Practical Probabilistic Slope Stability Analysis B. K. Low Associate Professor, School of Civil & Environmental Engineering, Nanyang Technological University, Singapore 639798

Abstract

A practical procedure is described for implementing Spencer’s method of slices with varying side force inclination. The search for the noncircular critical slip surface is accomplished using spreadsheet-automated constrained optimization. The deterministic formulation is then extended probabilistically to compute the Hasofer-Lind reliability index via constrained optimization of the dispersion hyperellipsoid in the original space of the random variables. Reliability analyses involving spatially correlated normal and lognormal random variables are illustrated for an embankment on soft ground, with search for the reliability-based noncircular critical slip surface. The probabilities of failure and the probability density functions inferred from reliability indices are compared with Monte Carlo simulations. It is demonstrated that the hitherto complicated problems of locating noncircular critical surface and reliability analysis involving implicit functions and correlated nonnormals can be solved with relative ease and transparency using cell-object-oriented constrained optimization in the ubiquitous spreadsheet platform.

Proceedings, Soil and Rock America 2003, 12th Panamerican Conference on Soil Mechanics and Geotechnical Engineering and 39th U.S. Rock Mechanics Symposium, M.I.T., Cambridge, Massachusetts, June 22-26, 2003, Verlag Glückauf GmbH Essen, Vol. 2, pp.2777-2784.

The next page shows Figure 1 of the paper, followed by two “guidance” sheets for creating the template of Figure 1, and Section 2 of the paper. (The guidance sheets are not part of the paper. They are provided here

by the author in response to readers’ enquiries.)

Going through the procedures in the “guidance” sheets and reading the paper’s section 2 will give a better

appreciation of the constrained optimization approach for Spencer’s method (with automatic search for critical circular and critical noncircular slip surfaces, for both deterministic and probabilistic approaches.)

To benefit from the experience, users should build up Figure 1 from scratch, and understand the physical meanings of the formulas in the first “guidance” sheet, by relating the formulas to the seven equations in

Section 2 of the paper. Emphasis is on understanding the mechanics and process, and acquiring versatile advanced spreadsheet techniques (including simple VBA skill, Solver optimization, array formulas ...) useful

beyond the slope stability context of the paper. The pedagogic objective is achieved when the worksheet is set-up with understanding. It also affords adaptability and refinement. Interfaces and further automation can

also be done, but with some loss of transparency.

Note: When analyzing another case with different soil profile or water regime, the VBA functions Slice_c(… )

and AveGamma(…) in the paper’s Section 2 and equations in columns φ and u can be modified accordingly.

This 5-page PDF is downloadable from http://www.ntu.edu.sg/home/cbklow, together with other information.

The full 8-page paper (PDF file) is also available at the same URL.

where v is a vector representing the set of random

Figure 1. Deterministic analysis of a 5 m high embankment on soft ground with depth-dependent undrained shear strength. The limit equilibrium method of slices is based on Spencer (1973), with half-sine variation of side force inclination.

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45464748

4950

A B C D E F G H I J K L M N O P Q Rλ′ F DumEq

0.105 1.287 1.287

ΣM ΣForces Varying λ0.00 0.00 TRUE

Ω H hc γw Pw Mw ru xc yc R xmin xmax27 5 1.73 10 15 61.518 0.2 4.91 7.95 13.40 -5.89 17.467

# x ybot ytop γave c φ W αrad u l P T E λ σ ' Lx Ly

0 17.47 3.27 5.00 15 0.0001 16.58 1.37 5.00 20.00 10.00 30 47.4 1.136 10.73 2.10 49.72 28.50 48.1 0.012 12.98 12.12 5.632 15.70 0.00 5.00 20.00 10.00 30 76.3 0.997 17.27 1.63 80.96 36.36 96.3 0.025 32.49 11.24 7.263 14.72 -1.18 5.00 19.58 35.26 0 107.4 0.879 21.89 1.54 112.01 42.14 155.7 0.038 50.94 10.30 8.544 13.74 -2.14 5.00 19.00 27.15 0 124.2 0.770 25.31 1.37 137.12 28.84 230.5 0.050 74.97 9.32 9.615 12.76 -2.92 5.00 18.66 22.52 0 137.8 0.673 28.09 1.25 149.26 21.95 306.3 0.062 90.90 8.34 10.476 11.78 -3.56 5.00 18.43 20.00 0 149.0 0.582 30.37 1.17 156.36 18.25 377.0 0.072 102.75 7.36 11.197 10.79 -4.10 5.00 18.27 20.00 0 158.3 0.496 32.25 1.12 160.90 17.34 438.4 0.082 111.92 6.38 11.788 9.81 -4.53 5.00 18.15 20.00 0 165.8 0.415 33.80 1.07 165.06 16.66 489.6 0.090 120.14 5.40 12.269 8.83 -4.87 4.50 18.01 20.00 0 167.0 0.336 34.04 1.04 164.03 16.15 528.5 0.096 123.77 4.42 12.6410 7.85 -5.13 4.00 17.84 20.00 0 161.9 0.259 33.00 1.02 158.16 15.78 553.8 0.101 122.79 3.43 12.9511 6.87 -5.31 3.50 17.67 20.66 0 155.6 0.184 31.71 1.00 152.37 16.03 565.9 0.103 120.93 2.45 13.1712 5.89 -5.42 3.00 17.51 21.10 0 148.1 0.110 30.18 0.99 146.54 16.19 565.9 0.105 118.26 1.47 13.3113 4.91 -5.46 2.50 17.34 21.32 0 139.4 0.037 28.41 0.98 140.44 16.27 554.8 0.104 114.61 0.49 13.3914 3.93 -5.42 2.00 17.17 21.32 0 129.6 -0.037 26.41 0.98 133.80 16.27 533.7 0.101 109.85 -0.49 13.3915 2.94 -5.31 1.50 16.98 21.10 0 118.6 -0.110 24.18 0.99 126.36 16.19 503.7 0.097 103.81 -1.47 13.3116 1.96 -5.13 1.00 16.77 20.66 0 106.5 -0.184 21.71 1.00 117.86 16.03 466.3 0.091 96.36 -2.45 13.1717 0.98 -4.87 0.50 16.52 20.00 0 93.2 -0.259 19.00 1.02 107.99 15.78 423.4 0.083 87.37 -3.43 12.9518 0.00 -4.53 0.00 16.20 20.00 0 78.7 -0.336 16.04 1.04 96.74 16.15 376.3 0.074 77.03 -4.42 12.6419 -0.98 -4.10 0.00 16.00 20.00 0 67.7 -0.415 13.80 1.07 89.12 16.66 325.1 0.064 69.32 -5.40 12.2620 -1.96 -3.56 0.00 16.00 20.00 0 60.1 -0.496 12.25 1.12 85.34 17.34 269.2 0.053 64.22 -6.38 11.7821 -2.94 -2.92 0.00 16.00 20.00 0 50.9 -0.582 10.37 1.17 79.74 18.25 210.1 0.040 57.52 -7.36 11.1922 -3.93 -2.14 0.00 16.00 22.52 0 39.7 -0.673 8.09 1.25 73.92 21.95 146.9 0.027 50.84 -8.34 10.4723 -4.91 -1.18 0.00 16.00 27.15 0 26.1 -0.770 5.31 1.37 68.39 28.84 78.6 0.014 44.70 -9.32 9.6124 -5.89 0.00 0.00 16.00 35.26 0 9.3 -0.879 1.89 1.54 67.12 42.14 0.0 0.000 41.75 -10.30 8.54

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

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

(kPa) (o) (kN/m3)

γclay

16 (kN/m3)

framed cells contain equations

-10

-5

0

5

10-10 -5 0 5 10 15 20

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

Array formulas

slope angle

Soft clay

Embankment

( )( )

−−′= πλλ

0

0sinxxxx

n

i

λ

x0xn

Center of Rotation

xn

x0

0

2

4

6

8

10

0 25 50

cu (kPa)

Dep

th

SPENCER method

Varying side-force angle λ

Ei

λiEi

λi-1Ei-1

Ei-1

Ti

Pi

Wi

αi

lybot

ytopΩ

Guidance for creating the paper's Figure 1 Excel template. cbklow@ntu.edu.sg

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495051

A B C D E F G H I J K L M N O P Q Rλ′ F DumEq

0.105 1.287 1.287

ΣM ΣForces Varying λ0.00 0.00 TRUE

Ω H hc γw Pw Mw ru xc yc R xmin xmax27 5 1.73 10 15 61.518 0.2 4.91 7.95 13.40 -5.89 17.467

# x ybot ytop γave c φ W αrad u l P T E λ σ ' Lx Ly

0 17.47 3.27 5.00 15 0.0001 16.58 1.37 5.00 20.00 10.00 30 47.4 1.136 10.73 2.10 49.72 28.50 48.1 0.012 12.98 12.12 5.632 15.70 0.00 5.00 20.00 10.00 30 76.3 0.997 17.27 1.63 80.96 36.36 96.3 0.025 32.49 11.24 7.263 14.72 -1.18 5.00 19.58 35.26 0 107.4 0.879 21.89 1.54 112.01 42.14 155.7 0.038 50.94 10.30 8.544 13.74 -2.14 5.00 19.00 27.15 0 124.2 0.770 25.31 1.37 137.12 28.84 230.5 0.050 74.97 9.32 9.615 12.76 -2.92 5.00 18.66 22.52 0 137.8 0.673 28.09 1.25 149.26 21.95 306.3 0.062 90.90 8.34 10.476 11.78 -3.56 5.00 18.43 20.00 0 149.0 0.582 30.37 1.17 156.36 18.25 377.0 0.072 102.75 7.36 11.197 10.79 -4.10 5.00 18.27 20.00 0 158.3 0.496 32.25 1.12 160.90 17.34 438.4 0.082 111.92 6.38 11.788 9.81 -4.53 5.00 18.15 20.00 0 165.8 0.415 33.80 1.07 165.06 16.66 489.6 0.090 120.14 5.40 12.269 8.83 -4.87 4.50 18.01 20.00 0 167.0 0.336 34.04 1.04 164.03 16.15 528.5 0.096 123.77 4.42 12.64

10 7.85 -5.13 4.00 17.84 20.00 0 161.9 0.259 33.00 1.02 158.16 15.78 553.8 0.101 122.79 3.43 12.9511 6.87 -5.31 3.50 17.67 20.66 0 155.6 0.184 31.71 1.00 152.37 16.03 565.9 0.103 120.93 2.45 13.1712 5.89 -5.42 3.00 17.51 21.10 0 148.1 0.110 30.18 0.99 146.55 16.19 565.9 0.105 118.26 1.47 13.3113 4.91 -5.46 2.50 17.34 21.32 0 139.4 0.037 28.41 0.98 140.44 16.27 554.8 0.104 114.61 0.49 13.3914 3.93 -5.42 2.00 17.17 21.32 0 129.6 -0.037 26.41 0.98 133.80 16.27 533.7 0.101 109.85 -0.49 13.3915 2.94 -5.31 1.50 16.98 21.10 0 118.6 -0.110 24.18 0.99 126.36 16.19 503.7 0.097 103.81 -1.47 13.3116 1.96 -5.13 1.00 16.77 20.66 0 106.5 -0.184 21.71 1.00 117.86 16.03 466.3 0.091 96.36 -2.45 13.1717 0.98 -4.87 0.50 16.52 20.00 0 93.2 -0.259 19.00 1.02 107.99 15.78 423.4 0.083 87.37 -3.43 12.9518 0.00 -4.53 0.00 16.20 20.00 0 78.7 -0.336 16.04 1.04 96.74 16.15 376.3 0.074 77.03 -4.42 12.6419 -0.98 -4.10 0.00 16.00 20.00 0 67.7 -0.415 13.80 1.07 89.12 16.66 325.1 0.064 69.32 -5.40 12.2620 -1.96 -3.56 0.00 16.00 20.00 0 60.1 -0.496 12.25 1.12 85.34 17.34 269.2 0.053 64.22 -6.38 11.7821 -2.94 -2.92 0.00 16.00 20.00 0 50.9 -0.582 10.37 1.17 79.74 18.25 210.1 0.040 57.52 -7.36 11.1922 -3.93 -2.14 0.00 16.00 22.52 0 39.7 -0.673 8.09 1.25 73.92 21.95 146.9 0.027 50.84 -8.34 10.4723 -4.91 -1.18 0.00 16.00 27.15 0 26.1 -0.770 5.31 1.37 68.39 28.84 78.6 0.014 44.70 -9.32 9.6124 -5.89 0.00 0.00 16.00 35.26 0 9.3 -0.879 1.89 1.54 67.12 42.14 0.0 0.000 41.75 -10.30 8.54

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

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

(kPa) (o) (kN/m3)

γclay

16 (kN/m3)

framed cells contain equations

-10

-5

0

5

10-10 -5 0 5 10 15 20

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

Array formulas

slope angle

Soft clay

Embankment

( )( ) ⎥

⎦

⎤⎢⎣

⎡−−

′= πλλ0

0sinxxxx

n

i

λ

x0xn

Center of Rotation

xn

x0

0

2

4

6

8

10

0 25 50cu (kPa)

Dep

th

SPENCER method

Varying side-force angle λ

Formulas: hc = 2*cm/(γm√Ka), Pw = 0.5*γw*hc^2, Mw = Pw*(yc−H+0.6667*hc)(Note: Ka =(1-sin(radians(phim)))/(1+sin(radians(phim))), in separate cell)xmin = −√(abs(R^2−yc^2))+xc, xmax = √(abs(R^2−(yc−H+hc)^2))+xc

x0 = xmax, x2 = xc+√(abs(R^2-yc^2)), x1 = (B16+B18)/2 x3 = B18-($B$18-xmin)/22, drag downwards to autofill. ybot0= −√(abs(R^2-(B16−xc)^2))+yc, autofill. (Columns ytop, γave and c invoke the user-defined functions shown in Fig. 2 of paper.)γave1 = AveGamma(0.5*(D16+D17),0.5*(C16+C17),γm,γclay)c1 = Slice_c(0.5*(C16+C17),max(dv),dv,cuv,cm), or Insert/Function/Category/UserDefined.

φ1 = IF(0.5*(C16+C17)>0, $Q$45,0), autofill (also for the formulas below .)W1 = (B16-B17)*0.5*((D16-C16)+(D17-C17))*E17, αrad1 =ATAN((C16-C17)/(B16-B17))u1 =ru*E17*0.5*(D16-C16+D17-C17), l 1 =(B16-B17)/COS(I17)Enter formulas for P and T as shown in paper. (Equation of P is long, may use VBA coding)E0 = Pw, E1 = N16+L17*SIN(I17)-M17*COS(I17)λ0=0, λ1=IF($R$5,$N$2*SIN(($B$16-B17)/($B$16-$B$40)*PI()),$N$2)σ′ = (P/l − u), Lx1 = 0.5*(B16+B17)−xc, Ly1 =yc−0.5*(C16+C17), DumEq = F*1

ΣM =SUM((M17:M40*SIN(I17:I40)+L17:L40*COS(I17:I40)−H17:H40)*Q17:Q40+ (M17:M40*COS(I17:I40)−L17:L40*SIN(I17:I40))*R17:R40)−F11, (Ctrl & Shift , then Enter ) ΣForces =SUM(M17:M40*COS(I17:I40)−L17:L40*SIN(I17:I40))−E11, (Ctrl & Shift , then Enter )Initially λ′ = 0, F = 1, then invoke Excel's built-in Solver, see next page.

Ei

λiEi

λi-1Ei-1

Ei-1

Ti

Pi

Wi

αi

lybot

ytopΩ

phim

dv cuv

Solver ParametersSolver Parameters

λ′ F0.105 1.287

xc yc R4.91 7.95 13.40

Run Excel’s built-in Solver, to automatically obtain the minimum F and critical slip circle shown in paper’s Fig. 1, within seconds:

Critical noncircular slip surface can also be searched, as explained in the paper, to obtain a minimum factor of safety (F) of 1.253.

The extension from deterministic factor-of-safety approach to the more logical probabilistic approach is (computationally) relatively straightforward and intuitive, as shown in the paper.

Guidance (continued)

2 DETERMINISTIC SPENCER’S METHOD The sketch in Figure 1 shows the forces acting on a slice that forms part of the potential sliding soil mass. Adopting the same assumptions as Spencer (1973), one can derive the following from Mohr-Coulomb criterion and equilibrium considerations:

( ) FluPlcT iiiiiii ⎥⎦⎤

⎢⎣⎡ ′−+′= φtan (1)

iiiiii TPEE αα cossin1 −+= − (2)

( )

( )

( )⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−′+

+

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−⎟⎠⎞⎜

⎝⎛ ′−′−

−−

=

−−

iiii

iii

iiiiiiii

iiii

i

F

lulcF

EW

P

αλαφ

ααλ

αλαφ

λλ

cossintan1cossin

cossintan111

(3)

[ ] 0sincos =−−∑ wiiii PPT αα (4)

( )( ) 0

sincoscossin

=−⎥⎦

⎤⎢⎣

⎡∗−+

∗−+∑ w

yiiiii

xiiiiii MLPT

LWPTαα

αα (5)

( ) ciixi xxxL −+= −15.0 (6)

( )15.0 −+−= iicyi yyyL (7)

where Pw is the water thrust in a water-filled tension crack, and Mw the overturning moment due to Pw. Equations 4 and 5 specify overall force and moment equilibrium. Equations 6 and 7, required for noncircular slip surface, give the lever arms with respect to an arbitrary center.

Figure 1 shows the spreadsheet set-up for stability analysis of a 5 m high embankment on soft ground. The undrained shear strength profile of the soft ground is defined in rows 44 and 45. Formulas need be entered only in the first or second cell (row 16 or 17) of each column, followed by autofilling down to row 40. The columns labelled ytop, γave and c invoke the functions shown in Fig. 2, created via Tools/ Macro/VisualBasicEditor/Insert/Module on the Excel worksheet menu. The dummy equation in cell P2 is equal to F*1. This cell, unlike cell O2, can be minimized because it contains a formula.

Initially xc = 6, yc = 8, R = 12 in cells I11:K11, and λ′ = 0, F = 1 in cells N2:O2. Microsoft Excel’s built-in Solver was then invoked to set target and constraints as shown in Fig. 3. The Solver option “Use Automatic Scaling” was also activated. The critical slip circle and factor of safety F = 1.287 shown in Fig. 1 were

obtained automatically within seconds by Solver via cell-object-oriented constrained optimization.

Noncircular critical slip surface can also be searched using Solver as in Fig. 3, except that “By Changing Cells” are N2:O2, B16, B18, B40, C17, and C19:C39, and with the following additional cell constraints: B16 ≥ B11/tan(radians(A11)), B16 ≥ B18, B40 ≤ 0, C19:C39 ≤ D19:D39, O2 ≥ 0.1, and P17:P40 ≥ 0.

Figure 4 tests the robustness of the search for noncircular critical surface. Starting from four arbitrary initial circles, the final noncircular critical surfaces (solid curves, each with 25 degrees of freedom) are close enough to each other, though not identical. Perhaps more pertinent, their factors of safety vary narrowly within 1.253 to 1.257. This compares with the minimum factor of safety 1.287 of the critical circular surface of Fig. 1.

Function Slice_c(ybmid, dmax, dv, cuv, cm)'comment: dv = depth vector, ‘cuv = cu vector, Fig. 1.If ybmid > 0 ThenSlice_c = cmExit FunctionEnd Ifybmid = Abs(ybmid)If ybmid > dmax Then 'undefined domain,Slice_c = 300 'hence assume hard stratum.Exit FunctionEnd IfFor j = 2 To dv.Count 'array size=dv.CountIf dv(j) >= ybmid Theninterp = (ybmid - dv(j - 1)) / (dv(j) - dv(j - 1))Slice_c = cuv(j - 1) + (cuv(j) - cuv(j - 1)) * interpExit ForEnd IfNext jEnd Function

Function ytop(x, omega, H)grad = Tan(omega * 3.14159 / 180)If x < 0 Then ytop = 0If x >= 0 And x < H / grad Then ytop = x * gradIf x >= H / grad Then ytop = HEnd Function

Function AveGamma(ytmid, ybmid, gm, gclay)If ybmid < 0 Then

Sum = (ytmid * gm + Abs(ybmid) * gclay)AveGamma = Sum / (ytmid - ybmid)Else: AveGamma = gm

End IfEnd Function

Figure 2 User-defined VBA functions, called by columns ytop, γave, and c of Fig. 1.

from Low (2003, 2001)