search for the smallest Fs, starting from this composite slip surface, is shown in the window of Fig. 8.17, p.88.

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C is the signal that you intend to enter the Circular slip surface, what you will do by defining the coordinates of the center and the coordinates of the PASSING POINT, as shown here in Fig.8.6 just in the same manner as in the application of BE.EXE. PASSING POINT. You should enter the pair of coordinates XP, and YP. The significance of this data is the same as in BE.EXE. If the value XP is different from 0 (zero), program will take that the pair of given coordinates (separated by ","), define the point through which the slip circle must pass. If XP is entered as 0 (zero) this value is accepted as the signal that the value YP defines the elevation "y" of the horizontal tangent for the slip-circle. Interactive search window for the critical slip circle is the same as described for BE.EXE, page 53, and Fig.7.8 to Fig.7.14, pages 54 to 57. For f(x) SHAPE No. see pages 77-78.

8.6 Entering Circular slip surface Some results of computation for this circular slip surface are shown in Fig. 8.13, p.82 and in Section 10, p.116.

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f(x) SHAPE No. is chosen afterwards. To simplify the definition of the function f(x) with small number of parameters, shapes numbered from 1 to 6 are shown in Fig. 8.7, p.78. HDE is the Horizontal Distance between Ends of the slip-surface. Shape 1 is constant that would yield the solution analogous to the Spencer (1967) method, which assumes constant inclination of inter-slice forces. This function shape does not require any parameters. It is usually used in practical cases. Examples are shown in Fig. 8.8-(a),p. 79, Fig. 10.1, p.105, Fig. 10.18, p.116 and Fig. 10.26, p.121. Shape 2 is a half sine wave. This is the second of two functions which do not require additional parameters. It is a default f(x), if you enter any SHAPE No. not in the range 1 to 6 or just press . It is usually used in most practical cases and used here for examples shown in Fig. 8.9 to Fig. 8.13 as well as in example shown in Fig. 8.7-(b). This is also a default f(x) assigned to a slip circle saved from BE.EXE. Shape 3 permits the definition of the single linear function between the extreme ends of the slip surface. It has 1 parameter. Highly recommended for pseudo static seismic analysis. Note that the mean value is 1.0. For example, if you enter the value for z1=0, at the opposite end of the slip surface the value of f(x) will be 2. If you enter the value z1=1, it would generate f(x)-constant. For example, see Fig. 10.27, p.122 and Fig. 11.12, p.144. Shape 4 can be interpreted as shape which includes Shapes No.1, No. 2 and No.3. It has 2 parameters. Note that the value in the middle is 1.0. This is rather general, because using the appropriate values for z1 and z2 you can generate all shapes listed above but the advantage of above shapes (1 to 3) is that they have no parameters at all or one parameter only. Shape 5 has 3 parameters. The values of zi shown as horizontal distances measured in horizontal direction are in meters. It permits the definition of rather wide variety of f(x) functions as some of the parameters can be positive values, zeroes and negative. Note, for example, that f(x) for the case shown in Fig. 8.8-(c) all three parameters are equal to zero. For the example shown in Fig. 8.8-(d) the parameters are z1=0, z3=0 and z2=HDE. The comparison of the theoretical solution for a bearing capacity problem shown in Section 10, in Fig. 10.33, p.125, uses this shape. Shape 6 permits the definition of arbitrary function by the set of straight lines in the interval defined by points of the POLYGONAL slip surface. In the case of the CIRCULAR slip surface this shape will imply a constant f(x), identical to Shape 1. It is possible but not convenient to use this shape for other types of slip surfaces. It is not advisable to use this shape for COMPOSITE or BEZIER type slip surfaces because it is impractical to type and enter up to 40-50 or 100 f(x) values. Note: f(x) can be easily changed or varied in order to examine the possible range of results in WORKING MENU (GE) as well as in menu CHANGES (GE).

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Fig. 8.7 f(x) shape numbers and shapes

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Fig. 8.8 Result Fs is usually not very sensitive on f(x) It is the general experience that the final result, the safety factor Fs, is not very sensitive to the shape of f(x) providing that reasonable line of thrust is obtained and that the solution is physically admissible. Figure 8.8 shows this point. Cases (a), (b) and (c) indicate the physically admissible line of thrust and in the case (d), the line of thrust seems rather high in the central part of the sliding body and towards the toe of the slope. In this case the factor of safety is in the rather narrow range for most practical purposes from Fs=1.415-1.422, the difference being less than 1 %. The function f(x) could be determined by the application of the finite element method (Maksimović 1979, Fredlund 1984, Fan et al. 1986), but such an approach can be justified in special cases only, and not in the practical and routine application. Note: Shapes 1 (constant) and 2 (sin wave) are the simplest for practical application as neither requires additional parameters for description. The shape 3 (linear) is also rather simple one and can be frequently used for the sensitivity analysis. Shape 5 uses the automatically generated half-wave sinus function for curved transitions between constants and is found useful in some particular cases, and the number of required parameters not equal to zero varies within 1 and 3. The default shape No. 2 is taken automatically when you do not select the number but just press . All data, which define the slip-surface, can be saved in a file for later use by choosing the option 12 SAVE SLIP-SURFACE in the INITIAL MENU (GE), Section 5.2, p.29. The extension to the file name will be automatically set to .SS. 9 COMPUTE ! is option which you will select from the INITIAL MENU if you are satisfied with the appearance of the slip surface. Though note the following: 79

8. GE.EXE

If PRINTING OPTION (Blank) option is used (the default at the start of the run and should be used in routine work), only the result (Fs) with graph will be shown, and that SPEEDS UP THE COMPUTATION to a fraction of the seconds only. If PRINTING OPTION IS (Screen) tables with intermediate results of iterative computation, described in Section 8.1, will appear on the screen. In order to cope with difficulty for the curious user to follow what is shown on the screen as the lines scroll rather fast, you will have to Hit the space bar to continue… frequently. Rarely used, probably only if you have some problems with convergence. If PRINTING OPTION IS (File), is active during computation, program will respond as above, intermediate results of computation will appear on the screen. Only after computation is completed you will have an opportunity to save results to a .TXT file. All three tables will be saved. In the case that the problem contains the nonlinear description of the soil shearing strength, computation will be rather slow. It is advisable to avoid this option (File) when actually computing. When the computation process is completed, the graph showing the cross section, the slip surface, the line of thrust, the slice limits and factor of safety Fs will appear on the screen, as shown in the following pages. The interslice section with the max. value of the horizontal interslice force (Emax) will be emphasized, as shown marked in Fig. 8.9.

Fig. 8.9 Graph after 9 COMPUTE for the polygonal surface shown in Fig.8.2, p.72.

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Fig. 8.10 Graph after 9 COMPUTE for the Bezier surface from Fig. 8.3, p.73

Fig.8.11 Graph after 9 COMPUTE for the Fixed plane surface from Fig. 8.4, p.74

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Fig. 8.12 Graph after 9 COMPUTE for the coMposite surface shown in Fig. 8.5, p.75

Fig.8.13 Graph after 9 COMPUTE for the Circle shown in Fig. 8.6, p.76

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The same graphs on the screen shown in two previous pages will appear when computation is completed. In the top left corner of the screen, computed Fs will be shown. After seeing graph with results, press any key to arrive at WORKING MENU which looks as follows: CHOICE W O R K I N G 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

MENU

( GE )

I N I T I A L M E N U. . . VIEW GRAPH PRINT RESULTS (Screen) or (File ) PRINTING OPTION IS: (Blank ) or (Screen) or (File ) COEFF. OF SEISMICITY kx, ky 0.000 0.000 f(x) SHAPE No. (any 1 to 6) EQUIVALENT BLOCK & kc ZOOM/ MOVE/ AXES/ MIRROR INTERACTIVE SEARCH for Fs,min. SWITCH to BE. . . (Bishop Method) C H A N G E S. . . SAVE .BGP FILE ( HP-GL & Printing ) SWITCH to BGP.. ( HP-GL & Printing ) TOLERANCES: Default or Small QUIT

CHOICE No.? Depending of results obtained and your estimate if the results obtained represent the smallest safety factor for the considered cross section, you may choose other possibilities which include further search for the critical slip surface by selecting option 9 or checking result against some other f(x) by selecting option 6. Options shown above are as follows: 1 I N I T I A L M E N U. . . self-explanatory 2 VIEW GRAPH Shows graph with results like in Fig.8.9 to Fig. 8.13. 3 PRINT RESULTS … with the appropriate 4 PRINTING OPTION … should be used if you wish to see results in detail on the screen or save results to the .TXT file. Note that in this repeated printing, after completion of the computation, the iteration cycles will not be shown. 4 PRINTING OPTION IS: … This option in routine computation should be kept Blank while computing in this menu for reasons explained earlier, page 31, unless you wish to use the previous option 3. 5 COEFF. OF SEISMICITY kx, ky can be altered. The convention on the sign of acceleration described in MAIN DATA preparation applies here. As new kx and ky are entered, computation starts automatically. Note that ky=0 (the vertical acceleration) is the usual entry. Only values Fs =1.0 and Fs>1 have physical meaning. 6 f(x) SHAPE No. … can be altered. Some examples are shown in Section 10.

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7 EQUIVALENT BLOCK & kc When chosen, the program will map all acting loads and resultant stresses on the slip surface into the system of resultant forces acting on an equivalent block on an inclined plane, give the equivalent inclination, show the average normal and shear stresses on the slip surface and give the initial estimate of the critical acceleration for which the safety factor might become equal to one. After that the following will appear on the screen: Type KC or DIS or press ENTER to continue ? KC if called repeatedly a few times in sequence, will compute the critical acceleration to desired number of significant digits or to obtain Fs=1.0000. The main purpose of this option is to provide data for an estimate of the rigid body displacements due to earthquake acceleration for which the safety factor is less than one by using the Newmark's (1965) principle. DIS in the same sub-menu will ask for the acting acceleration and the duration of the rectangular pulse T/2. The application of this option will give the magnitude of the components of the rigid body displacements. Note that the EQUIVALENT BLOCK option can be used to compute the point which represents the average normal and shear stresses on the slip surface. This point can be used for evaluation of the resistance envelope proposed by Casagrande and advocated by Janbu (1977).

8 ZOOM/ MOVE . . . option is basically described previously in Section 6, page 38. 9 INTERACTIVE SEARCH for Fs,min. This is one of the most important options in this program. It offers a possibility to vary the shape of the previously defined slip surface and immediately see the results. The interactive search is performed in an “interactive window” in which the definition of the change will depend on the TYPE OF THE SLIP SURFACE, though the basic principle is always the same. The windows for these changes are shown in the following pages. Previous result is shown in bottom right corner, second row from the bottom line as Old safety factor. Select and press the control point number which you wish to move. The selected control point number and the circle representing the point will become of red color and point is ready for moving using arrow ← ↑ → ↓ keys or +/- signs. After performing a change you can Compute! by pressing either the letter C or number 9. After the Compute option is executed, in the bottom right corner the result of computation as New factor of safety appears indicating the type of the change in FS. which can be either No change, (white color), or Increase (red color) or Decrease (green color). Green color encourages you to move the point (or to change the variable) in same direction until you get Increase in red in order to obtain minimum. It works like the traffic signal. The minimum is obtained when moving of any point in any direction indicates red color meaning that any change in slip surfaces causes the safety factor to increase for the current value of the Step. Step for the movement of a control point can be changed pressing S (S for Step) and entering the new desired value. The default Step value is 1.0 meter, but it is

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usually reduced to some smaller value at the final stages of iteration for the critical slip surface. From the practical experience, the Step should be reduced to 1/5 to 1/10 of the previous value for the next set of the interactive iterations. The interactive iterative procedure is based on the concept that you keep changing one variable while the safety factor is decreasing, and to skip to the other variable until the Fs is further decreasing. When all the variables (points which define the slip surface) arrive to such a position that any change causes the Increase of New Fs, the Step is reduced, and the procedure repeated. The minimum is obtained when moving of any point in any direction indicates red color meaning that any change in slip surfaces causes the safety factor to Increase. You can iterate to any desired accuracy or until all 5 visible significant digits stop decreasing and the only changes you can see is the change of the color in front of the New computed Fs. Press SPACE (bar) TO EXIT, and that will automatically compute again and automatically call option 2 VIEW GRAPH with the result, just as shown in Fig.8.9 to Fig.8.13, p. 80-82. Polygonal slip surface can be changed by selecting the number of point for which the coordinates will be changed, as shown in Fig. 8.14. Coordinates of the point which define the polygon are changed using arrow keys ← ↑ → ↓ .

Fig. 8.14. Interactive window for the Polygonal slip surface

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8. GE.EXE

Bezier line can be easily varied by moving the control points of the Bezier polygon. The interactive window for these changes is shown in Fig. 8.15. Select and press the control point number which you wish to move. The selected control point number and the circle representing the point will become of red color and point is ready for moving using arrow ← ↑ → ↓ keys. After performing a change, simply by moving the selected point in a trial direction, you should Compute! by pressing either the letter C or number 9. Decrease in green color encourages you to move the point in the same direction until you get Increase in red and than you move point back in order to obtain minimum. Pressing N you can change the number of points which approximate the Bezier curve. The default value for the number of poiNts N is 40, and the max. is 100, both values sufficiently large for a rather smooth and continuous appearance of the Bezier curve.

Fig. 8.15. Interactive window for the Bezier type slip surface The application of the Bezier curve for the description of the arbitrary slip surface with an interactive search for the critical slip surface and Fs,min. is probably, the unique and the most versatile feature of this software package. It is the author’s experience from many practical jobs that it almost always gives the slip surface with the safety factor smaller than the factor of safety of the critical surface of circular shape. In this manual an example is shown where the Bezier slip surface gives the smallest Fs, in comparison with all other shapes (circle, polygon and composite).

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8. GE.EXE

Fixed plane polygonal slip surface, see Fig. 8.16, can be varied using any of four arrow keys ← ↑ → ↓ for all points (1, 2, 5 and 6 in the example) except for two points which define the fixed plane. Point which define the fixed plane (points 3 & 4 in the example shown here) are moved along the defined plane using only ← → (horizontal) arrows, though the fixed plane can be inclined. The program shall not respond to ↑ ↓ (vertical) arrows when points which define the fixed plane are active. The amount of change of the coordinate is defined with Step which you can make active by pressing the letter S or s. After performing a change, simply by moving the selected point(s) in a trial direction, you should Compute! by pressing either the letter C or number 9. Observe the lower right corner. Decrease in green color encourages you to move the point in the same direction until you get Increase in red and than you move point back in order to obtain minimum. Results of the Interactive search are shown Fig. 10.29 and Fig. 10.30, page 123.

Fig. 8.16 Interactive window for the Fixed plane type slip surface

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8. GE.EXE

Composite slip surface can be varied by either changing the circle center coordinates using arrow keys ← ↑ → ↓ or changing the radius of the circular arcs using + or – keys which will MOVE PPT (meaning move the Passing PoinT), or both as shown in Fig. 8.17. By changing the number of points, pressing N previously, a set of polygonal slip surfaces can be generated and varied. The default value for the number of points is 50, and the max. is 100, both values sufficiently large for a rather smooth and continuous appearance of circular arc segments.

Fig. 8.17 Interactive window for the coMposite type slip surface Circular slip surface can be altered, and the initial slip circle for the interactive search will be the one from the previous run. The search is performed in an identical interactive window like the one used in BE.EXE, described and shown in Section 7, and shown from Fig.7.8 to Fig. 7.14, pages 53 to 57.

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10 SWITCH to BE…(Bishop Method) will save MAIN DATA to a temporary file (TEMP.BG), start running BE.EXE which reads this temporary file. MAIN DATA are transferred and any slip surface of general shape is lost unless it is a CIRCLE for which the safety factor will be automatically computed using Bishop Extended method. 11 C H A N G E S . . .Brings menu CHANGES (GE) (see next page) 12 SAVE .BGP FILE ( HP-GL & Printing )... is offered in WORKING MENU of both stability programs. This option is used to save file with MAIN DATA and results for later use by a program BGP.EXE for graphic presentation and/or printing. You have to give the name of the file only, and the extension will be automatically set to .BGP. 13 SWITCH to BGP ... ( HP-GL & Printing ) Alternatively to the previous option 12, you may select the option which will start running BGP.EXE and transfer MAIN DATA and results of computation for graphic presentation and/or printing. Note that in this case the file with results (TEMP.BGT) will be temporary only. You may save this file from BGP.EXE. You can return from BGP.EXE back to GE.EXE with results of computation and continue the analyses after, for example, saving (export) graphics in HP-GL format. 14 TOLERANCES: Default or Small. Works as a toggle. This option permits you to chose between two levels of the numerical tolerances, which differ for two orders of magnitude for forces and moments and one order of magnitude for normal stresses. The alternative tolerances are for the unbalanced horizontal force ∆En acting on the right hand side end of the sliding body i.e. or on the last slice, unbalanced moment for the last slice ∆Mn and stresses at the base of each slice ∆σi: The tolerance for Default Small

∆En(kN/m1)

∆Mn (kNm/m1)

0.000100 0.000001

0.00100 0.00001

∆σi(%) 0.10 0.01

For slopes of the height in the conventional range from, say, 5 m to 150 m, the option Default is suggested. For very small slopes or stability problems involving only a few meters of the size of the sliding body, the option Small might be required, though this tolerance is preferred in general. In the unlikely case that you encounter the instability in convergence in the iterative computation and/or during INTERACTIVE SEARCH for Fs,min, try to change (switch) the current tolerance. IN THE CASE THAT THE TOLERANCES ARE NOT MET AFTER 2000 ITERATIONS, DO NOT PANIC, SEE HINTS IN THE BOTTOM OF PAGE 92. 15 Q U I T You will be asked to confirm that you are leaving the program GE.EXE.

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Menu C H A N G E S (GE) contains 10 options (1 to 10) for MAIN DATA editing which are the same for both stability programs and described in more detail in Section 9, page 96. Other options in GE.EXE can be described as follows: 11 12 13 14 15 16 17 18

S-S: f(x) SHAPE No. __ ZOOM/ MOVE/ AXES/ MIRROR ( X0=__ Y0= __ ) SHIFT SECTION and/or CHANGE SCALE RMX=__ VIEW GRAPH (with axes) or (without axes) INITIAL MENU... WORKING MENU... COMPUTE !

Some other options are also given with identical purpose but have different number, in BE.EXE, and a few only are different to match the requirements of the particular method of the analysis, i.e.: 11 S-S: is the TITLE FOR SLIP SURFACE. It can be altered if you wish to do so for some book keeping purposes. 12 f(x) SHAPE No.__ can be changed here in CHANGES (GE) and in WORKING MENU (GE). In the former case, immediately as the new function shape is entered, computation does not restart automatically. If you wish to compute, you should use option 18. 13 ZOOM/ MOVE/ AXES/ MIRROR ( X0=__ Y0= __ ) option is described in Section 6, page 38 and Section 9, pages 101-102. 14 SHIFT SECTION and/or CHANGE SCALE RMX=__ (same as option 13 in BE.EXE) can be used to relocate the geometry of the cross section by shifting it for some increment in x and y directions with dx and dy values with respect to the origin. See Section 9, page 101-103, Fig. 9.4, Fig. 9.5, Fig. 9.6 and Fig.9.7. 15 VIEW GRAPH in both stability programs (BE & GE) will show the cross section with lines in different colors. Namely, each line describing the upper boundary of the same material zone will be of the same color. It is used to check if material zoning is described properly. 16, 17 & 18 are self-explanatory.

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8. GE.EXE

8.1 RESULTS FROM GE.EXE Details of results of computation for each slip-surface are shown in three tables. First two tables describe the values of computed constants and parameters for each slice, and the third table shows results obtained by iterative solution of the system of nonlinear equations. A sample output for the case shown in Fig. 10.25, p.121 and Fig. 11.11, p.143, is shown in pages 93 to 95 with the explanations for heading as follows: Heading of the first table: DATA 1 (See page 93) I XX YY b tanAlpha W/b u

Slice number (starting from 1) x of the vertical center line of the slice y of the center point at the base of the slice Width of the slice Tan of the angle of the base of the slice Ratio of the weight and the width of the slice Pore water pressure at the base of the slice.

Heading of the second table: DATA 2 (See page 94) I Slice number (starting from 1) RHor Horizontal component due to pore pressure, horizontal inertia force due to earthquake and external loading RVert Vertical component due to the weight of the slice, pore pressure, vertical component if inertia force due to earthquake and external loading RM Moment with respect to the center point at the base of the slice due to known forces Phi The angle of the shearing resistance at the base of the slice c The cohesion value at the base of the slice f(x) The value of f(x) in the interslice section. Note that in the printouts there is the difference between the Slice Number I in DATA 1 and DATA 2 which start from 1, and the Interslice section number in DATA 3, which starts from 0 (zero) at the left end of the slip surface. In DATA 3, I has dual meaning; for XML, R, E and tanBeta I represents the interslice section number, and for Sig and Tau, I represents the slice number. Results listed above are mainly used for checking the input data. It is recommended not to use F (File) option while COMPUTING with GE.EXE. Program prevents the possibility to form a .TXT file with all intermediate results of iterative computation because the amount of printout, showing all the iterations, would be huge. If you wish to produce the hard copy of all results, it is much more efficient to use the option from the WORKING MENU: 3 PRINT RESULTS (Screen) or (File) after the computation run under Blank printer option. As an alternative you might save results as .BGP file and perform printing to a text file (.TXT) of the same results from BGP.EXE.

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RESULTS OF INTERMEDIATE ITERATIVE COMPUTATION can be traced during computation as they are printed in an outline form only and contain the following: FS

Current value of the factor of safety, which satisfies the condition that interslice forces at both ends of the sliding body are equal to practical zero. A Current value of the unknown A ENL The unbalanced horizontal force at the end of the sliding body, rather small value, as it is printed after the convergence to the tolerable value is reached. DM The unbalanced moment for the last slice. The solution is reached after this value is smaller than the prescribed tolerance. This value monotonously decreases in the iteration process after the second iteration on moments. ITFS Iteration number in the internal iteration cycle with fixed value of A. These intermediate results of computation can be seen on the screen if PRINTING OPTION IS: (Screen) or (File). You will be able to control scrolling during iteration (as the File option will be automatically reverted to Screen during iterations) . It is rather unlikely that you will feel the need to see this iteration process unless you encounter some iteration problems. Results of intermediate iteration cannot be saved as the .TXT file. MAIN UNKNOWNS Fs and A are printed as result of computation after all the tolerance criteria are satisfied. (See page 95) Heading of the last, the third table: RESULTS OF COMPUTATION I Interslice section number (starting from 0) XML x of the interslice section R Position of the horizontal component of the interslice force defined as the vertical distance from the slip surface in the same section E Horizontal component of the interslice force T Vertical (shearing) component of the interslice force (frequently used symbol X) tanBeta tan of the inclination of the interslice force or simply T/E = A f(x) = tanBeta. Sig Effective normal stress on the base of the slice I Tau Shear stress on the base of the slice I These tables with described results appear on the screen during computation if the current PRINTING OPTION is (Screen) or (File). If the option is (Blank), these results can be recovered by using the option from the WORKING MENU: 3 PRINT RESULTS (Screen) or (File). Printed output in the next pages corresponds to the case shown in Fig. 10.25, p.121 and Fig. 11.11, p.143. Failure of convergence. Program stops computation sending the appropriate message if tolerances are not satisfied after 2000 iterations. Six suggestions on the screen what to try if that happens are: Swap TOLERANCES, Try some other f(x), Change the NUMBER OF SLICES, Change the number of poiNts (if Bezier or coMposite slip surface), Use miRror and then CHOOSE COMPUTE, CHECK YOUR DATA for errors or physical admissibility and PAUSE AND THINK if nothing of the above helps. If any of these suggestions do not help, the main reason is most frequently the physical inadmissibility of the defined problem with your (wrong) input data.

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Bezier 5 4b Nonlinear envelopes SLIP-SURFACE DATA I XX YY 1 5.24 17.50 2 5.83 16.56 3 6.45 15.62 4 7.15 14.60 5 7.80 13.68 6 8.45 12.80 7 8.88 12.24 8 9.48 11.50 9 10.01 10.86 10 10.40 10.43 11 10.97 9.80 12 11.57 9.19 13 12.10 8.68 14 12.70 8.15 15 13.66 7.39 16 14.55 6.77 17 14.98 6.50 18 15.39 6.29 19 16.18 5.92 20 16.93 5.64 21 17.63 5.44 22 18.31 5.29 23 18.96 5.20 24 19.58 5.14 25 20.19 5.11 26 20.79 5.10 27 21.39 5.09 28 21.98 5.09 29 22.58 5.08 30 23.19 5.08 31 23.81 5.07 32 24.46 5.05 33 25.13 5.03 34 25.84 5.02 35 26.50 5.00 36 26.88 5.00 37 27.37 5.00 38 28.20 5.03 39 29.01 5.07 40 29.47 5.10 41 30.02 5.17 42 31.02 5.31 43 31.73 5.44 44 32.28 5.57 45 33.23 5.82 46 34.41 6.20 47 35.04 6.43 48 35.41 6.59 49 36.08 6.88 50 36.77 7.20 51 37.49 7.57 52 38.19 7.96 53 38.89 8.37

b 0.63 0.55 0.70 0.70 0.61 0.69 0.17 1.03 0.02 0.78 0.36 0.85 0.21 0.99 0.92 0.87 0.00 0.81 0.77 0.73 0.69 0.66 0.64 0.62 0.60 0.60 0.59 0.60 0.60 0.62 0.64 0.66 0.69 0.72 0.60 0.16 0.81 0.86 0.78 0.13 0.97 1.03 0.38 0.73 1.18 1.18 0.08 0.67 0.67 0.71 0.71 0.70 0.70

tanAlpha 1.586619 1.586619 1.465218 1.465218 1.343105 1.343105 1.220693 1.220693 1.220693 1.098480 1.098480 0.977165 0.977165 0.857548 0.740668 0.627706 0.627706 0.520043 0.419193 0.326715 0.244196 0.173010 0.114245 0.068458 0.035600 0.014864 0.004737 0.003045 0.007187 0.014334 0.021702 0.026772 0.027437 0.022092 0.009640 0.009640 -0.010542 -0.038685 -0.074711 -0.074711 -0.118314 -0.169048 -0.226370 -0.226370 -0.289719 -0.358507 -0.358507 -0.432184 -0.432184 -0.510233 -0.510233 -0.592169 -0.592169

93

W/b 10.00 28.71 47.63 68.07 86.47 103.32 113.90 127.56 139.43 143.62 147.68 147.08 146.55 146.16 144.88 142.78 141.55 139.96 136.53 132.69 128.53 124.12 119.54 114.83 110.05 105.21 100.34 95.43 90.49 85.49 80.40 75.17 69.74 64.04 58.65 55.80 53.37 49.07 44.64 42.33 40.95 38.20 36.02 34.09 30.42 25.26 22.35 20.65 17.54 14.09 10.30 6.30 2.10

u 0.00 0.00 0.00 0.00 0.00 0.00 0.00 5.78 11.64 15.61 19.35 18.87 18.46 17.99 17.23 16.53 16.18 15.86 15.24 14.65 14.09 13.56 13.05 12.56 12.08 11.60 11.14 10.67 10.19 9.71 9.22 8.71 8.18 7.62 7.10 6.65 5.37 3.18 1.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

8. GE.EXE

I 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

RHor 0.0 0.0 0.0 0.0 0.0 0.0 0.0 7.3 0.2 13.3 7.6 15.7 3.7 15.2 11.8 9.0 0.0 6.7 4.9 3.5 2.4 1.5 0.9 0.5 0.3 0.1 0.0 0.0 0.0 0.1 0.1 0.2 0.2 0.1 0.0 0.0 -0.0 -0.1 -0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

RVert 6.3 15.8 33.2 47.5 52.6 71.6 18.9 125.8 2.0 99.6 45.6 109.1 26.4 126.6 118.0 109.3 0.0 100.9 93.0 85.6 78.9 73.0 67.7 63.1 59.1 55.7 52.8 50.4 48.4 46.7 45.2 43.8 42.4 40.7 30.9 8.0 38.7 39.3 33.8 5.7 39.7 39.5 13.6 24.7 35.8 29.8 1.7 13.8 11.8 10.1 7.4 4.4 1.5

RM 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

94

Phi 43.86 49.94 42.36 37.31 33.67 34.47 33.59 33.27 33.07 32.85 32.95 32.70 32.77 32.45 32.16 31.87 20.67 20.57 20.50 20.45 20.40 20.37 20.36 20.38 20.42 20.48 20.58 20.69 20.83 20.99 21.17 21.36 21.57 21.81 22.04 22.18 22.21 22.28 22.37 22.46 22.49 22.64 22.74 35.33 35.78 36.66 37.41 37.55 38.48 39.26 40.62 41.95 43.90

c 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

f(x) 0.058 0.108 0.171 0.233 0.287 0.347 0.362 0.448 0.449 0.512 0.539 0.603 0.618 0.687 0.746 0.796 0.796 0.839 0.875 0.905 0.930 0.951 0.967 0.980 0.989 0.996 0.999 1.000 0.997 0.992 0.982 0.969 0.952 0.930 0.908 0.902 0.868 0.826 0.784 0.776 0.717 0.648 0.621 0.568 0.476 0.379 0.372 0.315 0.256 0.192 0.127 0.064 0.000

8. GE.EXE

RESULTS OF COMPUTATION: I 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

XML 4.93 5.56 6.10 6.80 7.50 8.11 8.80 8.97 10.00 10.02 10.79 11.15 12.00 12.21 13.19 14.12 14.98 14.98 15.80 16.56 17.29 17.98 18.64 19.27 19.89 20.49 21.09 21.68 22.28 22.88 23.50 24.13 24.79 25.48 26.20 26.80 26.96 27.77 28.62 29.40 29.53 30.50 31.54 31.92 32.64 33.82 35.00 35.08 35.75 36.42 37.13 37.85 38.54 39.24

R 0.49 0.75 0.95 1.17 1.36 1.64 1.70 2.03 2.03 2.24 2.34 2.58 2.65 2.91 3.12 3.28 3.28 3.31 3.31 3.30 3.27 3.23 3.18 3.13 3.07 3.01 2.94 2.87 2.80 2.72 2.64 2.56 2.47 2.37 2.29 2.26 2.14 2.01 1.88 1.86 1.67 1.46 1.36 1.26 1.07 0.86 0.84 0.72 0.60 0.46 0.33 0.21

Fs= 1.4101

E 0.00 2.72 7.67 21.28 44.74 71.19 105.75 114.11 176.56 177.64 228.69 253.13 303.25 315.21 362.83 397.63 420.91 420.92 447.46 463.98 472.00 473.00 468.38 459.50 447.63 433.89 419.24 404.43 389.98 376.17 363.10 350.70 338.76 327.01 315.11 305.31 302.73 289.05 273.39 258.02 255.43 234.67 210.72 201.12 174.43 130.37 88.43 86.05 63.80 44.94 26.39 12.79 3.27 -0.00

T 0.05 0.25 1.09 3.12 6.10 10.96 12.32 23.61 23.82 34.92 40.75 54.60 58.16 74.36 88.49 100.00 100.01 112.03 121.16 127.51 131.31 132.89 132.62 130.90 128.11 124.59 120.61 116.36 111.96 107.45 102.82 98.00 92.91 87.43 82.75 81.48 74.83 67.39 60.36 59.17 50.23 40.76 37.30 29.57 18.53 9.99 9.56 5.99 3.43 1.51 0.49 0.06

95

A= 0.2984 tanBeta

Sig

Tau

0.0172 0.0321 0.0510 0.0697 0.0857 0.1037 0.1079 0.1337 0.1341 0.1527 0.1610 0.1801 0.1845 0.2049 0.2225 0.2376 0.2376 0.2504 0.2611 0.2701 0.2776 0.2837 0.2886 0.2924 0.2953 0.2972 0.2982 0.2984 0.2976 0.2959 0.2932 0.2893 0.2841 0.2775 0.2710 0.2691 0.2589 0.2465 0.2339 0.2316 0.2141 0.1934 0.1855 0.1695 0.1421 0.1130 0.1111 0.0939 0.0763 0.0573 0.0380 0.0190

4.8 12.1 23.8 36.4 49.9 58.2 67.2 70.7 73.0 75.7 74.4 77.5 76.7 80.6 84.5 88.5 92.5 96.0 98.4 100.6 102.3 103.5 103.8 103.2 101.6 99.1 95.8 91.8 87.4 82.8 78.0 73.2 68.4 63.6 59.2 56.8 56.4 55.2 53.8 52.3 52.0 49.9 48.4 50.5 46.7 40.1 34.9 34.0 28.2 23.8 17.0 11.1 3.7

3.3 10.2 15.4 19.6 23.6 28.3 31.6 32.9 33.7 34.6 34.2 35.3 35.0 36.3 37.7 39.0 24.8 25.6 26.1 26.6 27.0 27.2 27.3 27.2 26.8 26.3 25.5 24.6 23.6 22.5 21.4 20.3 19.2 18.0 17.0 16.4 16.3 16.0 15.7 15.3 15.3 14.7 14.4 25.4 23.9 21.1 18.9 18.5 15.9 13.8 10.3 7.1 2.5

9. CHANGES

9. MORE ON MENU OPTIONS - CHANGES Some menu options are described previously, (p.65 & p.90), while some, as self explanatory, do not need explanation. Other options offered in menus CHANGES are explained in this section. As a frequent rule, the first offered option is also the default option, and if you wish to select that one, you just press . After selecting the item from any of the menus, in certain cases some smaller sub-menu may be offered. In some cases just pressing or will cause the same effect as if you wished to cancel the started action. In the case that the entry is a letter, small and caps work equally well. Menus CHANGES in both stability programs, with the appropriate headings CHANGES (BE) & CHANGES (GE) shown in the next page have identical options numbered from 1 to 10 used for making changes in the MAIN DATA set which is identical for both stability programs (see page 97). The options 12 to 17 in CHANGES (BE) are identical to corresponding options 13 to 18 in CHANGES (GE) with identical purpose, though have different number, in BE.EXE and GE.EXE as explained in page 101. A few options only are different to match the requirements of the particular method of the analysis, as follows: The only option specific to CHANGES (BE) is: 11 LOCAL GRID Ratio H/S effective in WORKING MENU in LOCAL GRID… option can be changed. H/S Ratio can be selected in the range 0.2 to 5.0. If you only press without selecting the Ratio H/S, the default value of 1.0 will apply. Options specific to CHANGES (GE) are: 11 S-S: . . .meaning that you can change the title for the slip surface. It is practical, in certain cases, to use the title same as the file name in which the slip surface (*.SS) is saved. 12 f(x) SHAPE No... can be changed in WORKING MENU (GE) and here in menu CHANGES (GE). In a later case, immediately as the new function shape is entered, computation does not restart automatically.

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CHOICE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

C H A N G E S (BE) COORDINATES: 0 SOIL LIM. & PIEZ. LINES: 0 SOIL PARAMETERS: 0 LINE LOADS 0 TENS. CRACK Hc= 0 DISTRIBUTED LOADS 0 (TENS. CRACK) TAILWATER LEVEL 0.00 NUMBER OF SLICES 40 COEFF. OF SEISMICITY kx, ky 0.000 0.000 TIT: COM: LOCAL GRID Ratio H/S = 1.00 ZOOM/ MOVE/ AXES/ MIRROR ( X0= 320 Y0= 240 ) SHIFT SECTION and/or CHANGE SCALE RMX=10.00 VIEW GRAPH (with axes) INITIAL MENU... WORKING MENU... COMPUTE !

CHOICE No. ?

CHOICE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

C H A N G E S (GE) COORDINATES: 0 SOIL LIM. & PIEZ. LINES: 0 SOIL PARAMETERS: 0 LINE LOADS 0 TENS. CRACK Hc= 0 DISTRIBUTED LOADS 0 (TENS. CRACK ) TAILWATER LEVEL 0.00 NUMBER OF SLICES 40 COEFF. OF SEISMICITY kx, ky 0.000 0.000 TIT: COM: S-S: f(x) SHAPE No. 0 ZOOM/ MOVE/ AXES/ MIRROR ( X0=320 Y0= 240 ) SHIFT SECTION and/or CHANGE SCALE RMX=10.00 VIEW GRAPH INITIAL MENU... WORKING MENU... COMPUTE !

CHOICE No. ?

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THE FOLLOWING 10 OPTIONS ARE IDENTICAL IN BOTH MENUs CHANGES 1 COORDINATES: Number of points and the selected point coordinates can be changed. 2 SOIL LIMITS & PIEZ. LINES: Number of lines and line definitions for soil zones and piezometric lines can be changed. 3 SOIL PARAMETERS: All soil parameters, i.e. unit weight, pore pressure indicators as well as the shear strength parameters can be changed. 4 LINE LOADS… can be added, moved or varied, depending on previous line loads. Graph with an example of line loads is shown in Fig. 9.1 This option offers the possibility to edit (change) tension crack. In the case that the number if line loads was previously zero, the following sub-menu is offered: HOW DO YOU WANT LINE LOADS CHANGED 1 2 8 9 CHOICE No. ?

ENTER NEW SET OF LOADS NO CHANGES ADD ONE LINE LOAD EDIT TENSION CRACK (Hc=…)

However, if some set of line loads already exists, changes can be performed according to the following extended set of options: HOW DO YOU WANT LINE LOADS CHANGED 1 2 3 4 5 6 7 8 9 CHOICE No. ?

ENTER NEW SET OF LOADS NO CHANGES CLEAR LINE LOADS MULTIPLY EXISTING FORCES BY A COEFFICIENT SHIFT LINE LOADS BY dx & dy CHANGE ONE LINE LOAD REMOVE ONE LINE LOAD ADD ONE LINE LOAD EDIT TENSION CRACK (Hc=…)

Options 1 to 8 are self-explanatory. Option 4 is mainly used if you are dealing with anchor forces, to assess the sensitivity on Fs. Note that the water pressure in the crack, if present, is not affected by options 1 to 8. Option 9 EDIT TENSION CRACK implies editing the water pressure in the crack as a horizontal line load and other parameters which define the tension crack on the top of the slope. For details see Addendum No.1, page 149.

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Fig. 9.1 Option VIEW GRAPH (with axes) with line loads

Fig. 9.2 Option VIEW GRAPH (with axes) with distributed (surcharge) loading

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5

DISTRIBUTED LOADS (vertical surcharge) can redefined in the same manner as entered. Distributed loading will be shown in the graph as a layer on the top of the line which is loaded as shown in Fig. 9.2, p.99. Note that the soil/rock above level of the bottom of the crack has to be defined as the distributed load (see Addendum No.1).

6 TAILWATER LEVEL can be changed. This change is usually performed if piezometric line is changed. Usually, soil zoning has to be changed as well. 7 NUMBER OF SLICES is automatically set to the default value of 40 during the stage of entering the data via keyboard. This value is found reasonable by experience, but you can change it after entering all the data or at any other stage, by using this option. Number of slices is formally declared and can be entered as max. 100, but the program can handle up to 150 slices internally. You should note that the number of slices entered would not necessarily be identical to the number of slices used by the programs. It will be some value rather close to the selected number, as the program divides the sliding body automatically in such a way as to calculate required values using trapezoidal division and the condition that each base of the slice belongs to one material only. However, in the case of Bezier or coMposite slip surfaces, which include the number of poiNts for their definition (max.100), if you prescribe the NUMBER OF SLICES smaller than the number of poiNts, the later shall be used by the program GE.EXE. In the program GE.EXE the number of the above mentioned poiNts for Bezier or coMposite slip surfaces plus the number of coordinates should not exceed 140. If that happens, you will get the message TO MANY SLICES and you should reduce the number of poiNts for slip surfaces to about say 50 in an Interactive window. 8 COEFF. OF SEISMICITY kx, ky can be altered. The same convention on the sign of acceleration described previously in MAIN DATA preparation applies here. As new kx and ky are entered, but the computation WILL NOT start automatically and the control will return to the CHANGES menu. 9 TITLE: of the MAIN DATA can be changed. Pressing only will not alter the title for this set of data. 10 COMNT: is a comment text of the MAIN DATA, which can be changed. As in the previous case, pressing only will not alter the text. TEN ABOVE LISTED OPTIONS ARE IDENTICAL IN BOTH MENUS CHANGES IN BOTH STABILITY PROGRAMS

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OPTIONS in CHANGES WITH THE SAME FUNCTION IN BOTH PROGRAMS. The first shown option number is in CHANGES (BE) and the second in CHANGES (GE). 12 - 14 ZOOM/ MOVE ... This option from BE.EXE is described in Section 6, page 38, and shown in Fig. 6.4 to Fig. 6.9, pages 43-45. The only difference between these submenus in GE.EXE (window is shown here in Fig.9.3) is the option show Circles which exists in BE.EXE and BGP.EXE, and not in GE.EXE. The effect of miRror is shown in Fig.9.6 and Fig.9.7, p.103.

Fig. 9.3 ZOOM/ MOVE/ AXES/ MIRROR… window from GE.EXE 13 - 15 SHIFT CROSS-SECTION . . . can be used to relocate the geometry of the cross section by shifting it for some increment in x and y directions with dx and dy values with respect to the origin. An example is shown in Fig. 9.4 and Fig. 9.5, p.102. Handy if you change your mind about the relative position of the cross section with respect to origin after you have calculated and entered all the coordinates that define the crosssection in main data. Be careful to change the definition of the undrained strength variable with the depth (if any) if you have shifted the cross section in the vertical (up or down) direction as you may have to change the strength definition c=cu value as well. 14 - 16 VIEW GRAPH...(with axes) or (without axes) will show the geometry, (see Fig. 9.5) and slip surface(s) in a similar form as called from the INITIAL MENU. The soil limit lines will be shown in different colors; one color for one material. Useful for rough checking if the zoning of the cross section is properly described. 101

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Fig. 9.4 SHIFT CROSS-SECTION… window.

Fig. 9.5 VIEW GRAPH… after entries shown in Fig. 9.4.

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Fig. 9.6 ZOOM MOVE…window before R for miRror...

Fig. 9.7 Result of applying option R for miRror...

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10. EXAMPLES OF STABILITY COMPUTATIONS In this section some additional analyses of examples, beside those that were considered in the previous Sections, starting from Section 6, are performed. The geometry of the cross sections and other data for the examples presented in this manual are described by cross-sections given in Fig. 6.1, Fig. 6.2 and Fig 6.3 with input data in the pages starting from p.39, following the corresponding figures. The package is based on two methods and results, where applicable, are compared. Some results of computation are already presented in the previous Section 7 and Section 8, showing how different options can be used and what is the outcome of the particular analysis. These three mentioned cross section will be considered here in some detail. 10.1 EXAMPLE No. 1. (file: EX-1.BG) Example from Fig. 6.1, p.39, is a simple homogeneous slope and the soil shearing strength is described with Coulomb parameters (in terms of effective stresses) c’ =25 kPa and φ’=160. For the linear and the non-linear alternatives of these soil strength parameters see Fig. 4.3 and Fig. 4.4, p.15-16. This example was used for the description of the features of BE.EXE in Section 7 where it is found that Fs,min=1.6735 (Fig.7.22, p.62). Note that an asterisk sign * after the value of the safety factor indicates that some base of the slice is in tension. Tension was signaled in preliminary trials shown from Fig. 7.2 and for all the results presented in that Section. This is physically inadmissible, and that normally happens in the upper part of the slope with the soil which has a cohesion term in the soil strength description. In order to check the consequences, the more accurate, the General method was used by selecting option 10 SWITCH TO GE... from WORKING MENU (BE) and immediately for the same critical circle the safety factor is calculated with method which satisfies all equilibrium conditions, assuming, in this case, the default sine wave function f(x) for inclination of interslice forces. Result, for the default number of 40 slices is shown in Fig. 7.23, p.64, is not significantly different, being FS =1.6724 (compared to 1.6735 from Fig. 7.22, p.62). The number of slices with tension stresses at the base and the number of tension inter-slice forces is indicated for the initial default number of slices taken as 40 in Fig. 7.23. At the upper part of the slope some irregularities appear in the line of thrust, which should not, for the physically admissible solution, appear outside the sliding body. After some further refined iteration, the critical slipcircle is obtained using Bishop method (BE.EXE) the same Fs=1.6735 was found with the slightly different location of the slip circle and the computation repeated with GE.EXE using Spencer assumption (f(x)=const. and shown here in Fig. 10.1, p.105, and the result is Fs=1.6729. If the number of slices is increased to 90, the result shown in Fig. 10.2, p.105, indicates that the factor of safety has changed slightly to FS = 1.6730. The number of slices with tension at the bases and the number of inter-slice forces in tension has increased, and the irregularity in the line of thrust is much more emphasized. Theoretically, for the infinitesimally small slice width, in the zone where interslice forces change sign from tension to compression, the infinite values of lever arms of the opposite sign should be computed. Tension and irregularities disappear when the declared number of slices is reduced to 10, as shown in Fig. 10.3, p.106, and the safety factor is FS = 1.6644. The circular slip surface provides result close to though not equal to the absolute minimum, as shown by an example of the Bezier slip-surface in Fig. 10.4, p.106, with Fs=1.6527.

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Fig. 10.1 Critical slip-circle for the case No.1. Spencer assumption.

Fig. 10.2 Same circle from Fig. 10.1 and number of slices increased to 90. 105

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Fig. 10.3 Number of slices reduced to 10.

Fig. 10.4 Critical Bezier slip-surface. 106

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10.2 EXAMPLE No. 2. (file: EX-2.BG) Example from Fig. 6.1, p.39 is a simple homogeneous slope but in this case the soil shearing strength is described with the non-linear parameters in terms of effective stresses with the cohesion term c’ = 0. For the nonlinear shearing soil strength parameters see Fig. 4.3 and Fig. 4.4 (p.15-16). After some iteration, the critical slip-circle is obtained using Bishop method (BE.EXE) with the value of factor of safety FS = 1.3532 as shown in Fig. 10.5, p.108. Note that the tension stresses do not appear in this case. In order to run some comparisons, the more accurate, the General method is used by selecting option 10 SWITCH TO GE... from WORKING MENU (BE) and immediately for the same critical circle the safety factor is calculated with method which satisfies all equilibrium conditions, assuming the default “sine wave” function f(x) for inclination of interslice forces. Result, for the default number of 40 slices is shown in Fig. 10.6, p.108, is not significantly different, being FS =1.3516 (compared to 1.3532 from Fig. 10.5), but in the toe zone, the line of thrust is probably slightly to low. Bromhead (1986), suggests that in practical work, f(x)=const., (Spencer's assumption), should be tried first, and the result for this assumption is shown in Fig. 10.7, p.109, with the result FS =1.3576 with reasonable position of the line of trust. It can be seen that in this case of a simple and homogeneous slope, safety factors for the circular slip surfaces using distinctly different methods do not differ significantly. The circular slip surface provides result close to, but not equal to, the absolute minimum, as shown by an example of the Bezier slip-surface in Fig. 10.8, p.109, with Fs=1.3373 which is about 1% smaller than Fs obtained for the slip surface of circular shape. These two examples (No.1 and No.2) are designed to show that in the case that the nonlinear failure envelope is used and the cohesion is equal to zero (non-cemented soils and rock discontinuities), problems with tension at the top of the slope are removed. Moreover, the factor of safety using the nonlinear failure envelope is smaller than in the Example No.1, (compare Fs=1.66 approx. to Fs=1.34 approx.). The main reason for this difference of about 24 % on the unsafe side is the differences of the shearing strength in the low stress range indicated in Fig. 4.3 and Fig. 4.4, p 15-16. These two results are plotted on Fig.10.35, p.127, (points marked as M) to emphasize the importance of the non-linearity of the failure envelope in practice.

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Fig. 10.5 Critical slip-circle for the Example No.2

Fig. 10.6 Same as Fig. 10.5 analyzed by General method. 108

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Fig. 10.7 Spencer’s assumption.

Fig. 10.8 Critical Bezier slip-surface

109

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10.3 EXAMPLE No. 3. (files: EX-3A.BG & EX-3B.BG) To investigate the practical consequences of the cohesion and tension, an example of the same slope as in Example No.1 with the linear failure envelope and the cohesion term, but with introduced tension crack (empty or filled with water), defined with input data shown in Fig. 6.2, p.40, is analyzed using both methods. For the case of the empty crack Bishop Extended method gave FS=1.7056 as shown in Fig. 10.9, p.111. There is no asterisk after the value of the safety factor (no tension). The General method for the default “sine wave” gave similar result for the same slip-circle, FS=1.7049 as shown in Fig. 10.10, p.111. Tension at bases of slices or tension of interslice forces does not appear and the line of thrust is within reasonable limits. As in the both previous cases, the Bezier type slip surface is critical with Fs=1.6852, the difference being slightly more than 1% (Fig.10.11, p.112). For the case of the slope with the crack filled with water Bishop Extended method gave FS = 1.6542 as shown in Fig. 10.12, p.112, where 25 horizontal tangents (the LOCAL GRID option) have been used for the search of the critical slip-circle. There is no asterisk after the value of the safety factor (no tension). The same smallest safety factor was obtained using 25 PASSING POINTS in the toe zone of the slope though the critical slip-circle is not identical the difference being of the order of a few centimeters as shown in Fig. 10.13, p.113. The General method for the Bezier type slip surface again is found to be critical with FS = 1.63. As in the case without water in the crack, tension at bases of slices or tension of interslice forces do not appear, the line of thrust is reasonable starting from the point of application of the force due to the water pressure, as shown in Fig. 10.14, p.113. Results discussed above show that the Bishop Extended method and the General method give very close results for simple slopes and the circular slip-surfaces both for the non-linear as well as for the non-linear failure criterion, though results obtained different failure criterion differ significantly. When these results are compared to those that are shown for the Example No.1, it can be seen that the influence of the tension crack is of the order of a few percent only. However, the influence of the non-linearity of the failure envelope is much more significant.

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Fig. 10.9 Critical circle – Bishop method.

Fig. 10.10 Same circle as above, General method, default f(x) 111

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Fig. 10.11 Critical Bezier surface.

Fig. 10.12 Bishop extended method- 25 tangents. 112

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Fig. 10.13 Bishop extended method- 25 passing points in a toe zone.

Fig. 10.14 Critical Bezier surface. 113

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10.4 EXAMPLE No. 4. (file: EX-4A.BG) The version of the non-homogeneous cross-section and the soil shearing strength defined with the conventional linear failure envelopes described in Fig. 6.3, p.41, represents the cross section of the embankment dike on non-homogeneous soil, with an almost horizontal weak layer or lens (soil zone number 6). In all examples within this section slip surfaces of circular shape are assumed as tangential to the lower boundary of the weak layer at elevation y=5.0 m. The result of computation obtained using BE.EXE shown in Fig. 10.15, p.115 gives the critical slip-circle and smallest Fs=1.5530 using the Automatic search option with a rather remote initial center. The refined analyses using the LOCAL GRID option gave the same Fs with only slightly different position of the center of the critical slip-circle, as shown in Fig.10.16, p.115. Asterisk indicated that some slice(s) might be in tension at the base. In order to check the difference, the more accurate, the General method (GE.EXE) is used by selecting option 10 SWITCH TO GE... from WORKING MENU (BE) and immediately for the same critical circle the safety factor Fs=1.5493 is calculated with method which satisfies all equilibrium conditions, assuming, in this case, the default “sine wave” function f(x) for inclination of inter-slice forces. Result, for the default number of 40 slices shown in Fig. 10.17, p.116, is not significantly different, being FS =1.5493 (compared to 1.5530 from Fig. 10.16, p.115). Results are reasonable as there are no tensions at the base of the slice but one interslice force is in tension what causes some minor irregularity in the line of the thrust in the upper zone of the sliding body. Rather similar result is obtained for Spencer’s assumption shown in Fig.10.18, p.116, with Fs=1.5542. However, as the circular arc is not capable of following the layer with the low shearing strength, the Bezier type slip surface, after interactive iteration, was found to provide smaller safety factor Fs=1.4913 shown in Fig. 10.19, p.117. Even smaller safety factor Fs=1.4901 is obtained for the polygonal slip surface shown in Fig. 10.20, p.117. This example just illustrates the well known experience that, in the case of the nonhomogeneous cross sections, the circular slip surfaces might not necessarily give the result close to smallest Fs. The differences can be of practical importance. In this case, the approximate difference is 1.55 – 1.49 = 0.06, or about 4% approximately.

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Fig. 10.15 Bishop extended method. First run with Automatic search.

Fig. 10.16 Bishop extended method. LOCAL GRID and the critical slip circle. 115

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Fig. 10.17 General method and the critical slip circle.

Fig. 10.18 General method and the critical slip circle, Spencer’s assumption 116

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Fig. 10.19 Critical Bezier surface.

Fig. 10.20 Critical Polygonal slip surface. 117

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10.5 EXAMPLE No. 5. (file: EX-4B.BG) The version of the non-homogeneous cross-section with the geometry from the previous Section 10.4 and the soil shearing strength defined with the set of non-linear failure envelopes is analyzed. These non-linear failure envelopes are derived from the linear parameters considered in the previous Section using the assumption that the shearing strength is practically the same in the conventional stress range 100-300 kPa. Details of such a conversion are beyond the scope of this BGSLOPE Manual and are described in NENVE manual, which is the optional part of the package. In all examples, except for the first one shown in Fig. 10.21, p.119, slip surfaces of circular shape are assumed as tangential to the lower boundary of the weak layer at elevation y=5.0 m. The Grid option was used first for illustrative purposes assuming that all slip-circles have the common tangent at elevation y=6, or 1.0 m above the lower boundary of the weak layer. Contours shown in Fig. 10.21 indicate the possibility of the two local extremes. The result of computation obtained using BE.EXE shown in Fig. 10.22 gives the critical slipcircle and smallest Fs=1.4402 using the LOCAL GRID option. In order to check the difference, the more accurate, the General method (GE.EXE) is used by selecting option 10 SWITCH TO GE... from WORKING MENU (BE) and immediately, for the same critical circle, the safety factor Fs=1.4455 is calculated with method which satisfies all equilibrium conditions, assuming, in this case, the default “sine wave” function f(x) for inclination of inter-slice forces. Result, for the default number of 40 slices is shown in Fig. 10.23, p.120, is not significantly different, being FS =1.4455 (compared to 1.4402 from Fig. 10.22, p.119). Results are reasonable as there are no tensions at the base of the slice and there are no irregularities in the line of the thrust. Just as in the previous case in Section 10.4, the circular arc is not capable of following the layer with the low shearing strength, the Polygonal slip surface, after interactive iteration, was found to provide smaller safety factor Fs=1.4119 shown in Fig. 10.24, p.120. The smallest Fs was obtained for the Bezier type slip surface shown in Fig. 10.25, p.121, as Fs=1.4101. For Spencer’s f(x)=const., (SHAPE No.1) the safety factor shown in Fig. 10.26 is Fs=1.4219, though along the significant part of the sliding body the line of thrust in the lower part of the section, might be slightly too high. Two other f(x) shapes gave results within the range 1.41-1.42 as shown in Fig. 10.27 and Fig.10.28, p.122. The critical Fixed plane slip surface gave the safety factor Fs=1.4766 as shown in Fig. 10.29 and Fig.10.30, p.123. This slip surface, tangential to the upper boundary of the weak layer, is not critical though the plane along the lower boundary, considered here is more relevant. The critical coMposite slip surface gave the safety factor Fs=1.4295-1.4296 as shown in Fig. 10.31 and Fig.10.32, p.124. Two possibilities for prescribing the definition of the PASSING POINT are shown in these figures. The choice is up to the user, as it is a matter of the convenience, though the alternative in Fig.10.32, p.124, is probably slightly better as user knows that the passing point as tangent should be placed, at least initially, beneath the line which defines the planar part of the composite slip-surface.

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Fig. 10.21 Bishop extended method. Indication of possibly two local minim.

Fig. 10.22 Bishop extended method. Critical slip circle from Fig. 10.23 by General method. 119

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Fig. 10.23. General method and the critical slip-circle, see Fig.10.22

Fig. 10.24 Critical Polygonal slip surface with 6 points. 120

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Fig. 10.25 Critical Bezier slip surface with 5 points.

Fig. 10.26 Same as above using Spencer’s assumption. 121

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Fig. 10.27 Slip surface from Fig.10.25, 10.26 and f(x) SHAPE No.3 (linear).

Fig. 10.28 Slip surface as above and f(x) SHAPE No.5 (smooth wave). 122

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Fig. 10.29 Critical Fixed plane slip surface obtained after iteration started from Fig 8.17 in an INTERACTIVE SEARCH window.

Fig. 10.30 As in Fig 10.29 above in VIEW GRAPH window..

123

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Fig. 10.31 Critical coMposite surface with XP0.

Fig. 10.32 Critical coMposite surface with XP=0 (tangent). 124

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10.6

EXAMPLE No. 6. (files: PRANDTL.BG & PRANDTL.SS)

The validity of the General method is demonstrated by an example for which the closed form solution exists. An example shown in Fig. 10.33 below demonstrates the accuracy of the General method. The rigorous theoretical solution by Prandtl is available for the bearing capacity of the strip loading on the weightless soil with uniform surface loading. The bearing capacity qf is expressed by qf = q0 Nq in which q0 is the unit surface load and Nq is the Prandtl bearing capacity factor. The General method used here gave the exact result for f(x) SHAPE 5 (“smooth wave”) with zeros along active and passive Rankine wedges and the sinusoidal transitions along the spiral, as shown in Fig. 10.33. The unit load is applied aside the strip footing of unit width loaded by 18.4=Nq. The calculated value of the safety factor is exactly 1.000.

Fig. 10.33 Comparison of numerical solution with General method obtained by GE.EXE and the theoretical solution The comparison of stability analyses made by Hansen (1966), Turnbul and Hvorslev (1967) and Jiang & Magnan (1997), have shown that the differences from the correct value of the bearing capacity factor Nq for the Fellenius method is -69 %, for Janbu I +51 % and for the Bishop method + 69 %. These differences are rather excessive.

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10.7 INFLUENCE OF THE NON-LINEARITY OF THE ENVELOPE The most pronounced curvatures of the failure envelope are found for compacted rockfill, dense sands, overconsolidated clays, compacted clays and in most cases of the residual shearing strength of clays (Maksimović, 1989-a, b and c). Some examples of nonlinear failure envelopes are presented in Section 4. Due to the reason that the application of the nonlinear failure envelope in a routine work might be rather new, one comparative example is presented here. The homogeneous simple slope 9 meters high with an inclination 2/1 is selected for presentation in some detail. It is assumed that the soil has unit weight of 20 kN/m3 and that the failure law can be taken either as the conventional straight line or as a proposed curve, as discussed in the Section 4. The peak shearing strength of the compacted London Clay is based on the data by Atkinson and Farrar (1985) who devised a special stress-path controlled shear test to obtain the shape of the failure envelope at a low stress range. They report that in the conventional range of testing stresses (150-300 kPa), the failure envelope can be described by a linear relationship defined by the angle φ ′ = 16 0 and c ′ = 25kPa . But, in the low stress range (5-25 kPa), the envelope is curved and can be described by τ f = 2.72 σ 0.65 . Four points for mentioned values n of normal stresses, (5, 25, 150 and 300 kPa), are selected for the derivation of the proposed parameters, as shown in Fig. 4.3 and Fig. 4.4, p.15-16. It can be seen in Fig. 4.4 that in the conventional stress range from 100 to 400 kPa the failure envelope is very close to a straight line, but at the lower stress range, linear extrapolation toward the zero normal stress gives the cohesion value and the unsafe, higher values of the shearing strength as shown in Fig. 4.3. High value of the maximum angle difference of 48.10 indicates that at very low level of normal stresses the compacted clays exhibit extremely high dilation. To examine the differences due to the non-linearity, computations are carried out using the conventional linear and the proposed nonlinear description of the same test assuming that the failure surface is of the circular shape. The results of application of BE.EXE are shown in Fig. 10.34 and Fig. 10.35, p.127. It is previously assessed that results for the same shearing strength and pore pressure assumptions and the same slip-circle are practically identical using both methods. However, the main difference is due to alternative descriptions of the failure law. For the accuracy of the smallest FS to the second decimal place, results can be summarized as follows: It can be seen that the error due to the application of the conventional interpretation ( φ ′ and c ′ ) is particularly unsafe for the larger value of the pore pressure ratio. The position of critical circles is significantly different. With the conventional description, the tension stresses could occur, unless the tension crack is introduced. In the presented example of the simple slope and the nonlinear failure envelope, tensions do not appear and the solution is physically admissible. Similar example of comparison for the fixed value ru=0.4 can be made considering the results of cases described in Section 10.1 and 10.2, the only difference being that the slope is flatter with an inclination 1 V : 2.67 H. These results are shown as two small circles, points M which denote the slope inclination in Fig. 10.35, p.127.

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Fig. 10.34 Slip-circle and results for the linear and non-linear failure criterion.

Fig. 10.35 Pore pressure - FS relationships for cases shown in Fig. 10.34.

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10.8 MISCELLANEOUS REMARKS and HINTS 1. The nonlinear failure envelope should be used generally in almost all slope stability analyses in terms of effective stresses. Otherwise, in many cases, drastic differences can occur and misleading unsafe results for the safety factor and wrong location of the critical slip-surface can be obtained. (For example, see Day & Maksimovic, 1994) 2. Results of stability analyses for the non-homogeneous sections considered here illustrate that the Circular slip surfaces have the largest value of FS, next smaller are coMposite surfaces and the Polygonal, and the smallest FS,min is most frequently, obtained by using Bezier type slip surface, providing that results are physically admissible. From the extensive use of the program as well as from the examples shown here, it seams that the Bezier type slip surfaces are the most versatile for the description of the smooth arbitrary slip surfaces in soil slopes without planes of weakness. However, in slopes with thin weak layer, Fixed plane type slip surface is more relevant. 3. As it is generally known, some complex as well as well as not very complicated slopes might have more than one local minima. Options NEW INITIAL CENTER and/or NEW PASSING POINT facilitate systematic search for the global smallest safety factor and the critical slip circle. 4. Previously reported results show that the typical difference between two methods (Bishop Extended vs. General method) for the circular slip-surfaces is not more than 1% - 2% approximately. This finding is commonly reported in literature. 5. It can be seen that for the SHAPE and the f(x) type, within the reasonable and acceptable solutions for the slip surfaces of a general shape, the difference rarely exceeds 2%-4% approximately, taking for the reference the smallest Fs is obtained for the default f(x) SHAPE No.2 (sine wave). This finding about the significance of f(x) is commonly reported in literature. 6. It is suggested in literature that in practical work, f(x) = const, SHAPE No.1, (Spencer's assumption), should be tried first. It is authors experience that most often the default SHAPE No.2 (sine wave) is more appropriate than the Spencer’s f(x)=const. assumption, when considering the simple functions without explicitly defined parameters.

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11. PROGRAM BGP.EXE This program reads .BGP type file with MAIN DATA and RESULTS OF COMPUTATION. In the case that SWITCH to BGP… option is used in stability programs, the temporary file TEMP.BGT for the transfer of data and results is transparent. Input file for BGP.EXE is formed and saved from stability programs BE.EXE and/or GE.EXE. File with extension .BGP contains input data and results. The program BGP.EXE is used to present the results of computation as exported plotter graphics files (*.PLT), HP-GL files and/or text files (*.TXT) for printed outputs. Running of this program can be initiated from: A. DOS or WINDOWS, usually in the practical work, B. BE.EXE by selecting 15 SWITCH TO BGP.EXE...from WORKING MENU (BE) C. GE.EXE by selecting 13 SWITCH TO BGP.EXE…from WORKING MENU (GE) D. BGSETUP.EXE, rarely in practical work. 11.1 MENU The full menu that drives the program offers the following options. The options actually offered on the screen will depend on the origin of the file which is being red: CHOICE

MENU

(BGP)

1 LOAD DATA FILE (.BGP) 2 PRINT DATA and/or RESULTS (Screen) or (File) 3 VIEW GRAPH (with axes) or (no axes) 4 EXPORT HP-GL FILE (.PLT) 5 SAVE DATA and RESULTS FILE (.BGP) 6 PRINTING OPTION IS: (Screen) or (File) 7 TIT: 8 COM: 9 SS: 10 ZOOM/ MOVE/ AXES ( X0=320 Y0= 240 ) 11 SHIFT SECTION and/or CHANGE SCALE RMX=10.00 12 CONTOUR INTERVALS CI=0.02…..(alternative generated by BE…) 12 SAVE SLIP SURFACE... (.SS)………(alternative generated by GE…) 13 SWITCH TO BE ... (Bishop Method) 14 SWITCH TO GE ... (General Method) 15 CHANGE HEADINGS and/or LOAD SCALE 20 16 SHIFT BASE OF f(x) GRAPH 0 17 Q U I T CHOICE No. ? If you start running BGP.EXE from DOS or WINDOWS or BGSETUP, initially, options 2-10 will be omitted as there are no data to deal with. As soon as you LOAD DATA FILE (.BGP), or you have initiated the program using the SWITCH option, the corresponding alternatives of the menu will be offered. The option 12 shown here twice, will depend on the data file origin. If *.BGP file is formed by BE.EXE, then 12 CONTOUR INTERVALS…will appear, and at the other hand if the file originates from GE.EXE, this option 12 offers the possibility to SAVE SLIP SURFACE (.SS) of the arbitrary shape.

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11. BGP.EXE

1 LOAD DATA FILE - Loads files with extension *.BGP. After reading the file, graph will appear as shown in Fig. 11.1. The graph on the screen approximately corresponds to the appearance of the plotted graph. All three HEADINGS and the Caption start will show in position, which approximately correspond to their position on the graph produced using HP-GL (*.PLT) format (option 5). In the case that the program has been started via SWITCH TO BGP.EXE from stability programs this option will not be used because the temporary file TEMP.TMP formed by any of the stability programs will be loaded automatically.

Fig. 11.1 VIEW GRAPH…Graph appears automatically after loading *.BGP file. 2 PRINT DATA (Screen) or (File) - Results as well as input data can be printed after selecting the option from the sub-menu, on the screen or to the file, depending on what is mode after 6 PRINTING OPTION (Screen) or (File). 3

VIEW GRAPH (with axes) or (no axes) The graph on the screen approximately corresponds to the appearance of the .PLT type graph and it is the same as shown in Fig. 11.1. Adjustments can be made using options 10 and/or 11. Caption, after Caption start, or HEADINGS can be entered later, as described under option 15.

4

EXPORT HP-GL FILE (.PLT) Forms and saves a HP-GL file in .PLT format on the disk. You will have to enter the file name and extension *.PLT is added automatically. If for the file name you enter “=” (equality sign), the name of the .PLT file will be the

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same as the loaded .BGP file, though only the extensions will differ. This simplifies bookkeeping. The graph will appear on the screen as shown in Fig. 11.2.You should enter the figure CAPTION which will appear on the drawing (starting from the lower left corner of the drawing, see examples pages 135-148) preceded by the text entered using BGSETUP, option 4, Caption start. (For example: Fig. or Figure or Drawing No…). Max. number of characters that can be entered for the neat drawing, without overlapping letters, will be shown and should not be exceeded. See Fig. 11.2. If you do not type the caption, and only press , Caption start will be omitted as well. The resulting graph for cases shown in Fig.11.1 and Fig.11.2 is shown in Fig.11.11, p.143. This option 4 is the most important one because HP-GL files are of *.PLT type and can be handled by different commercially available programs for the final graphical presentation in your report.

Fig. 11.2 Entering caption for plotting or for HP-GL (*.PLT) file. 5 SAVE DATA and RESULTS FILE (.BGP) Saves file with extension *.BGP formed by any of two stability programs. You may use this option if you wish to save the file on your disk for some later use. If for the file name you enter “=” (equality sign), the name of the new saved .BGP file will be the same as the name of the LOADed FILE. This might simplify bookkeeping. One of the reasons for this choice is that you might have made some changes for graphic presentation that you wish to save. Note that in the case that the program has been initiated via SWITCH TO BGP.EXE from stability programs, only temporary file TEMP .BGT is formed and you might use this option to save the normal, permanent, file entering the appropriate new file name.

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6 PRINTING OPTION - (Screen) and (File). Works as a toggle. 7 TIT: Title can be modified. 8 COM: Comment can be modified. 9 SS: Title of the slip surface can be modified. 10 ZOOM/ MOVE/ AXES - You can change the scale and the position of the origin, but you cannot rotate the picture. Options for displaying or omitting the coordinate axes are the same as in both stability programs, but the change the position of the origin (MOVE) works in smaller steps in order to facilitate the precise positioning of the graph. 11 SHIFT SECTION and/or CHANGE SCALE RMX =. . . - Shifts the section with respect to coordinate system that was valid when the file was formed (saved). Same as in both stability programs. Provides the possibility to enter new value of the SCALE. 12 CONTOUR INTERVALS CI= (altern. BE.EXE) Changes Fs contour intervals (CI) for results obtained for slip circles with grid of centers. Same as option 7 in BE.EXE, p.52 & p.63. It is advisable not to use extremely small contour interval, (the default value of 0.02 is usually adequate, though the values in the range 0.01 to 0.05 might be OK in some cases. The suggested values for the final presentation are CI=0.2 or 0.1 or 0.05 or 0.025 or 0.02 or 0.01 or 0.005. Avoid unnecessary dense contours (CI too small for the Fs range within the grid) because you may get too many contours in HP-GL (*.PLT) file which may exceed the size of the buffer and cause error. 12 SAVE SLIP-SURFACE – (altern. GE.EXE). When this item is selected, a list is obtained with all the existing *.SS files in the directory. Type the name of the slip surface file you wish to save and press the key. Do not type extension. If you enter the name of the already existing data file, it will be overwritten by the new one. So pay attention, there is no warning message. Same as option 12 in INITIAL MENU (GE), p.29 & p.31. 13 SWITCH to BE...(Bishop Method) - Transfers back MAIN DATA and RESULTS with all slip-circles if you have reached BGP.EXE from BE.EXE. However, if you have previously switched from GE.EXE, or you have loaded .BGP file generated by GE.EXE, which does not contain the circular slip surface, only MAIN DATA will be transferred, without noncircular slip-surface. 14 SWITCH to GE...(General Method) - Transfers MAIN DATA and SLIP-SURFACE, and starts the run of General Method if you have reached BGP.EXE from GE.EXE. If you have previously SWITCHed from BE.EXE or you have read .BGP file formed by BE.EXE with slip circles, GE.EXE will automatically carry out computation for the last critical circular slip-surface. 15 CHANGE HEADINGS and/or LOAD SCALE - Three headings, and Caption start, initially entered in BGSETUP.EXE, option 3, page 4, and the graphic scale for the surcharge load can be changed. These three headings appear in the top right corner of the graph formed by and Caption start precedes the caption (see next Section). If you press 15, the submenu identical to one shown in page 4 will appear and the procedure 132

11. BGP.EXE

for changing these labels is just the same as the following submenu, (though with the different labels if you have changed them before) will appear: HEADINGS ARE 1: 2: 3: Caption start is 4:

(left blank = omitted) Manual - examples (left blank = omitted) Fig.

OK (Y/N) ? If you enter Y or y or just press , that would mean YES, it is OK, confirming that you wish that these four labels on your drawings produced by BGP.EXE. If you enter N or n, that would mean NO, it is NOT OK for these four labels, the possibility to change any of these, one by one, is offered in this form: CHANGE No. (1/ 2/ 3/ 4) : ? Press the number and type the corresponding new heading and . Again you will have on the screen: OK (Y/N) ? The new loop offering 4 options starts until you make all the changes which you wish ending with Y for OK?. The approximate appearance of HEADINGS can be seen using option 3, see Fig. 11.1, and as they appear on graphs formed using HP-GL graphic format. See in the Section 11.3, examples from Fig.11.3 to Fig. 11.15. After you have entered Y or y for OK, the possibility to change LOAD SCALE, the graphic scale for the surcharge load, will be offered. The default value is 20 meaning that 20 kPa of surcharge corresponds to 1 meter on the graph, adopted here as an approximate soil unit weight. Such a value is usually appropriate, unless in some bearing capacity problems you may wish to change (increase) this value. The entered changes are automatically saved for the further application when you QUIT the program BGP.EXE or SWITCH TO… any of the two stability programs. 16 SHIFT BASE OF f(x) GRAPH … The function f(x) is shown on x axis in stability program GE.EXE meaning that the SHIFT f(x) is fixed as zero, as shown, for example in Fig. 11.2. However, it is sometimes neater and convenient for graphical presentation only to show f(x) below or above the x-axis. You may choose the shift the base line of f(x) some meters above or below the x-axis. Compare Fig.11.18 to Fig. 11.19 and explanation for these figures in pages 140-141. The entered change is automatically saved for the further application in GE.EXE and BGP.EXE when you QUIT the program or SWITCH TO… any of the two stability programs. 17

QUIT…Exit the program

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11. 2 NOTES ON GRAPHIC POST-PROCESSING Most of the figures shown in this manual are black & white hard copies of the screen obtained in graphic VGA mode (640 x 480 pixels). *** Suggested pen thickness and/or colors for plotting/printing are as follows: No.1 No.2 No.3 No.4

0.2 mm 0.3 mm 0.5 mm 0.2 mm

black for most of the lines and text black for the frame and some text red or black for the slip surface blue or black for piezometric lines ***

In the case that you are using dot matrix or laser printer, which emulate plotter using HP-GL file and uses different pen thickness expressed in pixels, the following pen sizes can be tried first: Pen No.1 1-2 pixels Pen No.2 2-3 pixels Pen No.3 3-4 pixels Pen No.4 1-2 pixels *** HP-GL files (*.PLT) can be handled by a number of commercially available programs like: PRINTAPLOT of Insight Dev. Corp., PRINTGL of Rawitz Software Inc., AUTOCAD of Autodesc Inc., CorelDRAW of COREL Systems Corp., and many others. Instead of the graphic plotter, dot matrix or laser printer, for the final presentation, you might be able to capture i.e. copy the graph from the screen to a file using some commercially available software, though this option is not feasible an all platforms. 11.3 EXAMPLES Examples of graph produced from HP-GL files with extension .PLT are shown in the pages which follow. Note that the file names in the top right hand corner of the graph are shown in order to facilitate bookkeeping. File with extension .BGP is the file name loaded by BGP.EXE, and the file name with the extension .PLT is the HP-GL file used for making the drawing. Note that file names (though with the different extensions) can be made by using “=” (equality sign) when using option 4 for exporting file in HP-GL format from BGP.EXE and that might simplify bookkeeping in most practical cases.

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11. BGP.EXE

Fig. 11.3 Graph generated by producing .BGP file from BE.EXE, when the option of the Automatic search was used. The graph shows the path from the initial center to the final position that corresponds to the location of the critical circle for the selected tangential level. The same case was shown in Fig.10.15, p.115.

135

11. BGP.EXE

Fig. 11.4 Graph generated by producing .BGP file from BE.EXE, when the LOCAL GRID option was used with 11 passing tangents. The graph shows the location of the circle with the smallest Fs of all 11x11x11=1331 slip circles, contours for interval CI=0.02, and values of the factors of safety at each corner of the grid. 136

11. BGP.EXE

Fig. 11.5 Graph generated by producing .BGP file from BE.EXE, when the LOCAL GRID option was used with 8 passing points in the toe zone of the slope. The graph shows the location of the circle with the smallest Fs of all 11 x 11x8=968 slip-circles, Fs contours, and values of the factors of safety at corners of the grid.

137

11. BGP.EXE

Fig. 11.6 Graph generated by producing .BGP file from GE.EXE, when the CRITICAL CIRCLE was considered using SWITCH TO GE… option in BE.EXE. The graph shows the circular slip-surface, the line of thrust and f(x) SHAPE No.1.

138

11. BGP.EXE

Fig. 11.7 Case with the undrained strength varying with depth. Slip circles generated for 25 passing points along an inclined line defined by two limiting passing points.

139

11. BGP.EXE

Fig. 11.8 Critical circle from Fig.11.7 analyzed by GE.EXE. f(x) drown on x-axis, though the x-axis is omitted. The LOAD SCALE differs from the one shown in Fig. 11.9.

140

11. BGP.EXE

Fig. 11.9 Critical Bezier surface for previous cases. Note f(x) shown 5 m above x-axis using option 16 SHIFT BASE OF f(x) GRAPH (UP) with the entered value 5. The scale of the surcharge load is also changed using option 18…LOAD SCALE. Compare to Fig. 11.8.

141

11. BGP.EXE

Fig. 11.10 Graph generated by producing .BGP file from GE.EXE, when the POLYGONAL slip surface was considered. The graph shows the line of thrust and f(x) SHAPE No.2.

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11. BGP.EXE

Fig. 11.11 Graph generated by producing .BGP file from GE.EXE, with the Bezier type slip-surface as shown in Fig.11.1 and Fig.11.2. Note that the Bezier polygon is not drawn.

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11. BGP.EXE

Fig. 11.12 Graph generated by producing .BGP file from GE.EXE, when the BEZIER type slip surface was considered with the different f(x) SHAPE No. 3.

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11. BGP.EXE

Fig. 11.13 Graph generated by producing .BGP file from GE.EXE, when the BEZIER type slip surface was considered with the different f(x) SHAPE No.5.

145

11. BGP.EXE

Fig. 11.14 Graph generated by producing .BGP file from GE.EXE, when the FIXED PLANE type slip surface was considered with the f(x) SHAPE No.1 (constant).

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Fig. 11.15 Graph generated by producing .BGP file from GE.EXE, when the COMPOSITE slip surface was considered. The graph shows the line of thrust and f(x) SHAPE No.2.

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Fig. 11.16 The cross section with point numbers and WITHOUT SLIP SURFACES if you wish to check your data. In order to produce such a graph you can start from any of the two stability programs, after entering MAIN DATA or reading .BG file, save *.BGP or SWITCH TO BGP… It is essential that you must not have any slip surface(s) present.

148

Addendum 1. TENSION CRACK

Addendum No. 1 INPUT DATA RELATED TO THE TENSION CRACK This Addendum describes the treatment of the tension crack at the top of the slope which can be either empty or fully filled with water. Before starting entering of MAIN DATA described in Section 6.1, calculate the depth of the tension crack HC. The following equation is suggested: HC =

2c

γ

tan(45 0 + φ / 2) =

2c 1 + sin φ γ 1 − sin φ

The value of γ is the unit weight of soil or, more frequently, the rock, in which the tension crack is described. In the case of the nonlinear failure envelope, which might contain cohesion term, (p.13, Fig. 4.2, p.14), you can take the angle φ = φ0 = φB + ∆φ or some slightly smaller value. Compute the vertical stress at the lowest point of the crack as γ HC . This value of the distributed (surcharge) loading must be entered as the first distributed load when entering input data, as described in paragraph (xiv) in page 36. In the case that you wish to enter the pressure due to the water filling the crack to the top of the crack, inducing the horizontal water pressure γ H C2 /2 , it will be computed

automatically and the value will be further treated as a FIRST horizontal line force described in (xiii) in page 36. Programs will treat this force as a free vector moving the force horizontally in such a manner that it always acts on the first slice at the top of the slip surface. When defining the geometry of the cross section, the soil mass above the depth of the possible tension crack is omitted as a part of the soil zone, and substituted by a distributed (surcharge) loading with a magnitude that corresponds to γ HC . You should examine the simple example shown in Fig. 6.2, page 40. ENTERING TENSION CRACK IN MAIN DATA

Entering the input data is the same to the point where you define material properties. As soon as you complete entering the data described in (x) and (xi) the question (xii) from the screen appears as: DEPTH OF THE TENSION CRACK ?

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Addendum 1. TENSION CRACK

Enter the depth of the crack Hc. In the enclosed example in page 40 in Fig. 6.2, it is 3 meters. In the case that you enter 0 or just press it will mean that there is no tension crack and you will be asked to proceed further (xiii). The value of the depth of the tension crack does not enter into computation if the crack is not filled with water (empty). If that is the case, it is only used in graphical presentation. After that, if you entered the depth of the crack larger than zero, the following appears on the screen: CRACK TOP y = ?

Enter this elevation. (In the example mentioned above it is 18 meters). This value also does not enter directly in computation, but it is used for graphical presentation and it is used for placing the water pressure automatically in the crack if you answer positively to the next question that follows: IS TENSION CRACK FILLED WITH WATER - Yes/No (Y/N) ?

In the case that you enter Y or y (meaning Yes, tension crack is fully filled with water), the value of the horizontal hydrostatic force will be calculated automatically from the crack depth defined previously and placed at the elevation (2/3) Hc below the …TOP y = defined previously and automatically entered as the first LINE LOAD acting in the horizontal direction. If you enter N or n, you are saying that the crack is not filled with water and the horizontal force will not be generated. See example in page 40 where the force is slightly changed, as mentioned at the bottom of that page. Only in the case that the tension crack is filled with water, you will be asked to answer the question: IS SLOPE DIPPING Right or Left (R/L) ?

In the case that your slope is dipping left, the water pressure must have a negative sign. Note that in any case, when you have a crack, you will have to enter the distributed (surcharge) loading (xiv). The explanations (xiii) and (xiv) in page 36 are rewritten here and extended as follows: (xiii) NUMBER OF LINE LOADS if larger than zero, you will be asked to enter the pair of the coordinates for the point at which the line load is acting, and the intensity of each component in kN/m. In the case that you wish to apply the water pressure in the crack, this force will be automatically entered as the first force by a program. Note that "y" coordinate of the water pressure force in the crack will be held constant, while "x" coordinate of the point where the force acts will be initially set at x=0.01. The horizontal hydrostatic force will be moved horizontally during computations in such a way that it always acts at the top of the slope, on the first slice. Line loads will appear as dark red arrows on VGA color monitor.

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Addendum 1. TENSION CRACK

(xiv) NUMBER OF DISTRIBUTED LOADS in vertical direction are the surcharge loads (in kPa). This option has to be used if you want to describe the vertical tension crack. Soil above the depth of the crack is treated as the surcharge loading acting on the level of the tip of the crack. If larger than zero, you will be asked to enter two point numbers, for which the location is defined in the list of coordinates (viii) such that elevations of these points correspond to the elevation of the crack tip. In the case that you have entered the depth of the tension crack >0, the surcharge due to the vertical pressure on the level coinciding with the tip of the crack must be entered as the first distributed (surcharge) load γ HC. In the current version of the software, the surcharge at the crack tip level is not calculated automatically.

EXAMPLE An example of a homogeneous slope with soil parameters φ ′ , c ′ is shown in Fig. 6.2 and input data in the same page following the figure. Data are in files EX-3A.BG and EX3B.BG (page 40) Results of computation for these two examples (empty crack and crack filled with water) shown and discussed in Section 10.3, starting from page 110. EDITING TENSION CRACK

Menu CHANGES in both stability programs offers a possibility to add, remove or change the tension crack description under the following options: 4 LINE LOADS… 5 DISTRIBUTED LOADS…

where the following option in a sub-menu is offered after 4 LINE LOADS…is: 8 EDIT TENSION CRACK (Hc=…TOP y =…)

Essentially, the new tension crack definition is entered in an identical manner as in the input stage, described here in the previous page. Note that when dealing with the tension crack you have to consider line loads, distributed (surcharge) loading, as well as the geometry of the cross section in most cases.

151

REFERENCES

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Day, R.W., (1997) State of the Art: Limit Equilibrium and Finite-Element Analysis of Slopes, Discussion, Journal of Geotechnical and Geoenviromental Engineering, ASCE, Vol.122, No.9, September, pp. 894 De Mello, V.F.B., (1977) Reflections on design decisions of practical significance to embankment dams, Geotechnique 27, (3), pp. 281-354. Duncan, J.M., (1996) State of the Art: Limit Equilibrium and Finite-Element Analysis of Slopes, Journal of Geotechnical Engineering ASCE, Vol.122, No.7, July, pp. 577-596. Duncan, J.M., (1997) State of the Art: Limit Equilibrium and Finite-Element Analysis of Slopes, Closure, Journal of Geotechnical and Geoenviromental Engineering, ASCE, Vol.122, No.9, September, pp. 894. Fan, K.,Fredlund, D.G., and Wilson,G.W.,(1986). An interslice force function for limit equilibrium slope stability analysis. Can. Geotech. J. 23. Fredlund, D.G. (1984). State of the Art Lecture-Analytical methods for slope stability analysis. IV Int. Symposium on Land-slides, Toronto Hansen, J.B., (1966). Comparison of Methods for Stability Analysis, Bulletin No.21, The Danish Geotechnical Institute, Copenhagen, Denmark. Hoek, E. and Bray J.W., (1981). Rock slope engineering. The Institution of Mining and Metallurgy, London. Hoek, E. (1995) Strength of Rock and Rock Masses, ISRM News Journal, Vol.2, pp. 4-16 Janbu, N., (1954). Application of the composite slip surfaces for stability analysis. Proc. of the Europ. Conf. on Stability of Earth Slopes, Stockholm, Vol.3, pp. 43-49. Janbu, N.,Bjerrum,L., and Kjaernsli,B., (1956). Veiledning ved Losning av Fundamen tenteringsoppgaver. NGI Publication No.16. 93 p. Janbu, N., (1973). "Slope stability computation" in Embankment Dam Engineering", Casagrande Memorial Volume, eds. Hirschfeld and Poulos. John Wiley, New York, pp. 47-86 Janbu, N. (1977). State of the Art Report 9th Int. Conf. SMFE, Vol.2, Tokyo Jiang, G.L. and Magnan, J.P, (1997). Stability analysis of embankments: comparison of limit analysis with methods of slices. Geotechnique No.4, pp.857-872. Kenney, T.C., (1956). An examination of the methods of calculating the stability of slopes. M.Sc. Thesis, Imperial College. Lambe, T.W. (1985). Amuay landslides. Proc. 11 th Int. Conf. on SMFE, Golden Jubilee Volume, San Francisco. Leps, T.M. (1970). Review of shearing strength of rockfill. Journ. of the Soil Mech. and Found. Div. ASCE, Vol.96, No.1., pp.1159-1170 Little, A.L. and Price, V.E., (1958). The use of an electronic computer for slope stability computation, Geotechnique, 8, pp.113-120. Lupini, J.F., Skinner,A.E.& Vaughan,P.R., (1981). The drained residual strength of cohesive soils, Geotechnique 31, No.2, 181-213. Maksimović, M. (1970). A new method of slope stability analysis. Computer program and notes. (Unpublished) Imperial College, London. Maksimović, M. (1977). Discussion, Proc. 9 th. International Conf. SMFE Tokyo, Vol.3, pp.414. Maksimović, M. (1979). Limit equilibrium for nonlinear failure envelope and arbitrary slip surface. Third Int. Conf. on Numerical Methods in Geomechanics, Aachen, pp. 769-777. Maksimović, M., (1988). General slope stability software package for micro-computers. 6 th Int. Conf. on Num. Meth in Geomechanics, Vol.3, Innsbruck, pp. 2145-2150. Maksimović, M., (1989-a). Nonlinear failure envelope for soils. Journal of Geotechnical Engineering, ASCE, Vol.115 No.4, April, pp. 581-586.

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Maksimović, M., (1989-b). On the residual shearing strength of clays. Geotechnique 39, No.2, June, pp.347-351. Maksimović, M., (1989-c). Nonlinear failure envelope for coarse-grained soils. XII Int. Conf. SMFE, Vol.1, Rio de Janeiro, August, pp. 731-734. Maksimović, M., (1992). New description of the shear strength for rock joints, Rock Mechanics and Rock Engineering, 25 (4), pp.275-284 Maksimović, M., (1993). Nonlinear failure envelope for the limit state design, International Symp. on Limit State Design, Copenhagen, Vol.1, pp.131-140. Maksimović, M. (1995) Drained residual strength of cohesive soils. Discussion on paper by Stark&Hisham, , Journ. of Geot. Eng. ASCE, Vol.121,No.9, Sept. pp. 670-672 Maksimović, M. (1996). A Family of Non-linear Failure envelopes for Non-Cemented Soils and Rock Joints, WWW, Electric Journal for Geotechnical Engineering, ppr 9607 Maksimović, M. (1996). Shear Strength Components of a Rough Rock Joint, Int. Journal of Rock Mechanics and Mining Sciences, Pergamon, Vol.13, No.8, Dec. pp.769-783. Marachi, N.D., Chan,C.G. & Seed, H.B. (1972). Evaluation of properties of rockfill materials. Journ. of the Soil Mech. and Found. Div. ASCE, Vol.98, SM1. Marsal, R.J. (1973). Mechanical properties of rockfill, Embankment Dam Engineering, Casagrande Memorial Volume, Hirschfield & Poulos. ed. John Willey, pp.109-200. Morgenstern, N.R. & Price, V.E. (1965). The analysis of the stability of general slip surfaces, Geotechnique 15. Morgenstern, N.R. & Price, V.E., (1967). A numerical method for solving the equations of stability of general slip surfaces. Computer Journal, 9, 388-393 Newmark, N.M., (1965). Effects of earthquakes on dams and embankments. Geotechnique, 15. Singh, R., Henkel, D.J., Sangrey, D.A. Shear and Ko Swelling of Overconsolidated clay, Proc. of the Eight Int. Conf. on SMFE, Moskow, 1973, Vol.1.2, pp. 367-376 Skempton, A,W. & Petley, D.J., (1967). The strength along structural discontinuities in stiff clays. Proc. of the Oslo Geot. Conf. on the Shear Strength Prop. of Natural Soils and Rocks. Vol.2. Skempton, A.W., (1985). Residual strength of clays in landslides, folded strata and the laboratory. Geotechnique 35, pp. 3-18. Spencer, E., (1967). A method of analysis of the stability of embankments assuming parallel interslice forces, Geotechnique, 17 Stark, D.T. & Eid T. Hisham, (1994). Drained residual strength of cohesive soils. ASCE Journal of Geotechnical Engineering, Vol.120, No.5, May 1994, pp. 856-871 Turnbul, W.J. & Hvorslev, M.J., (1967).Special Problems in Slope Stability, Journal ASCE, Soil Mech. and Found. Div., Vol.93, No.SM4, July. Walker, B. & Fell,R., (1987). Slope instability and stabilization, A.A. Balkema, Rotterdam Brookfield. Whitman, R.V. and Bailey, W.A., (1967). Use of computers for slope stability analysis. Journal ASCE, 93, No.SM4, pp.475-498. Wright, S.G. & Kulhawy, F.H., (1973) Accuracy of equilibrium slope stability methods, Journal of SMFE, Proc. ASCE, 99. SM10,pp. 783-791 Wright, S.G. (1974). SSTAB1-A General computer program for slope stability analyses. Research Report No. GE-74-1, Dept of Civ. Eng. The Univ. of Texas.

154

INDEX

non-linear envelope, 12, 36 number of lines, 34

arbitrary surfaces, 30,71 asterisk (*), 47 automatic search, 24, 46

number of line loads, 36, 98 number of materials, 34 number of points, 32, 34 number of slices, 100

Bezier slip surface, 29, 73 Bishop method, 1, 23, 46 caption 4, 130 changes in BE, 65, 101 changes in GE, 90 circular surface, 23, 29, 76 composite surface, 29, 75 contours, 63 convergence, 92 critical acceleration, 84

origin, 4, 43, 102 passing point, 23, 46, 75 pen thickness, 134 piezometric line, 35 PLT or HP-GL files, 11, 132 polygonal surface, 72 pore pressures, 35

distributed load, 36, 151 dummy lines, 33, 35

residual strength, 17 rockfill, 19 rock discontinuity, 19 rock mass, 20

factor of safety, 12 file types, 11 f(x) functions, 77, 78 fixed plane, 71, 74, 81, 123

scale, 38, 65, 101 seismic coefficient, 37 shearing strength, 12, 13 shifting cross section, 101 soil limits, 35 soil parameters, 35 Spencer method, 77 spider, 24, 46 stress level ratio (SLR), 21 steepest descent, 24, 46 step, 53, 84 surcharge load, 36, 151 switch to BE, 83, 89 switch to BGP, 52, 64 switch to GE, 63

general method, 1, 29 graphic modes, 3 gravel, 18 grid of centers, 26, 49 headings, 4, 133 HP-GL or *.PLT files, 11, 132 H/S Ratio, 57, 96 interactive search, 53, 84 Bezier, 86 circle, 53 composite, 88 fixed plane, 87 polygon, 85

tailwater level, 34 tension stresses, 51, 64 tension crack, 40, 110 title, 34 title for slip surface, 71 tolerances, 66, 89

line of thrust, 79 line loads, 36 load scale, 132 local grid, 26

water filled crack, 36, 110 water level, 34

main data, 33 mirror, 38, 103 Morgenstern-Price, 29 move, 38

zoom, 38, 43, 101 155