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BASIC GEOTECHNICAL ANALYSIS

Jeen-Shang Lin University of Pittsburgh

jslin@engrng.pitt.edu

Ching S. Chang University of Massachusetts

(Last update February 2001) (Copyright material, no reproduction without written consent from authors.)

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CHAPTER 6 SLOPE STABILITY ANALYSIS

6.1 Introduction In this chapter we will work on the important topic of stability analysis. Generally, we may classify a soil stability analysis technique into one of the following categories: 1) limiting analysis approach; 2) limiting equilibrium approach; and, 3) displacement-based approach.

By limiting analysis we mean the so-called upper bound solution and lower bound solution techniques. They are derived from classical plasticity theory using associated flow rule. Their application is very much limited to ideal material with simple geometry. On the other hand, the displacement based approach is a more recent development which includes the finite elements, the boundary elements, and the discrete element methods. The discrete element methods is particularly useful to rock slope stability analysis.

We will concentrate on the limiting equilibrium approach for now. Although our emphasis will be on Simplified Bishop method, we will first cover some basics of the limiting equilibrium methods that are in use today. Specifically, we will address the assumptions involved, and how the various methods differ. This is then followed by the development of a general program for Simplified Bishop method. We will briefly demonstrate how we can set up a spreadsheet solution for a particular slip surface. This then serves as a means for validating the accuracy of the computer program developed. 6.2 Factor of Safety The goal of a slope stability analysis is to find out safety margin of a slope. Toward this goal, in a limiting equilibrium analysis, one investigates a soil mass bounded by the ground surface and as many as necessary the potential failure surfaces. For each of the potential failure surface, a safety margin is determined. Conventionally, a safety margin is reflected by a single number-- the factor of safety (FS), which is defined as follows,

where, fτ is the average shear strength of the soil along the potential failure surface, whereas τ the average shear stress on the same surface. The minimum factor of safety obtained for all the failure surfaces investigated is then denoted as the factor of safety for the problem. It is therefore crucial that failure surface used be

ττ f=FS 1

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physically realizable and that all potential failure surfaces be investigated. 6.3 Basic Considerations Unless undrained stability is to be considered, the normal forces from a soil upon a potential failure surface is needed in evaluating the shear strength of the soil. In order to obtain this piece of information, a soil mass bounded by the slope surface and a potential slip surface is often divided into a number of slices. The underlying consideration is that one may then find out the forces associated with each of the slice and consequently the fraction of the strength that is mobilized along the potential failure surface. As it turns out, however, the slice methods actually produce statically-indeterminate problems.

Figure 1 Slice method and the forces on a slice

Let's examine a simple case as depicted in Fig. 1, in which the slope is divided into n slices. The total number of unknowns and the available equations are summarized in Table 6.1. For this problem we have (5n-2) unknowns, while only (3n) equations are available. Since n is generally in excess of 20, we have quite significant indeterminancy here. As we all learned from "Strength of Material", to solve a statically indeterminate problem, we need to introduce additional equations based upon deformation considerations. In the early stage of soil mechanics development, there is no tool for such an undertaking and an ingenious engineering approach was developed, which was later

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called "Limiting Equilibrium" approach. In application, the slices we cut are rather thin, and the normal force at the base of each slice can be reasonably assumed to act at the center of the base, the total unknowns of a problem can be viewed as 4n-2.

TABLE 1 TOTAL NUMBER OF EQUATIONS VERSUS KNOWNS

Unknowns associated with n slices

source assumptions number of unknowns

normal resultant force on the base of a slice 2 unkowns for each slice (magnitude and location)

2n

2n can be reduced to n if slice is thin and the normal force is assumed to act at the center of each slice base.

factor of safety One unknown for each slice

Often assume constant for all slices for regular analysis. (For a progressive failure, this assumption no longer applies.)

n

1

Inter-slice forces on n-1 inter-faces 3 unkowns on each face (magnitude, direction and location)

3(n-1)

5n-2 or 4n-2 for thin slice

(For progressive failure, the above become

6n-3 and 5n-3, respectively.)

Available Equations

3 equations for each slice (ΣFx=0, ΣFy=0,ΣM=0) Total available Equations= 3n

The basic idea behind limiting equilibrium is really simple: we have a situation involves more unknowns than we can handle, why not just try to reduce the number of the unknowns. But how? By assumptions! Of course, making unjustifiable assumption may be disastrous to your analysis. In Swedish circle method (also known as the ordinary method of slice or Fellenius method), it is assumed that the resultant of the forces from the two sides of a slice is parallel to the bottom of the slice. That is, n assumptions are introduced into the solutions. As a result the total number of unknowns is reduced to (3n-2), while the number of equations remains (3n). The Swedish method therefore has created a scenario of more equations than unknowns. We have now an "over-determined" system. A new problem is created that--We are not able to satisfy all the equations. The engineering answer is simple: We simply neglect some equations. This is how it is done:

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1. Use only equilibrium equation in the normal direction of each slice, ΣFn=0. This

allows for finding the normal force, N, on the bottom of each slice. (This can be done, because the resultant side force is perpendicular to N.)

2. Use only the global moment equilibrium along the failure surface. Notice that the

inter-slice forces, although internal forces, may not be in equilibrium.

Figure 2 Assumption of the Swedish method

Many equations are neglected, not all the moment equilibrium and force equilibrium are satisfied. Among the 3n equations, only n+1 are being used. How good are the solutions so obtained is a crucial question. Experiences (Lambe and Whitman, 1969) show that the factor safety so obtained is generally underestimated, the error may be 10 % to 15% lower than from other methods for some problem, and for others the error may be as high as 60%. This conservatism comes mainly because the side forces have been placed on the worst possible direction. Swedish method should not be used in practice. It can be used at best only as a way to get initial guess for other methods. The most frequently used slice method is the Simplified Bishop method. The method assumes that the side forces lie in the horizontal direction, see Fig. 3. This is equivalent to introduce (n-1) assumptions. The number of unknowns is therefore reduced to (3n-1). The result is again an "over-determined" system. The method also neglects one equation to reach a solution. The steps involved in the Simplified Bishop method is basically the same as in the Swedish method except that in order to find N the equilibrium equation in vertical direction has to be used. The Simplified Bishop method, however, is capable of giving reasonable answers for most cases (Spencer, 1967 ). In a way, assuming lateral force on each side acts horizontally, in an average sense may not be too much off from reality. Again, this method only applies to circular slip surface.

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Figure 3 Simplified Bishop: Assumption and Forces

For non-circular slip surfaces, more rigorous methods were also proposed. A method is called a rigorous method if the final number of unknowns equals the number of equations. It is apparent from the above discussions that all it takes is to increase the number of unknowns. The method put out by Morgenstern and Price is considered to be the most general limiting equilibrium method to date. Basically, they introduce an additional unknown. What they proposed is as follows, 1. Consider a different angle for each the resultant force on each interface: (n-1)

assumptions are introduced. Let x be a horizontal axis starts from the toe toward the slope, this assumption can be written as,

where, Xi, Ei are the vertical and horizontal components of the side force acting on

interface i that has a coordinate of xi. f(x) is an arbitrary function which should be

θ iii

i =)xf(=EX tan 2

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specified in advance. f(x) can take any shape 2. Assume all these angles are related through an unknown scale factor, λ-- one

additional unknown. In equation form, this becomes,

Morgenstern and Price's method therefore poses a slope stability analysis as a statically determinate problem. For a problem involving n slices, 3n equations have to be solved. Moreover, these simultaneous equations are highly non-linear, the numerical solution is quite tedious, and frequently numerical divergence may be encountered. The benefit of this method is that it is quite versatile. Of interest here is to briefly mention Spencer's work. Spencer has simplified the Morgenstern and Price's method by assuming that f(x) is a constant. In other words, Spencer poses the equation as,

That is, the angles of all resultant interslice forces are the same that have to be solved. So, one proceed by assuming an arbitrary angle, and then check if all the equilibrium conditions are satisfied, if not, iterations are carry out by modifying the angle using a solution strategy such as the Newton-Ralphson's scheme. Within the framework of Spencer's solution, the simplified Bishop method can be viewed as a partial solution in that (1) the angle of the inter-slice force is assumed beforehand a fixed number, tanθ=0 that is; and (2) only moment equilibrium is satisfied. Spencer has shown that the factor of safety from moment equilibrium is not very sensitive to the angle of the inter-slice force when a constant value is used. Thus Bishop was able to obtain a reasonable solution. The common belief on the accuracy of slope stability analysis is that the better the statics are satisfied, the more reliable is the solution. And as such the solution from rigorous methods are often desired. But as the displacement based methods become more accessible, it may be before long that the rigorous solutions will be replaced by methods like discrete elements altogether. In the following, we will concentrate on the simplified Bishop method. 6.4 Basic Equations For a typical slice, say the slice i, in theSimplified Bishop method, the normal force, Ni, at

)xf(=EX

ii

i λ 3

θλ tan==EX

i

i 4

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the base of a slice can be obtained from the vertical equilibrium equation as follows,

On the other hand, the shear force, Ti, equals in magnitude the mobilized shear strength, which is the shear strength divided by the factor of safety, Fs,

where c and φ are the effective cohesion and the effective internal friction angle of soils. The overall moment equilibrium requires,

by combining Eqs. (5) through (7), the factor of safety, Fs, can be further expressed as

where,

)T-W(1=N iiii

i θθ

sincos

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)]lu-N[+lc(F1=T iiiii

si φtan∆∆ 6

θ

φ

ii

n

1=i

iiiii

n

1=i

W

)]lu-N[+lc(=FS

sin

tan

∑

∑ ∆∆ 7

θ

θφ

sin

tan

ii

n

1=i

iiiiiii

n

1=is

W

)](M][1/)xu-W(+xc[=F

∑

∑ ∆∆ 8

)F

+(1=)(Ms

iiii

φθθθ

tantancos 9

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Eq. (7) shows that the factor of safety, FS, to be solved appear on both sides of the equation. It, however, can easily be solved through iterations. To illustrate the general steps involved, we will go through the exercise of tabular solutions in the next section. In Swedish method, the normal force, Ni, is found, on the other hand, using the force equilibrium equation in Ni direction as follows,

and by substituting Eq. (10) into Eq. (7) the factor of safety of Swedish method can be found as,

As posed by Eq. (11) Swedish method does not require iteration in determining the factor of safety of a slope. Also, all the terms used in Eq. (11) also appeared in Eq. (7), this suggests that one may set up the calculation for Simplified Bishop method, but using Swedish method first to find the initial factor of safety to initiate the iteration process. 6.5 Tabular Solutions for the Simplified Bishop Method To manually carry out a slope stability analysis for a given slip surface through Simplified Bishop method, we generally begin with setting up a table with the equations embedded. Table 2 shows such a typical calculation table for a slice method. This calculation is very tedious. But a closer look tells us that because of the repetitive nature of the calculation, we may reduce the trauma via the use of spreadsheet. For this, we still have to prepare some input data. We do not want to make the spreadsheet too complicated by having it automatically calculate these input data. We might as well just develop a general program if we want to go that extra step. Let us first work on a very simple problem in which the soil is homogeneous, and that no water is present. The problem is depicted in Fig. 5. To begin with, we first draw a scaled sketch of the problem. For simplicity let's only divide the slope into 6 slices. We have to find out for the input of each slice the average base angle, the slice width, and the average height for each slice, and the average density. It is rather easy to measure tangent of the base slope angle, and therefore we incorporate that tangent into the table rather than the angle itself. The dimensions are shown in Fig.

θθ iiiiii /xu-W=N coscos ∆ 10

θ

θθφθ

ii

n

1=i

iiiiiiiii

n

1=is

W

xuWxc=F

)/-(+/

sin

coscostancos

∑

∆∆∑ 11

10

6.

11

Figure 5 Example problem layout

Figure 6. Dimensions of slices

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TABLE 2 Typical Tabular Form for Calculating Factor of Safety by a Method of Slices slice no.

(1) (2) (3) (4) (5) (6) (7) (8)

Wi θ i sinθ i Wisinθ i ∆ xi ui∆ xi Wi-(6)

Wicosθ i-(6)/cosθ i

1

2

.

n

Σ(4)

slice no. (9) (10) (11) (12) (13) (14)

(7) tanφ i

ci ∆ xi ci∆ xi/cosθ i (9)+(10) Mi(θ) (12)/(13)

1

2

.

n

∑ )(11 ∑ )(14

Fs initial (Swedish) =(∑ )(8 + ∑ )(11 )/∑ )(4

Afterwards, Fs =∑ )(14 /∑ )(4 , only the calculation of columns 13 and 14 is repeated.

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The input spreadsheet is explained in Table 3. Here, we have taken the advantage of the fact that no pore water pressure presents. To facilitate the check, first column is the slice number, which is followed by x∆ , and the mid-height of a slice. TABLE 3 Speadsheet input for the example problem A1>TITLE A2>START: B2>0 C2>ITER: D2>=IF(B2=0,0,D2+1) A4>SLICE B4> x∆ C4>H D4>γ E4>W F4>tanθ G4> θ H4>W*Sinθ I4>φ J4>c K4>c* x∆ L4>c* x∆ /cosθ M4> Wcosθ tanφ N4>W*tanφ O4>Mi P4>RESIST A5>1 B5>16.38 C5>2.15 D5>120 E5>D5*C5*B5 F5>=TAN(G5) G5> -14 H5>E5*SIN(Radians(G5)) I5>20 J5>100 K5>J5*B5 L5>J5*B5/COS(Radians(G5)) M5>E5*COS(Radians(G5))*TAN(Radians(I5)) N5>E5*TAN(Radians(I5)) O5>=IF($B$2=0,0,COS(Radians(G5))*(1+(TAN(Radians(G5))*TAN(Radians(I5))/$M$15)) P5>=IF($B$2=0,0,(K5+N5)/O5) INPUT DATA FROM A6>..D10>, G6>..G10>,I6>..J10> COPY THE REST OF THE EQUATIONS FROM ROW 5 TO ROW 6..ROW10 G12> DR. M= H12>=SUM(H5..H10) J12>SUM= L12>=SUM(L5..L10) M12>=SUM(M5..M10) N12>=SUM(N5..N10) P12>=SUM(P5..P10) J13>RM(SWEDISH)= M13>L12+M12 J14>FS(SWEDISH) M14>M13/H12 J15>FS(SIMPLIFIED) M15>=IF($B$2=0,M14,P12/H12) GoTo Tools->Options>set iteration to manual, and one iteration after each pressing of F9.To start, input a non-zero number into cell B2. Iteration is carried out each time F9 key is pressed.

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TABLE 4 Results of the analysis after 4 iterations A D E F G H

1 Slope Stability Analysis Example

2 start> 1Iter= 4 3 4 SLICE x∆ H r W tanθ θ (deg) W*sinθ 5 1 16.38 2.15 120 4226.04 -0.24933 -14 -1022.37161 6 2 16.38 4.51 120 8864.856 0.017455 1 154.71307 7 3 16.38 34.63 120 68068.728 0.212557 12 14152.2843 8 4 16.38 47.23 120 92835.288 0.466308 25 39233.888 9 5 16.38 36.42 120 71587.152 0.809784 39 45051.2545

10 6 16.38 14.68 120 28855.008 1.664279 59 24733.5693 11 12 Dr. M= 122303.338 13 14 15

I J K L M N O P

1 2 3 4φ c c* x∆ c* x∆ /cosθ Wcosθ tanφ Wtanφ Mi Resist 5 20 100 1638 1688.145 1492.46306 1538.153 0.869147 3654.3356 20 100 1638 1638.25 3226.0523 3226.544 1.007145 4830.0357 20 100 1638 1674.594 24233.5979 24774.99 1.065077 24799.148 20 100 1638 1807.333 30623.489 33789.28 1.083007 32711.959 20 100 1638 2107.712 20248.9985 26055.59 1.040269 26621.57

10 20 100 1638 3180.347 5409.11735 10502.36 0.873426 13899.7111 12 Sum= 12096.38 85233.718 MR= 106516.713 RM(Swedish)= 97330.1

14 FS(Swedish)= 0.795809

15 FS(Simplified)= 0.870923 6.6 Equilibrium Check We will use the results of the spreadsheet calculation from the simplified Bishop method to show that the force equilibrium is not satisfied. We will calculate the normal and shear force of each

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slice first. By plugging Eq. 6 into 5, the normal force, Ni , can be found as

]sin1[)(

1iii

ii lc

FSW

MN θ

θ∆−= (12)

With Ni available, Ti can be found from Eq. 6. For the slice at either end of the slope, which has only one interface with another slice, there is only one unknown lateral side force that lies presumably in the horizontal direction. From the horizontal force equilibrium consideration, this force can readily be found. Here we will solve the unknown force first on slice 1, then progress toward the slice 6. The force polygon on slice 1 is shown below,

Figure 7 Lateral force on a block

By denoting horizontal force to be positive in x direction, the horizontal component in x direction is found as,

θ

θsin

cosNN

TT

x

x

−==

(13)

The unknown lateral force on slice 1 becomes, xx NTE 111 += (14) For block 2, it can readily be seen that the lateral force between slices 2 and 3 is, 1222 ENTE xx ++= (15) Repeating this procedure, the net lateral force or unbalanced lateral force on the slice 6 becomes,

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xxx NTEF 666 ++=∑ (16) For a general n slice problem, the unbalanced horizontal force can be found as,

∑∑=

+=n

iixixx NTF

1

)( (17)

This procedure is applied to the example spreadsheet, and the results clearly show the simplified Bishop method does not satisfy horizontal force equilibrium by leaving an unbalanced horizontal force of 12105 lbs. This error amounts to about 10% of ∑ iiW θsin . Q R S T U

2 3

4 (+to right)

5 N Nx T Tx Fx

6 5401.40

2 1306.717 4195.227 4070.6115377.32

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7 8769.41

2 -153.047 5545.334 5544.495391.44

2

8 63537.3

1 -13210.2 28472.94 27850.7414640.5

9

9 84918.0

3 -35887.9 37559.71 34040.65 -1847.25

10 67361.7

2 -42392.1 30568.31 23756.04 -18636.1

11 29460.0

8 -25252.2 15961.82 8220.943 -17031.312

13 Total unbalanced horizontal force -12105.2

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15 ∑∑ iix WF θsin/ -0.09898Q6=(E6-L6/$L$16*SIN(G6*PI()/180))/O6 R6=-Q6*SIN(G6/180*PI()) S6=1/$L$16*(L6+Q6*TAN(I6/180*PI())) T6=S6*COS(G6/180*PI()) U6=R6+T6 U13=SUM(U6..U16) U15=U13/SUM(H6..H11) 6.7 Water and slopes

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6.7.1 A completely submerged slope The preceding discussion of dry slopes can readily be applied to a completely submerged slope. A slope completely submerged underneath a static water table is depicted in Fig. 8. The problem is redrawn and is shown in Fig. 9. In which the effect of water is separated. As such, it can easily be seen that the slope is almost the same as a dry slope except now that the submerged unit weight of soil be used. There is no need to consider the pore water pressure.

Figure 8. The modeling of a submerged slope

6.7.2 A partially Submerged Slope Most frequently, however, a slope is only partially submerged. Fig. 9 depicts a slope is partly submerged to a water level marked as A-A. In addition, field observation or flow net construction allows one to come up with a piezometric surface 1-1.

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Figure 9. A partially Submerged slope

Here we will follow the procedure employed by Terzaghi and Peck (1967). Basically, the approach incorporated the fact that water below A-A level has to be in equilibrium, and therefore can be taken out of the analysis just as what has been done to a completely submerged slope. This is further illustrated in Fig. 10 1) For all soils below A-A level, the effective weight be used. 2) For all soils above A-A, total weight be used. 3) For all soils between A-A and a piezometric surface, the total weight be used. Because only the pore water pressure up to A-A has been taken out, the pore water pressure in excess of A-A still has to be accounted for. Three distinct slices can therefore be distinguished for Fig. 9:

Figure 10 Three different modeling considerations in a submerged slope

1)Slice 3: The weight of soil, W, can be expressed as

where, Wa is the total weight above A-A, Wb is the submerged weight below A-A, and zb wγ is the water pressure in excess of that caused by A-A. The additional pore water pressure here is z wγ , and z is the height of the piezometric surface above A-A. 2)Slice 1: This slice is above A-A. Wb term is zero, the total weight of a slice W equals to Wa. The pore water pressure u equals wzγ .

γ wba zb+W+W=W 128

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3)Slice 5: This slice is completely submerged under A-A the only weight is the submerged weight Wb. No pore water pressure needs to be found. Now, we may substitute the proper W and u for each slice into the calculation of the factor of safety. The driving moment becomes,

whereas, the mobilized resisting moment becomes,

6.7 Pore Water Pressure For a slope under steady state seepage, we may estimate the location of the phreatic surface and obtain the pore water pressure for stability calculation. But often we have pore water pressure measurements in the field, and we instead construct the piezometric surface. This later is much easier to use. In that the pore water pressure below a piezometric surface is simply the depth multiplied by the unit weight of water. As a result, we also often approximate a phreatic surface to be a piezometric surface. If a piezometric surface is known, the program asks for input the nodes that require to define this surface. Since the slope is required to face left, the elevation of water nodes can only increase to ward the right. Pore pressures are also often defined by a pore pressure ratio, ru , which is defined as,

where, u is the pore pressure at a point, and γz is the total overburden pressure above that point. ru is used mostly for evaluating the construction induced pore water pressure within an embankment. We may assume an average ru value in determining the stability of a embankment right after it is completed.

θ ibar )RW+W(=M sin∑ 139

θ

φγ

iiba

iiwiiii

R )W+W()/FS]lz-N[+lcR(

=FSsin

tan∑

∆∆∑ 20

zu=ru γ 21

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If no water nodes are input, the program asks for ru for each soil stratum involved.

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Program CH6 is used in the computation of the following examples. A Windows version is currently being written. Click to Download CH6. The program asks for the nodal points that are required for defining the geometry. These coordinates are input first. Each soil stratum is defined by its surface. On top of that, the ground surface is nodes are required. For instance, Fig.11 gives an example in which the ground geometry is defined by 13 nodes, while water by 3. After input the x,y coordinates of all these three nodes. The stratum surfaces are input as follows: Ground surface: 5 nodes=>1,4,8,11,12. Stratum 1:3 nodes=>8,11,12 Stratum 2:4 nodes=>4,8,9,10 Stratum 3:4 nodes=>4,5,6,7 Stratum 4:5 nodes:=>1,4,2,3,13. Sometimes, we will input the bedrock surface as the surface for the last stratum, then specify it as the impenetrable surface. A screen shot for example 3 input is provided below.

Figure 11. Input explanation

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Example 1

EXAMPLE 1

INPUT DATA FOR GROUND NODES: NODE POINT X-COOR Y-COOR 1 100.00 112.20 2 417.50 255.90 3 500.00 255.90 4 188.90 112.20 5 426.00 250.00 6 500.00 250.00 7 100.00 105.20 8 500.00 105.20 INPUT DATA FOR WATER NODES: NODE POINT X-COOR Y-COOR 1 100.00 242.60 2 500.00 242.60 ----------------------------------------------------------------------------

INPUT DATA FOR GEOMETRY: NODES FOR WATER SURFACE ARE: 1 2 NODES FOR GROUND SURFACE ARE: 1 2 3 FOR THE SURFACE OF STRATUM 1 ARE 1 2 3 FOR THE SURFACE OF STRATUM 2 ARE 1 4 5 6 FOR THE SURFACE OF STRATUM 3 ARE 7 8 ----------------------------------------------------------------------------

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GENERAL GEOTECHNICAL INFORMATION: PIZOMETRIC SURFACE CONSISTS OF THE FOLLOWING WATER NODES: 1 2 UNIT WEIGHT OF WATER= 62.4 ---------------------------------------------------------------------------- STRATUM COHESION PHI(DEG) UNIT WT saturate ---------------------------------------------------------------------------- 1 0.00 38.00 133.00 133.00 2 300.00 29.00 133.00 133.00 THE DEEPEST STRATUM IS CONSIDERED AS UNPENETRATABLE ---------------------------------------------------------------------------- OUTPUT OF THE ANALYSIS:

CENTER OF SLIP CIRCLE: X-COOR= 146.6 Y-COOR= 278.6 RADIUS= 168.7 XL= 105.9183 XU= 296.4518 FS( 1 )= 2.02021 FS( 2 )= 2.015524

DIAGNOSTIC RESULT

TRIAL FS= 2.02021 ------------------------------------------------------------------------- UNIFORM SLICE WIDTH = 9.526676 SLICE COS PHI C WT-STATIC u MI DR M RE M 1 0.98 0.66 0.00 2.197E+03 0.000E+00 8.947E-01 -4.678E+02 1.918E+03 2 0.99 0.51 300.00 6.301E+03 0.000E+00 9.448E-01 -9.858E+02 6.722E+03 3 0.99 0.51 300.00 1.003E+04 0.000E+00 9.676E-01 -1.003E+03 8.700E+03 4 1.00 0.51 300.00 1.339E+04 0.000E+00 9.871E-01 -5.825E+02 1.042E+04 5 1.00 0.51 300.00 1.639E+04 0.000E+00 1.003E+00 2.126E+02 1.190E+04 6 1.00 0.51 300.00 1.902E+04 0.000E+00 1.017E+00 1.321E+03 1.318E+04 7 0.99 0.51 300.00 2.130E+04 0.000E+00 1.027E+00 2.681E+03 1.428E+04 8 0.98 0.66 0.00 2.320E+04 0.000E+00 1.054E+00 4.231E+03 1.720E+04 9 0.97 0.66 0.00 2.471E+04 0.000E+00 1.063E+00 5.903E+03 1.816E+04 10 0.96 0.66 0.00 2.584E+04 0.000E+00 1.070E+00 7.631E+03 1.887E+04 11 0.94 0.51 300.00 2.655E+04 0.000E+00 1.033E+00 9.339E+03 1.702E+04 12 0.91 0.51 300.00 2.681E+04 0.000E+00 1.025E+00 1.095E+04 1.729E+04 13 0.89 0.51 300.00 2.660E+04 0.000E+00 1.013E+00 1.236E+04 1.738E+04 14 0.85 0.51 300.00 2.587E+04 0.000E+00 9.964E-01 1.348E+04 1.726E+04 15 0.82 0.51 300.00 2.455E+04 0.000E+00 9.748E-01 1.418E+04 1.689E+04 16 0.77 0.51 300.00 2.257E+04 0.000E+00 9.472E-01 1.431E+04 1.622E+04 17 0.72 0.66 0.00 1.979E+04 0.000E+00 9.903E-01 1.367E+04 1.562E+04 18 0.66 0.66 0.00 1.606E+04 0.000E+00 9.536E-01 1.200E+04 1.315E+04 19 0.60 0.66 0.00 1.107E+04 0.000E+00 9.060E-01 8.895E+03 9.546E+03 20 0.51 0.66 0.00 4.327E+03 0.000E+00 8.428E-01 3.721E+03 4.011E+03 ------------------------------------------------------------------------- TOTAL DRIVING MOMENT/R= 131845.5 TOTAL RESISTING MOMENT/R= 265737.8 FACTOR OF SAFETY= 2.015524

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Example 2

EXAMPLE 2

INPUT DATA FOR GROUND NODES: NODE POINT X-COOR Y-COOR 1 100.00 100.00 2 300.00 100.00 3 500.00 200.00 4 700.00 300.00 5 900.00 300.00 6 900.00 200.00 INPUT DATA FOR WATER NODES: NODE POINT X-COOR Y-COOR 1 100.00 150.00 2 400.00 150.00 3 600.00 200.00 4 900.00 250.00 ----------------------------------------------------------------------------

INPUT DATA FOR GEOMETRY: NODES FOR WATER SURFACE ARE: 1 2 3 4 NODES FOR GROUND SURFACE ARE: 1 2 3 4 5 FOR THE SURFACE OF STRATUM 1 ARE 3 4 5 FOR THE SURFACE OF STRATUM 2 ARE 1 2 3 6 ----------------------------------------------------------------------------

GENERAL GEOTECHNICAL INFORMATION: PIZOMETRIC SURFACE CONSISTS OF THE FOLLOWING WATER NODES: 1 2 3 4 UNIT WEIGHT OF WATER= 62.4 ---------------------------------------------------------------------------- STRATUM COHESION PHI(DEG) UNIT WT saturate ---------------------------------------------------------------------------- 1 1500.00 20.00 126.00 126.00 2 1000.00 33.00 130.00 130.00 ----------------------------------------------------------------------------

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OUTPUT OF THE ANALYSIS:

CENTER OF SLIP CIRCLE: X-COOR= 364.3 Y-COOR= 600 RADIUS= 500.2 XL= 306.3389 XU= 764.55 FS( 1 )= 1.607154 FS( 2 )= 1.573335 FS( 3 )= 1.568754

Example 3. This example is taken from Reference 3. It is used to demonstrate the use of ru. In this example ru=0.5. Spencer obtained a FS of 1.039 for this case. The input screen shots are,

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A mistake was made in input for the surface node, which can be corrected as shown:

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The results are also contained in the output file. The input file saved can be used for analysis again without going through the tedious input process.

EXAMPLE 3

INPUT DATA FOR GROUND NODES: NODE POINT X-COOR Y-COOR 1 0.00 0.00 2 10.00 0.00 3 210.50 100.00 4 240.00 100.00 INPUT DATA FOR WATER NODES: NODE POINT X-COOR Y-COOR

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----------------------------------------------------------------------------

INPUT DATA FOR GEOMETRY: NODES FOR WATER SURFACE ARE: NODES FOR GROUND SURFACE ARE: 1 2 3 4 FOR THE SURFACE OF STRATUM 1 ARE 1 2 3 4 ----------------------------------------------------------------------------

GENERAL GEOTECHNICAL INFORMATION: UNIT WEIGHT OF WATER= 62.4 ---------------------------------------------------------------------------- STRATUM COHESION PHI(DEG) UNIT WT saturate ---------------------------------------------------------------------------- 1 240.00 40.00 120.00 120.00 ---------------------------------------------------------------------------- OUTPUT OF THE ANALYSIS:

CENTER OF SLIP CIRCLE: X-COOR= 41.6 Y-COOR= 209.2 RADIUS= 211.6 XL= 9.820691 XU= 222.8455 FS( 1 )= .8489703 FS( 2 )= .9889693 FS( 3 )= 1.028287 FS( 4 )= 1.038059

REFERENCES 1. Bishop A. W., "The Use of the Slip Circlwe in the Stability Analysis of Slopes," GEOTECHNIQUE, 5, pp7-17, 1955. 2. Lambe W. and R. V. Whitman, SOIL MECHANICS, John Wiley & Sons Inc., 1969. 3. Spencer E., "A Method of Analysis of the Stability of Embankments Assuming Parallel Inter-Slice Forces," GEOTECHNIQUE, 17, pp11-26, 1967. 4. Terzaghi K. and R. B. Peck, Soil Mechanics in Engineering Practice, John Wiley & sons Inc., 1967. 5. Whitman R. V. and W. A. Bailey, "Use of Computers for Slope Stability Analysis," Conference on Stability and Performance of Slopes and Embankments, ASCE, pp519-542, l966.