slope spencer formulated

Spencer_forSARA2003

l¢ F DumEq0.105 1.287 1.287

### ###

W H R27 5 1.73 10 15 61.518 0.2 4.91 7.95 13.40 -5.89 17.467

# x c f W u l P T E l0 17.47 3.27 ### 15 0.000

1 16.58 1.37 ### ### ### 30 ### 1.136 ### 2.10 ### ### ### 0.012 ### 12.12

2 15.70 0.00 ### ### ### 30 ### 0.997 ### 1.63 ### ### ### 0.025 ### 11.24

3 14.72 -1.18 ### ### ### 0 ### 0.879 ### 1.54 ### ### ### 0.038 ### 10.30

4 13.74 -2.14 ### ### ### 0 ### 0.770 ### 1.37 ### ### ### 0.050 ### 9.32

5 12.76 -2.92 ### ### ### 0 ### 0.673 ### 1.25 ### ### ### 0.062 ### 8.34

6 11.78 -3.56 ### ### ### 0 ### 0.582 ### 1.17 ### ### ### 0.072 ### 7.36

7 10.79 -4.10 ### ### ### 0 ### 0.496 ### 1.12 ### ### ### 0.082 ### 6.38

8 9.81 -4.53 ### ### ### 0 ### 0.415 ### 1.07 ### ### ### 0.090 ### 5.40

9 8.83 -4.87 ### ### ### 0 ### 0.336 ### 1.04 ### ### ### 0.096 ### 4.42

10 7.85 -5.13 ### ### ### 0 ### 0.259 ### 1.02 ### ### ### 0.101 ### 3.43

11 6.87 -5.31 ### ### ### 0 ### 0.184 ### 1.00 ### ### ### 0.103 ### 2.45

12 5.89 -5.42 ### ### ### 0 ### 0.110 ### 0.99 ### ### ### 0.105 ### 1.47

13 4.91 -5.46 ### ### ### 0 ### 0.037 ### 0.98 ### ### ### 0.104 ### 0.49

14 3.93 -5.42 ### ### ### 0 ### -0.037 ### 0.98 ### ### ### 0.101 ### -0.49

15 2.94 -5.31 ### ### ### 0 ### -0.110 ### 0.99 ### ### ### 0.097 ### -1.47

16 1.96 -5.13 ### ### ### 0 ### -0.184 ### 1.00 ### ### ### 0.091 ### -2.45

17 0.98 -4.87 ### ### ### 0 ### -0.259 ### 1.02 ### ### ### 0.083 ### -3.43

18 0.00 -4.53 ### ### ### 0 ### -0.336 ### 1.04 ### ### ### 0.074 ### -4.42

19 -0.98 -4.10 ### ### ### 0 ### -0.415 ### 1.07 ### ### ### 0.064 ### -5.40

20 -1.96 -3.56 ### ### ### 0 ### -0.496 ### 1.12 ### ### ### 0.053 ### -6.38

21 -2.94 -2.92 ### ### ### 0 ### -0.582 ### 1.17 ### ### ### 0.040 ### -7.36

22 -3.93 -2.14 ### ### ### 0 ### -0.673 ### 1.25 ### ### ### 0.027 ### -8.34

23 -4.91 -1.18 ### ### ### 0 ### -0.770 ### 1.37 ### ### ### 0.014 ### -9.32

24 -5.89 0.00 ### ### ### 0 ### -0.879 ### 1.54 ### ### ### 0.000 ### -10.30

Undrained shear strength profile of soft clay Embankmentdepth 0 1.5 3 5 7 10 (m)

cu 40 28 20 20 26 37 (kPa) 10 30

(kPa)

16

SM SForces Varying l

hc gw Pw Mw ru xc yc xmin xmax

ybot ytop gave arad s ' Lx

cm fm

(o)

gclay

(kN/m3)

framed cells contain equationsframed cells contain equations

-10 -5 0 5 10 15 20

-10

-5

0

5

10

Units: m, kN/m3, kN/m2, kN, or other consistent set of units.

Array formulas

slope angle

Soft clay

Embankment

λ=λ ' sin[ (x i−x0 )( xn− x0 )

π ]

l

x0xn

Center of Rotation

xn

x0

0 25 500

2

4

6

8

10

cu (kPa)

De

pth

SPENCER method

Varying side-force angle l

Ei

liEi

li-1Ei-1

Ei-1

Ti

Pi

Wi

ai

lybot

ytop

W

A B C D E F G H I J K L M N O P Q1

23

4

567

8

9

10

11

1213

14

15

16

1718192021222324252627282930313233343536373839404142

43

44

45

464748

49

50

Spencer_forSARA2003

0 25 500

2

4

6

8

10

cu (kPa)

De

pth

A B C D E F G H I J K L M N O P Q51

Spencer_forSARA2003

1

5.63

7.26

8.54

9.61

10.47

11.19

11.78

12.26

12.64

12.95

13.17

13.31

13.39

13.39

13.31

13.17

12.95

12.64

12.26

11.78

11.19

10.47

9.61

8.54

Embankment

20

Varying l

Ly

gm

(kN/m3)

λ=λ ' sin[ (x i−x0 )( xn− x0 )

π ]x0


R1

23

4

567

8

9

10

11

1213

14

15

16

1718192021222324252627282930313233343536373839404142

43

44

45

464748

49

50

from Spencer_forSARA2003

l¢ F DumEq

0.103 1.253 1.253

### ### 1

W H27 5 1.73 10 15 62.321 0.2 8.00 8.00

Lever arms

# x c f W u l P T E l0 18.07 3.27 ### 15.00 0.000

1 16.88 1.68 ### ### ### 30 ### 0.931 ### 1.98 ### ### ### 0.016 ### 9.47 5.53

2 15.70 0.00 ### ### ### 30 ### 0.958 ### 2.06 ### ### ### 0.031 ### 8.29 7.16

3 14.72 -0.95 ### ### ### 0 ### 0.767 ### 1.36 ### ### ### 0.044 ### 7.21 8.47

4 13.74 -1.84 ### ### ### 0 ### 0.738 ### 1.33 ### ### ### 0.055 ### 6.23 9.39

5 12.76 -2.63 ### ### ### 0 ### 0.682 ### 1.26 ### ### ### 0.066 ### 5.25 10.24

6 11.78 -3.37 ### ### ### 0 ### 0.641 ### 1.22 ### ### ### 0.076 ### 4.27 11.00

7 10.79 -4.01 ### ### ### 0 ### 0.583 ### 1.18 ### ### ### 0.084 ### 3.29 11.69

8 9.81 -4.63 ### ### ### 0 ### 0.562 ### 1.16 ### ### ### 0.091 ### 2.30 12.32

9 8.83 -5.15 ### ### ### 0 ### 0.484 ### 1.11 ### ### ### 0.097 ### 1.32 12.89

10 7.85 -5.52 ### ### ### 0 ### 0.360 ### 1.05 ### ### ### 0.100 ### 0.34 13.33

11 6.87 -5.78 ### ### ### 0 ### 0.261 ### 1.02 ### ### ### 0.103 ### -0.64 13.65

12 5.89 -5.94 ### ### ### 0 ### 0.168 ### 1.00 ### ### ### 0.103 ### -1.62 13.86

13 4.91 -6.02 ### ### ### 0 ### 0.076 ### 0.98 ### ### ### 0.102 ### -2.60 13.98

14 3.93 -6.00 ### ### ### 0 ### -0.019 ### 0.98 ### ### ### 0.099 ### -3.58 14.01

15 2.94 -5.88 ### ### ### 0 ### -0.121 ### 0.99 ### ### ### 0.094 ### -4.57 13.94

16 1.96 -5.65 ### ### ### 0 ### -0.233 ### 1.01 ### ### ### 0.088 ### -5.55 13.76

17 0.98 -5.28 ### ### ### 0 ### -0.360 ### 1.05 ### ### ### 0.081 ### -6.53 13.46

18 0.00 -4.73 ### ### ### 0 ### -0.507 ### 1.12 ### ### ### 0.072 ### -7.51 13.00

19 -0.98 -4.08 ### ### ### 0 ### -0.587 ### 1.18 ### ### ### 0.062 ### -8.49 12.41

20 -1.96 -3.39 ### ### ### 0 ### -0.615 ### 1.20 ### ### ### 0.051 ### -9.47 11.73

21 -2.94 -2.61 ### ### ### 0 ### -0.668 ### 1.25 ### ### ### 0.039 ### -10.45 11.00

22 -3.93 -1.81 ### ### ### 0 ### -0.684 ### 1.27 ### ### ### 0.026 ### -11.43 10.21

23 -4.91 -0.93 ### ### ### 0 ### -0.734 ### 1.32 ### ### ### 0.013 ### -12.42 9.37

24 -5.89 0.00 ### ### ### 0 ### -0.757 ### 1.35 ### ### ### 0.000 ### -13.40 8.46


cu 40 28 20 20 26 37 (kPa) 10 30 20

(kPa)

16

SForces SM Varying l

hc gw Pw Mw ru xc yc

yb yt gave arad s ' Lx Ly

cm fm gm

(o) (kN/m3)gclay

(kN/m3)



Array formulas

slope angle

Center of Rotation

Ei

liEi

li-1Ei-1Ei-1

Ti

Pi

Wi

ai

l

-20 -15 -10 -5 0 5 10 15 20

-10

-5

0

5

10

Soft clay

Embankmentxn

x0

yb

ytyt

initial

critical

0 25 50

0

2

4

6

8

10

cu (kPa)

de

pth

SPENCER: Search for noncircular slip surface

l

x0xn

λ=λ ' sin[ (xi−x0 )( xn− x0 )

π ]



0 25 50

0

2

4

6

8

10

cu (kPa)

de

pth


Ka0.33

hc1.73205

xmin xmax-18.12 15.37 -2.944 17.028

xc yc R -15.180 -1.661

-2 8 18.00

xi ybi ytiEquilCheck 15.37 3.27 #VALUE!

### #VALUE! 14.75 1.40 #VALUE!

### #VALUE! 14.12 0.00 #VALUE!

### #VALUE! 12.66 -2.45 #VALUE!

### #VALUE! 11.19 -4.25 #VALUE!

### #VALUE! 9.73 -5.66 #VALUE!

### #VALUE! 8.26 -6.79 #VALUE!

### #VALUE! 6.80 -7.70 #VALUE!

### #VALUE! 5.33 -8.44 #VALUE!

### #VALUE! 3.86 -9.02 #VALUE!

### #VALUE! 2.40 -9.45 #VALUE!

### #VALUE! 0.93 -9.76 #VALUE!

### #VALUE! -0.53 -9.94 #VALUE!

### #VALUE! -2.00 -10.00 #VALUE!

### #VALUE! -3.47 -9.94 #VALUE!

### #VALUE! -4.93 -9.76 #VALUE!

### #VALUE! -6.40 -9.45 #VALUE!

### #VALUE! -7.86 -9.02 #VALUE!

### #VALUE! -9.33 -8.44 #VALUE!

### #VALUE! -10.80 -7.70 #VALUE!

### #VALUE! -12.26 -6.79 #VALUE!

### #VALUE! -13.73 -5.66 #VALUE!

### #VALUE! -15.19 -4.25 #VALUE!

### #VALUE! -16.66 -2.45 #VALUE!

### #VALUE! -18.12 0.00 #VALUE!

Read the Low (2003) paper "Practical Probabilistic Slope Stability Analysis", downloadable from http://alum.mit.edu/www/bklow.For hands-on:(1) Click the button "InitialCircle"(2) Invoke Solver (Menu bar: Tools/Solver); click Solve. Then click OK to accept the solution found. May run Solver one more time.


l¢ F DumEq

0.098 1.000 1.000

### ### 1

W H27 5 1.78 10 15.869 66.455 0.2 8.00 8.00

Lever arms

# x c f W u l P T E l0 17.22 3.22 ### 15.87 0.000

1 16.19 1.66 ### ### ### 30.949 ### 0.990 ### 1.87 ### ### ### 0.015 ### 8.71 5.56

2 15.17 0.00 ### ### ### 30.949 ### 1.016 ### 1.95 ### ### ### 0.029 ### 7.68 7.17

3 14.28 -0.86 ### ### ### 0 ### 0.769 ### 1.24 ### ### ### 0.041 ### 6.72 8.43

4 13.38 -1.69 ### ### ### 0 ### 0.744 ### 1.21 ### ### ### 0.052 ### 5.83 9.27

5 12.49 -2.43 ### ### ### 0 ### 0.697 ### 1.16 ### ### ### 0.062 ### 4.94 10.06

6 11.60 -3.11 ### ### ### 0 ### 0.648 ### 1.12 ### ### ### 0.071 ### 4.04 10.77

7 10.71 -3.72 ### ### ### 0 ### 0.600 ### 1.08 ### ### ### 0.079 ### 3.15 11.41

8 9.81 -4.30 ### ### ### 0 ### 0.578 ### 1.07 ### ### ### 0.086 ### 2.26 12.01

9 8.92 -4.79 ### ### ### 0 ### 0.505 ### 1.02 ### ### ### 0.091 ### 1.37 12.55

10 8.03 -5.14 ### ### ### 0 ### 0.370 ### 0.96 ### ### ### 0.095 ### 0.47 12.97

11 7.14 -5.37 ### ### ### 0 ### 0.256 ### 0.92 ### ### ### 0.097 ### -0.42 13.26

12 6.24 -5.53 ### ### ### 0 ### 0.174 ### 0.91 ### ### ### 0.098 ### -1.31 13.45

13 5.35 -5.62 ### ### ### 0 ### 0.097 ### 0.90 ### ### ### 0.097 ### -2.20 13.57

14 4.46 -5.63 ### ### ### 0 ### 0.016 ### 0.89 ### ### ### 0.094 ### -3.10 13.63

15 3.57 -5.57 ### ### ### 0 ### -0.069 ### 0.89 ### ### ### 0.090 ### -3.99 13.60

16 2.67 -5.43 ### ### ### 0 ### -0.162 ### 0.90 ### ### ### 0.084 ### -4.88 13.50

17 1.78 -5.18 ### ### ### 0 ### -0.268 ### 0.93 ### ### ### 0.077 ### -5.77 13.30

18 0.89 -4.78 ### ### ### 0 ### -0.424 ### 0.98 ### ### ### 0.069 ### -6.66 12.98

19 0.00 -4.17 ### ### ### 0 ### -0.600 ### 1.08 ### ### ### 0.059 ### -7.56 12.47

20 -0.90 -3.41 ### ### ### 0 ### -0.703 ### 1.17 ### ### ### 0.048 ### -8.45 11.79

21 -1.79 -2.59 ### ### ### 0 ### -0.744 ### 1.21 ### ### ### 0.037 ### -9.34 11.00

22 -2.68 -1.79 ### ### ### 0 ### -0.732 ### 1.20 ### ### ### 0.025 ### -10.23 10.19

23 -3.57 -0.91 ### ### ### 0 ### -0.777 ### 1.25 ### ### ### 0.013 ### -11.13 9.35

24 -4.47 0.00 ### ### ### 0 ### -0.796 ### 1.28 ### ### ### 0.000 ### -12.02 8.46


cu 34.07 22.74 15.91 15.85 22.46 34.82 (kPa) 10.47 30.95 20.76

(kPa)

16

SForces SM Varying l

hc gw Pw Mw ru xc yc

yb yt gave arad s ' Lx Ly

cm fm gm

(o) (kN/m3)gclay

(kN/m3)



Array formulas

slope angle

Center of Rotation

Ei

liEi

li-1Ei-1Ei-1

Ti

Pi

Wi

ai

l

-10 -5 0 5 10 15 20 25

-10

-5

0

5

10

Soft clay

Embankmentxn

x0

yb

yt

yt

initial

critical

0 25 50

0

2

4

6

8

10

cu (kPa)

de

pth

SPENCER: Search for noncircular slip surface

l

x0xn

λ=λ ' sin[ (xi−x0 )( xn− x0 )

π ]



Ka hc

0.32 1.78151

mean StDev

DistName s

lognormal 10.472107 10 1.50 ### ###

lognormal 30.948505 30 3.00 ### ###

lognormal 20.757609 20 1.00 ### ###

lognormal 34.072981 40 6.00 ### ###

lognormal 22.735506 28 4.20 ### ###

lognormal 15.906807 20 3.00 ### ###

lognormal 15.852003 20 3.00 ### ###

lognormal 22.463107 26 3.90 ### ###

lognormal 34.81741 37 5.55 ### ###

EquilCheck Array formula: Ctrl+Shift, then Enter

### #VALUE!

### #VALUE!

### #VALUE!

### #VALUE!

### #VALUE! DeterministicCritical Normals, reliability-based critical### #VALUE! x yb yt x yb yt

### #VALUE! 18.07 3.27 5.00 17.2543 3.2144 5.0000

### #VALUE! 16.88 1.68 5.00 16.2395 1.6480 5.0000

### #VALUE! 15.70 0.00 5.00 15.2246 0.0000 5.0000

### #VALUE! 14.72 -0.95 5.00 14.3227 -0.8758 5.0000

### #VALUE! 13.74 -1.84 5.00 13.4208 -1.7092 5.0000

### #VALUE! 12.76 -2.63 5.00 12.5188 -2.4603 5.0000

### #VALUE! 11.78 -3.37 5.00 11.6169 -3.1408 5.0000

### #VALUE! 10.79 -4.01 5.00 10.7150 -3.7526 5.0000

### #VALUE! 9.81 -4.63 5.00 9.8131 -4.3359 5.0000

### #VALUE! 8.83 -5.15 4.50 8.9111 -4.8270 4.5404

### #VALUE! 7.85 -5.52 4.00 8.0092 -5.1699 4.0809

### #VALUE! 6.87 -5.78 3.50 7.1073 -5.3990 3.6213

### #VALUE! 5.89 -5.94 3.00 6.2054 -5.5524 3.1618

### #VALUE! 4.91 -6.02 2.50 5.3034 -5.6343 2.7022

### #VALUE! 3.93 -6.00 2.00 4.4015 -5.6452 2.2427

### #VALUE! 2.94 -5.88 1.50 3.4996 -5.5803 1.7831

### #VALUE! 1.96 -5.65 1.00 2.5976 -5.4312 1.3236

### #VALUE! 0.98 -5.28 0.50 1.6957 -5.1821 0.86400.00 -4.73 0.00 0.7938 -4.7721 0.4045-0.98 -4.08 0.00 -0.1081 -4.1579 0.0000

-1.96 -3.39 0.00 -1.0101 -3.4079 0.0000

-2.94 -2.61 0.00 -1.9120 -2.5921 0.0000

-3.93 -1.81 0.00 -2.8139 -1.7930 0.0000

-4.91 -0.93 0.00 -3.7158 -0.9136 0.0000

-5.89 0.00 0.00 -4.6178 0.0000 0.0000

vi mi mN sN

cm

fm

gm

cu1

cu2

cu3

cu4

cu5

cu6

=SQRT(MMULT(TRANSPOSE(vn),MMULT(MINVERSE(crmat),vn)))

EqvN(…, 1) EqvN(…, 2)

Read the Low (2003) paper "Practical Probabilistic Slope Stability Analysis", downloadable from http://alum.mit.edu/www/bklow.

For hands-on: Read the section in paper related to Figure 6.


vn

#VALUE! 1 -0.3 0.5 0 0 0 0 0 0

#VALUE! -0.3 1 0.5 0 0 0 0 0 0

#VALUE! 0.5 0.5 1 0 0 0 0 0 0

#VALUE! 0 0 0 1 0.6065 0.3679 0.1889 0.097 0.0357 0

#VALUE! 0 0 0 0.6065 1 0.6065 0.3114 0.1599 0.0588 1.5

#VALUE! 0 0 0 0.3679 0.6065 1 0.5134 0.2636 0.097 3

#VALUE! 0 0 0 0.1889 0.3114 0.5134 1 0.5134 0.1889 5

#VALUE! 0 0 0 0.097 0.1599 0.2636 0.5134 1 0.3679 7

#VALUE! 0 0 0 0.0357 0.0588 0.097 0.1889 0.3679 1 10

0 1.5 3 5 7 10 dv

b PrFail

Array formula: Ctrl+Shift, then Enter ### ###

LogNormals, reliability-based critical

17.22 3.218 5

16.19 1.655 5

15.17 0 5

14.28 -0.86 5

13.38 -1.69 5

12.49 -2.43 5

11.6 -3.11 5

10.71 -3.72 5

9.813 -4.3 5

8.921 -4.79 4.545

8.028 -5.14 4.091

7.136 -5.37 3.636

6.243 -5.53 3.181

5.351 -5.62 2.726

4.458 -5.63 2.272

3.566 -5.57 1.817

2.674 -5.43 1.3621.781 -5.18 0.908

0.889 -4.78 0.453

0 -4.17 0

-0.9 -3.41 0

-1.79 -2.59 0

-2.68 -1.79 0

-3.57 -0.91 0

-4.47 ### 0

Correlation matrix, "crmat"

(v-mN)/sN cm fm gm cu1 cu2 cu3 cu4 cu5 cu6

-10 -5 0 5 10 15 20

-8

-6

-4

-2

0

2

4

6

=SQRT(MMULT(TRANSPOSE(vn),MMULT(MINVERSE(crmat),vn)))

Equation 9 autofilled

Probability of failure

EqvN(…, 2)

deterministic critical

=normsdist(-b)

Case1

2 12

Case1

6 21 24

Case1

1.940582171

Case2

2 14

Case2

9 19 24

Case2

1.230582584

slope spencer formulated

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