low and duncan 2013 geo-congress

Testing bias and parametric uncertainty in analyses of a slope failure in San Francisco Bay mud

B. K. Low1, F. ASCE and J. Michael Duncan2, Hon. M. ASCE

1Assoc Prof, School of Civil and Environmental Engineering, Nanyang Technological University, Singapore 639798; [email protected]

2Distinguished Professor Emeritus, Department of Civil and Environmental Engineering, Virginia Tech., Blacksburg, VA 24061; [email protected]

ABSTRACT: An underwater slope in San Francisco Bay Mud that failed during excavation will be analyzed, first deterministically using data from field vane shear and laboratory triaxial tests, then probabilistically, accounting for parametric uncertainty and correlation of the undrained shear strength and soil unit weight. In the deterministic analysis, the factors of safety will be compared based on field vane shear test data, and triaxial tests on 35 mm test specimens and 70 mm Shelby tube specimens. The principal factors influencing these measures of undrained strength are shown to be sample disturbance and testing rate. Computed factors of safety are also affected by extrapolation of measured strengths to depths greater than the actual depths of sampling and vane shear testing. The deterministic analysis is extended into reliability analysis to account for parametric uncertainty and correlation among input parameters. The results from the first-order reliability method (FORM) and Monte Carlo simulation method are compared and discussed. The effects of different input probability distributions on the probability of slope failure are investigated. INTRODUCTION The failure of a slope excavated underwater in San Francisco Bay has been described in Duncan and Buchignani (1973), Duncan (2000, 2001), and Duncan and Wright (2005). The slope was part of a temporary excavation and was designed with an unusually low factor of safety to minimize construction costs. During construction a portion of the excavated slope failed. A drawing of the slope cross section is shown in Fig. 1. The undrained shear strength data (from in situ and lab tests) for the upper 21 m of the 30.5 m deep excavation are presented in Fig. 2. Duncan and Buchignani calculated a short-term factor of safety equal to 1.17 based on field vane and laboratory triaxial tests. In this paper the slope failure is revisited, and analyzed using a reformulated Spencers procedure of slices in an adaptable spreadsheet platform, both deterministically using average values of input parameters, and probabilistically accounting for the uncertainties in the values of input parameters. The reformulated Spencer procedure is presented next, as it will be used in both the deterministic and probabilistic analyses.

937Geo-Congress 2013 ASCE 2013

FIG. 1. Underwater excavated slope in San Francisco Bay mud described in Duncan and Wright (2005)

FIG. 2. Undrained shear strength data from in situ vane shear and UU triaxial tests

30.5 m

6.1 m Debris dike

San Francisco Bay mud

Firm soil

12

0

12

24

36

Dep

th

m

0

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25

30

0 10 20 30 40 50

Dep

th b

elo

w

0.0

-m

su from in situ vane shear tests, kPa

0

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25

30

0 10 20 30 40 50

Dep

th b

elo

w 0

.0 -

m

su from UU tests on 35 mm trimmed, kPa

0

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0 10 20 30 40 50

Dep

th b

elo

w 0

.0 -

m

su from UU tests on 70 mm untrimmed, kPa

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0 10 20 30 40 50

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th b

elow

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-m

su from all data, kPa


SPENCER METHOD REFORMULATED FOR SPREADSHEET

The derivations and procedures in this section are based on Low (2003), using the notations in Nash (1987). The sketch at the top of Fig. 3 shows the forces acting on a slice (slice i) that forms part of the potential sliding soil mass. The notations are: weight Wi, base length li, base inclination angle i, total normal force Pi at the base of slice i, mobilized shearing resistance Ti at the base of slice i, horizontal and vertical components (Ei, Ei1, iEi, i1Ei1) of side force resultants at the left and right vertical interfaces of slice i, where i1 and i are the tangents of the side force inclination angles (with respect to horizontal) at the vertical interfaces. Adopting the same assumptions as Spencer (1973), but reformulated for spreadsheet-based automatic constrained optimization approach, one can derive the following from Mohr-Coulomb criterion and equilibrium considerations:

( ) FluPlcT iiiiiii

+= tan (Mohr-Coulomb criterion) (1)

iiiiiiiii TEEWP sincos 11 += (Vertical equilibrium of slice i) (2) iiiiii TPEE cossin1 += (Horizontal equilibrium of slice i) (3)

( )( )

( )

+

+

=

iiii

iii

iiiiiiii

iiii

i

F

lulcF

EW

P

cossintan1

cossin

cossintan1

11

(From above equations) (4)

[ ] 0sincos = wiiii PPT (Overall horizontal equilibrium) (5) ( )

( ) 0sincoscossin

=

+

+ wyiiiii

xiiiiiiM

LPT

LWPT

(Overall moment equilibrium) (6)

( ) ciixi xxxL += 15.0 (Horizontal lever arm of slice i) (7) ( )15.0 += iicyi yyyL (Vertical lever arm of slice i) (8)

where ci, i and ui are cohesion, friction angle and pore water pressure, respectively, at the base of slice i, Pw is the water thrust in a water-filled vertical tension crack (at x0) of depth hc, and Mw the overturning moment due to Pw. Equations 7 and 8, required for noncircular slip surface, give the lever arms with respect to an arbitrary center. For circular slip surfaces, Lxi = Rsini and Lyi = Rcosi, and Eq. (6) for overall moment equilibrium reduces to: [ ] 0sin = wii MWRRT . The use of both i and i-1 in Eq. 4 allows for the fact that the right-most slice (slice #1) has a side that is adjacent to a water-filled tension crack, hence 0 = 0 (i.e., the direction of water thrust is horizontal), and for different values (either constant or varying) on the other vertical interfaces.


F0.484 1.148

0.00 0.00

c0 b

2.23 1.21

H hc w Pw Mw ru xc yc R xmin xmax48.8 30.5 0.00 10 0 0 0 0.13 -20.25 56.85 0.00 50.50

# x ybot ytop b c W rad u l P T E 0 50.50 6.10 6.10 0 0.000

1 46.32 12.88 6.10 5.72 13.7 0 81.1 1.019 0.00 7.97 9.89 95.18 -41.5 0.124

2 42.65 17.49 6.10 5.72 20.6 0 191.1 0.897 0.00 5.89 159.35 105.73 17.0 0.227

3 39.41 20.84 6.10 5.72 25.4 0 241.8 0.804 0.00 4.66 205.52 103.25 93.3 0.308

4 36.56 23.39 6.10 5.72 29.0 0 260.9 0.729 0.00 3.82 219.35 96.43 167.6 0.369

5 34.06 25.36 6.10 5.72 31.7 0 262.0 0.668 0.00 3.19 219.59 88.18 234.3 0.413

6 31.85 26.92 6.10 5.72 33.9 0 252.8 0.616 0.00 2.70 213.06 79.68 292.3 0.443

7 29.91 28.17 6.10 5.72 35.6 0 238.1 0.572 0.00 2.31 202.81 71.48 341.9 0.463

8 28.20 29.18 6.10 5.72 36.9 0 220.5 0.534 0.00 1.98 190.48 63.82 384.0 0.475

9 26.70 30.01 6.10 5.72 38.0 0 202.0 0.501 0.00 1.71 177.08 56.78 419.3 0.482

10 24.79 30.97 8.28 5.72 39.1 0 254.3 0.468 0.00 2.14 227.04 72.83 456.6 0.483

11 22.90 31.84 10.44 5.72 40.2 0 238.2 0.431 0.00 2.08 218.08 72.83 481.4 0.478

12 21.03 32.62 12.57 5.72 41.2 0 221.6 0.394 0.00 2.03 209.08 72.73 494.6 0.467

13 19.18 33.31 14.69 5.72 42.1 0 204.7 0.359 0.00 1.98 199.61 72.54 496.9 0.450

14 17.35 33.93 16.78 5.72 42.9 0 187.5 0.325 0.00 1.93 189.31 72.28 488.8 0.426

15 15.54 34.47 18.85 5.72 43.6 0 170.0 0.291 0.00 1.89 177.86 71.96 470.9 0.398

16 13.74 34.95 20.91 5.72 44.2 0 152.4 0.258 0.00 1.86 165.04 71.57 443.8 0.365

17 11.96 35.35 22.94 5.72 44.8 0 134.6 0.226 0.00 1.82 150.67 71.14 408.2 0.328

18 10.20 35.70 24.95 5.72 45.2 0 116.7 0.194 0.00 1.79 134.68 70.67 364.8 0.287

19 8.46 35.99 26.94 5.72 45.6 0 98.7 0.163 0.00 1.77 117.08 70.16 314.6 0.243

20 6.73 36.22 28.91 5.72 45.9 0 80.7 0.132 0.00 1.74 97.97 69.61 258.4 0.197

21 5.02 36.39 30.86 5.72 46.2 0 62.7 0.101 0.00 1.72 77.54 69.03 197.6 0.149

22 3.33 36.51 32.79 5.72 46.3 0 44.7 0.071 0.00 1.70 56.08 68.43 133.3 0.100

23 1.66 36.58 34.71 5.72 46.4 0 26.8 0.042 0.00 1.68 33.90 67.80 67.0 0.050

24 0.00 36.60 36.60 5.72 46.5 0 8.9 0.012 0.00 1.66 11.38 67.15 0.0 0.000

framed cells contain equations

slope angle

x0xn

Center of Rotation

Varying side-force angle ( = tan)

Ei

iEii-1Ei-1

Ei-1

Ti

Pi

Wi

i

li

Reformulated Spencer method0

10

20

30

40

0 20 40 60

Units: m, kN, kN/m2 (kPa), kN/m3.

( )( )

= 0

0sinxx

xx

n

i

2H

Forces3H

M

Note: The column labels are to be interpreted in context. For the undrained analysis of the submerged slope here, b denotes buoyant unit weight, c means cu, means u, and u is shown as zero because it is not used in total stress analysis. Unit weight of seawater = 10 kN/m3.

FIG. 3. General template for reformulated Spencer method in spreadsheet


The algebraic manipulation that results in Eq. 4 involves opening the term (Pi uili)tani of Eq. 1, an action legitimate only if (Pi uili) > 0, or, equivalently, if the effective normal stress i ( = Pi/li ui) at the base of a slice is non-negative. Hence, after obtaining the critical slip surface, one needs to check that i > 0 at the base of all slices and Ei > 0 at all the slice interfaces. Otherwise, one should consider modeling tension cracks for slices near the upper exit end of the slip surface. Figure 3 shows the spreadsheet set-up for deterministic stability analysis of the underwater excavated slope in San Francisco bay. The automatic search for critical slip surface using Microsoft Excels built-in Solver routine is described in Low (2003), available at http://alum.mit.edu/www/bklow, including a downloadable Excel file for hands-on to enhance understanding of the concepts and procedures. The reformulated Spencer method (Fig. 3) enables effective stress analysis with search for noncircular slip surfaces in heterogeneous soil characterized by shear strength parameters c and . For the underwater slope in hand, undrained analysis is conducted, for which the column labelled c shows the undrained shear strength, and the column labelled shows the u values, which is 0 for saturated clay. Zeros are entered in the column labelled u because the pore pressures are not used in total stress analysis. Same template for Spencer, Bishop Simplified and Force Equilibrium Methods but Different Constraints

Equations 1 to 8 are more straightforward than the original Spencer (1973) formulations. The procedure of Fig. 3, as elaborated in Low (2003), also differs from the Spencer method of solution, which was formulated for use in an age prior to personal computers. The present formulation and constrained optimization procedure also allows convenient investigations of the difference in results among Spencer, Bishops simplified, and force equilibrium (wedge) methods. The following may be noted:

- The solution for the critical circular slip surface in Fig. 3 (and, if desired, that for the critical noncircular slip surface) are as rigorous as the Spencer (1973) method and the Chen-Morgenstern (1983) method (Low et al., 1998), but is operationally simpler and conceptually more transparent.

- If cell is set to zero, and Excel Solver is invoked to change cell F, subject to the constraint that the cell Moment be equal to 0, Solver will obtain a factor of safety but with a net horizontal unbalanced force in the cell labelled Forces. This F corresponds to Bishops simplified method, which assumes horizontal side-forces, and satisfies overall moment and vertical force equilibrium but not horizontal equilibrium.

- The critical circular slip surface of Fig. 3 is for an assumed half-sine variation of the side-force inclination parameter . If constant side-force inclination is imposed, the factor of safety corresponding to the critical circular slip surface is also 1.15 for the case in hand, with a constant side-force inclination parameter = 0.398.

- If Solver is invoked to change F but with fixed at some assumed value, and with constraints on satisfying overall force equilibrium (Forces = 0) but not overall moment equilibrium, the factor of safety obtained would correspond to the force equilibrium (wedge) method.


The following table summarizes the above procedures for conducting limit equilibrium methods using different optimization settings. Table 1. Implementing Spencer, Bishop Simplified and Force Equilibrium methods

on the same spreadsheet template by using different optimization settings

Method Assumption for or

By automatic changing cells

Solver Constraints regarding equilibrium

Spencer with varying side-force inclination, Morgenstern and Price

Varying , F, Forces = 0 Moments = 0

Spencer with constant side-force inclination

Constant , F, Forces = 0 Moments = 0

Bishop simplified method (Circular slip surface)

Set = 0 F, Moments = 0

Force equilibrium or wedge method

Example: = (tan)/F

F, Forces = 0

Note: and are defined in Figure 3.

DETERMINISTIC STABILITY ANALYSIS OF THE UNDERWATER SLOPE The San Francisco Bay underwater slope was analyzed in Fig. 3 based on search for the critical circular slip surface, using the mean value trend of all data combined (vane tests and two types of triaxial tests, last plot of Fig. 2), for which c0 = 2.23 kPa and b = 1.21 in the linear trend su = c0 + by. An Fs of 1.15 was obtained. Since u = 0, the same Fs will be obtained by rotational analysis of a cylindrical block, the Ordinary Method of Slices, Bishop Simplified Method, and Spencer circular slip surface. However, the reformulated Spencer method affords search for the critical noncircular surface and a lower Fs, as shown in Table 2. It is seen that while for circular slip surfaces the Fs ranges from 1.03 to 1.23, for noncircular critical slip surfaces the Fs ranges from 1.00 to 1.20. (Row 2 results ignored, as explained later.) In other words, the Fs for the critical noncircular slip surface is about 2.5% to 3% lower than the Fs for critical circular slip surface. Rows 2 and 3 differ in the manner of interpreting the vane shear test data. As can be seen in Figure 2, the values of vane shear strength at 7 m depth and at 11.3 m depth are relatively very low, and are very likely erroneous, probably as a result of having been performed in mud that was disturbed in the process of drilling the borehole or inserting the vane. Row 2 of Table 2 reflects calculations that include these two tests. Row 3 of Table 2 reflects calculations in which these two values were excluded. Paradoxically, eliminating the low strengths had the effect of reducing the computed factor of safety. This illogical result is a consequence of extrapolating the measured strengths from the upper 21 m of Bay mud, where the tests were performed, to 31 m, which is the bottom of the Bay mud deposit. This is discussed more completely in the following section.


Table 2. Deterministic analysis based on mean values of su, using Spencer method for circular and noncircular slip surfaces.

Tests c0 (kPa) Slope of

trend line, b

Fs, critical circular surface

Fs, critical noncircular surface

All data combined 2.23 1.21 1.15 1.12 In situ vane shear including two bad low-values 2.71 1.35 1.29 1.26 In situ vane shear excluding two bad tests 9.11 1.04 1.23 1.20 35 mm trimmed triaxial test specimens 1.10 1.30 1.19 1.16 70 mm untrimmed triaxial test specimens 0.05 1.17 1.03 1.00

Figure 4 shows the critical noncircular slip surface for the all-data case, obtained using Microsoft Excels built-in constrained optimization Solver routine, by varying 24 degrees of freedom which eventually deforms the critical circular surface of Fig. 3 into the critical noncircular surface of Fig. 4. The su profile is the mean trend of all data, for which su = 2.23 + 1.21y, in kPa, where y is the depth in meter below 0.0.

r0.950

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40

0 10 20 30 40 50 60

Critical noncircular slip surface: Fs = 1.12,based on the mean sutrend of all data.

0

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0 10 20 30 40 50

y -m

eter

su - kPa

Stiff layer

FIG. 4. The reformulated Spencer method obtains the critical noncircular slip surface as shown and a factor of safety of 1.12, compared with a Fs of 1.15 based on critical circular slip surface. BIASES IN STRENGTH MEASUREMENTS AND INTERPRETATIONS

The results of both the deterministic and the probabilistic analyses are affected by biases in the strength measurements and interpretations. The sources of these biases include disturbance resulting from sampling and handling the triaxial test specimens and from inserting the vanes; rates of shearing in the triaxial and vane shear tests that are much more rapid than in the field; and extrapolating the measured strengths to the full depth of the deposit. These three sources of bias are discussed below. Disturbance Effects. It is well understood that inserting a tube sampler below the bottom of a borehole disturbs the soil to some extent, even if a thin-walled sampler is used. Similarly, inserting a vane below the bottom of the borehole also causes some amount of disturbance, and in both cases the effect of disturbance is to reduce the


undrained strength of a normally consolidated clay like San Francisco Bay mud. In the site investigation for the LASH terminal, an attempt was made to minimize the amount of disturbance in some of the triaxial test specimens by trimming away the outer 18 mm of the 70-mm Shelby tube samples, because it was thought that the outer part of the samples would be more disturbed than the inner part which is further from the wall of the sampling tube. Other test specimens were prepared by extruding the entire 70-mm samples. It can be noted in Figure 2 that the strengths measured using the 35-mm trimmed test specimens were consistently higher than those measured using 70-mm (untrimmed) test specimens. The effect of disturbance reducing undrained strength is reflected in the fact that the minimum factors of safety calculated using the data from the tests on 35-mm specimens are about 15 percent higher than those calculated using data from the 70-mm specimens. While it can be seen that trimming away the outer portion of the tube samples served to reduce the strength loss due to disturbance, it is likely that it did not completely eliminate the effect of disturbance. Rates of testing. Undrained strengths of saturated clays like San Francisco Bay mud depend to an important extent on the length of time required to reach failure. Duncan and Buchignani (1973) showed that the undrained strengths measured in tests where the time to failure was a week or so was about 30 percent lower than in tests where the time to failure was 10 to 20 minutes. The triaxial tests performed on both the 35-mm and the 70-mm test specimens were performed at normal rates of loading, where failure was reached in 10 to 20 minutes. The vane shear tests were also performed at normal rates of rotation of the vane (one degree of rotation per 10 seconds), and failure was reached in one to two minutes. Thus the Bay mud was brought to failure in both the triaxial tests and the vane shear tests in much less time than the excavated slope at the LASH terminal was expected to remain stable. This shorter time to failure would increase the measured strength as compared to that mobilized in the field, where failure occurred after a period of days or weeks. Extrapolation of Measured Strengths to Greater Depths. Due to the difficulties in sampling and performing vane shear tests at greater depths with the equipment available in 1970, tests were not performed below a depth of 21 m beneath the mudline. The fact that su/p is constant for normally consolidated clays like San Francisco Bay mud had been well established, and was supported by the increase in strengths down to 21 m. It seemed logical to expect that the strength should continue to increase linearly below 21 m, and it still does. However, extrapolating the strengths measured above a depth of 21 m to the bottom of the profile resulted in a biased strength profile because of a secondary influence of disturbance. Both sampling and vane shear testing became more difficult, and probably resulted in greater disturbance effects as the depth increased. This would result in too small a rate of increase of strength with depth, which would affect all of the measured strengths. Ironically, the factor of safety calculated for the linear variation of strength with depth with two low values of vane shear strength eliminated resulted in a lower factor of safety than when the erroneous low strengths were included. The reason for this paradoxical result is that the erroneous low strengths were at relatively shallow depths. Eliminating these low strength values resulted in a higher average strength within the upper 15 m of the Bay mud, but it also reduced the apparent rate of strength increase with depth. The average strength over the full 31 m thickness of the Bay mud turned out to be about the same with and without the two low value strengths, as can be seen in


Figure 5. However, because the portion of the slip surface in the lower 16 m of the Bay mud is significantly longer than the portion in the upper 15 m (Figs. 3 and 4), the extrapolated su profile with the two erroneous low strengths omitted results in a lower factor of safety than if the two erroneous low strengths were not removed. It is interesting that using a nonlinear trend line for the variation of strength with depth, as shown at the right in Figure 5, does not lead to the paradox that eliminating low strengths results in lower factors of safety. However, this nonlinear variation of strength with depth is not consistent with the expected behavior of saturated clays like San Francisco Bay mud, and hence will not be used for the case in hand. Importance of Judgment. The preceding discussion shows that none of the means of measuring the undrained strength is perfect. All involve bias of one type or another: lower strength due to disturbance, higher strength due to faster rate of testing, and complex effects due to extrapolating strengths beyond the depths where they were measured. As a result, in this situation as in many others, geotechnical engineers face the necessity of working with imperfect data. Probabilistic analyses are helpful for understanding the effects of random variations in measured properties, but even probabilistic analyses are affected by biases in the underlying data. To cope with the effects of bias requires an understanding of the behavior of the soil, and the application of engineering judgment. Engineering judgment indicates, for example, that a factor of safety of 1.17 is too low for a project like the LASH Terminal excavated slope.

FIG. 5. Excavation is from depth 6.1 m to 36.6 m (a) Linear regression in the top 2/3 of the 30.5 m excavation, and extrapolation into the lower 1/3 of the excavation. (b) Second- degree polynomial model for the top 2/3 of the excavation. Probabilistic analyses are conducted in the following sections on three sets of data for which the linear su-profile assumption seems to be good approximations (at least for the depth where su data are available), namely (i) field vane test data with two bad points eliminated, (ii) UU triaxial test data on 35 mm specimens trimmed from 70 mm Shelby tube samples, (iii) UU triaxial test data on untrimmed 70 mm specimens. The mean-value trend and the 1 standard deviation lines for the three types of data are shown in Fig. 6.

e data Ignore two bad points

0

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0 15 30 45

Dep

th b

elow

0.0

-m


0

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0 15 30 45 60

Dep

th b

elo

w

0.0

-m


Inflexible linear trend plus extrapolation lead to misleading su trend and higher Fs.


The undrained shear strength profile is modeled as a random variable by

( )bycsu += 0 (9)

where y = depth in meter (> 6.1 m), is the ratio of measured su values to trendline values. The values of c0 and b are given in Fig. 6. The mean of is equal to 1. The standard deviations of are the values in Fig. 6. Another quantity treated as a random variable is the buoyant unit weight of the Bay mud, with a mean value of 5.72 kN/m3 and a standard deviation of 0.52 kN/m3.

0

5

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20

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30

0 15 30 45

Dep

th b

elow

0.0

-m

su f rom UU tests on 70mm untrimmed, kPa

0

5

10

15

20

25

30

0 15 30 45D

epth

bel

ow 0

.0 -

m

su f rom UU tests on 35 mm trimmed, kPa

0

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10

15

20

25

30

0 15 30 45

Dep

th b

elow

0.0

-m

su f rom in situ vane shear tests, kPa

c0 = 9.11b = 1.04

= 0.18c0 = 1.10b = 1.30

= 0.21

c0 = 0.05b = 1.17

= 0.21

Two bad points removed

FIG. 6. The average su profile, described by su = c0 + by, and the 1 Std. Dev. lines. EFFICIENT APPROACH FOR FIRST-ORDER RELIABILITY METHOD The Hasofer-Lind index (for correlated normal random variables) and the first-order reliability method (FORM, for correlated nonnormals) can be better understood by adopting an intuitive perspective and a spreadsheet-based reliability approach as described in Low and Tang (2004, 2007), Low (2008) and Low et al. (2007). The procedures can be applied to stand-alone numerical (e.g. finite element) packages via the response surface method. Hence the applicability of the reliability approach is not confined to models which can be formulated in the spreadsheet environment. The established matrix formulation of the Hasofer-Lind index is:

( ) ( )xCxx

=

1min TF

(10a)

or, equivalently:

=

i

ii

T

i

ii

F

xx

1min Rx

(10b)

where x is a vector representing the set of random variables xi, the vector of mean values i, C the covariance matrix, R the correlation matrix, i the standard deviations,


and F the failure domain. Low and Tang (1997) used Eq. (10b) instead of Eq. (10a), because the correlation matrix R is easier to set up, and conveys the correlation structure more explicitly than the covariance matrix C. The point denoted by the xi values, which minimize Eq. (10) and satisfies Fx is the design point. This is the point of tangency of an expanding dispersion ellipsoid with the limit state surface (LSS), which separates safe combinations of parametric values from unsafe combinations. The quadratic form in Eq. (10) appears also in the negative exponent of the established probability density function of the multivariate normal distribution. As a multivariate normal dispersion ellipsoid expands from the mean-value point, its expanding surfaces are contours of decreasing probability values. Hence, to obtain by Eq. (10) means maximizing the value of the multivariate normal probability density function, and is graphically equivalent to finding the smallest ellipsoid tangent to the LSS at the most probable failure point (the design point). For correlated nonnormals, the ellipsoid perspective still applies in the original coordinate system, except that the nonnormal distributions are replaced by an equivalent normal ellipsoid, centered not at the original mean values of the nonnormal distributions, but at the equivalent normal mean-value point. Details are in Low and Tang (2004). An alternative FORM computational procedure is given in Low & Tang (2007), which uses the following equation for the reliability index :

nRnx

1min

=T

F (11)

The reliability index and the numerical values of the dimensionless equivalent standard normal vector n at the design point (denoted as n*) are obtained via spreadsheet-automated constrained optimization. The original random variables xi (on which the performance functions are formulated) are computed automatically from the probabilistic connections between ni and xi. RELIABILITY ANALYSIS OF THE SLOPE IN SAN FRANCISCO BAY MUD The computation and concepts will be illustrated first for uncorrelated normal random variables, then for correlated normal random variables, and finally for correlated lognormals.

Simple illustration of FORM for uncorrelated normal random variables and b

The FORM reliability index is obtained easily using the Low and Tang (2007) spreadsheet procedure, with search for the reliability-based critical slip circle. The probability of failure is then computed from Pf = 1 () = 1 NormSDist(). Comparisons are made with three Monte Carlo simulations each with 3000 trials and different initial seed, using the commercial software @RISK (http://www.palisade.com), based on the reliability-based critical slip circle. The three Monte Carlo Pf values are shown below the tables for comparison with the FORM Pf values. The Pf values from reliability index are practically identical to the Pf from Monte Carlo simulations. The computation time for each FORM is only a second or two. The computation time for Monte Carlo simulation is about 500 times higher.


Table 3a. In situ vane shear excluding two bad tests Distribution x n Correlation matrix Pf

Normal 1 0.18 0.836 -0.913 1 0 0.99 16.2%Normal b 5.72 0.52 5.914 0.373 0 1

Compare: Pf from three Monte Carlo simulations: 16.1%, 15.9%, 16.1%.

Table 3b. 35 mm diameter UU triaxial trimmed specimens Distribution x n Correlation matrix Pf

Normal 1 0.21 0.859 -0.669 1 0 0.71 23.8%Normal b 5.72 0.52 5.847 0.243 0 1


Table 3c. 70 mm diameter UU triaxial untrimmed specimens Distribution x n Correlation matrix Pf

Normal 1 0.21 0.969 -0.149 1 0 0.16 43.6%Normal b 5.72 0.52 5.752 0.062 0 1


FORM analysis for correlated normal random variables and b The undrained shear strength su in Eq. (9) increases with increase in the parameter . Physical considerations then suggest that the parameter should be positively correlated with the buoyant unit weight b, because high su values are likely to go with high unit weights, and vice versa. In this section reliability analysis is performed assuming a positive correlation coefficient of 0.5 between and b. The results are shown below. It can be seen that when a strength parameter () is positively correlated with a load parameter (b), the probability of failure is lower than the uncorrelated case of Table 3.

Table 4a. In situ vane shear excluding two bad tests Distribution x n Correlation matrix Pf

Normal 1 0.18 0.799 -1.118 1 0.5 1.22 11.1%Normal b 5.72 0.52 5.653 -0.129 0.5 1


Table 4b. 35 mm diameter UU triaxial trimmed specimens Distribution x n Correlation matrix Pf

Normal 1 0.21 0.831 -0.807 1 0.5 0.86 19.4%Normal b 5.72 0.52 5.650 -0.134 0.5 1

Compare: Pf from three Monte Carlo simulations: 18.6%, 19.8%, 19.4%


Table 4c. 70 mm diameter UU triaxial untrimmed specimens Distribution x n Correlation matrix Pf

Normal 1 0.21 0.961 -0.183 1 0.5 0.20 42.0%Normal b 5.72 0.52 5.710 -0.019 0.5 1


FORM analysis for correlated lognormal random variables and b Input random variables are often modeled by the lognormal distribution, which provides some mathematical convenience and also avoids the negative domain. For the case in hand where the coefficient of variation is 0.21, the failure probabilities for correlated lognormals (Table 5) do not differ much from those for correlated normals (Table 4).

Table 5a. In situ vane shear excluding two bad tests Distribution x n Correlation matrix PfLognormal 1 0.18 0.806 -1.120 1 0.5 1.30 9.7%Lognormal b 5.72 0.52 5.703 0.012 0.5 1


Table 5b. 35 mm diameter UU triaxial trimmed specimens Distribution x n Correlation matrix PfLognormal 1 0.21 0.833 -0.778 1 0.5 0.86 19.4%Lognormal b 5.72 0.52 5.664 -0.063 0.5 1

Compare: Pf from three Monte Carlo simulations: 19.4%, 19.8%, 19.2%

Table 5c. 70 mm diameter UU triaxial untrimmed specimens Distribution x n Correlation matrix PfLognormal 1 0.21 0.959 -0.100 1 0.5 0.11 45.6%Lognormal b 5.72 0.52 5.692 -0.008 0.5 1


It seems logical to assume that the strain rate effect is more dominant than the disturbance effect in the field vane tests and in the UU triaxial tests on 35 mm trimmed specimens, thereby resulting in overestimated undrained shear strengths and hence underestimated probabilities of failure. In any case, a computed (and underestimated) Pf of about 10% (Table 5a, lowest of Tables 3, 4 and 5) would have made the slope design unacceptably risky, since reliability-based design typically aims at a of at least 2.5 (i.e., Pf < 1%). SUMMARY AND CONCLUSIONS

An underwater slope excavated in San Francisco Bay mud has been analyzed first deterministically using a reformulated Spencer method, then probabilistically using the


first-order reliability method (FORM). Factors of safety and reliability indices are computed separately for three types of measured undrained shear strength data, namely field vane tests, UU triaxial tests on trimmed 35 mm specimens, and UU triaxial tests on untrimmed 70 mm specimens. The principal factors affecting the measured strength values are disturbance and rate of loading effects, together with subtle errors caused by extrapolating measured strength data into untested lower depth of the Bay mud. On the basis of this study, the following conclusions are justified: (i) The results of both the deterministic and the probabilistic analyses are affected by

biases in the strength measurements and interpretations, and by extrapolating the measured strengths to the full depth of the deposit.

(ii) To cope with the effects of bias requires an understanding of the behavior of the soil, and the application of engineering judgment in both the deterministic and the probabilistic analysis.

(iii)For the slope in hand, the computed factors of safety are 1.20, 1.16 and 1.00, based on field vane, trimmed 35 mm diameter and untrimmed 70 mm diameter specimens in UU triaxial tests. The FORM analyses produce probabilities of failure between 10% and 46%. An unusually low factor of safety = 1.17 was deemed acceptable for design of this slope because good quality tests had been performed and because the excavated slope only had to be stable temporarily. Conventional design would have required a higher factor of safety (about 1.3) at the end of construction.

(iv) Given the fact that the slope failed, the right factor of safety is 1.0. It is ironic that the strength measurements that result in a factor of safety closest to 1.0 are those that were considered to be of lowest quality the tests performed on the 70 mm samples, where the most disturbed outer portion of the samples was not trimmed away. However, for these samples, the effect of the greater degree of disturbance reducing the undrained strengths was compensated very closely by the rate of loading increasing the undrained strengths. This shows clearly the significant effect of rate of loading, which is commonly ignored.

(v) That judgment is required in estimating the statistical inputs and in interpreting the outputs of a probabilistic analysis should not deter one from adopting the probabilistic approach (which is more rational and has richer output information), but should serve as an incentive to find ways to improve the approach, including more thought given to estimation of statistical inputs and their probabilistic modeling.

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Chapter 2 (pages 76-133) of Reliability-Based Design in Geotechnical Engineering- Computations and Applications, Taylor and Francis, ed. K.K.Phoon.

Christian, J.T., Ladd, C.C. and Baecher, G.B. (1994). Reliability applied to slope stability analysis. J. Geotech. Eng. ASCE, 120(12): 2180-2207.

Duncan, J.M. (2000). Factors of safety and reliability in geotechnical engineering. J. Geotechnical and Geoenvironmental Engrg. 126(4): 307-316.


Duncan, J.M. (2001). Closure to Discussions on Factors of safety and reliability in geotechnical engineering. J. Geotechnical and Geoenvironmental Engrg. 127(8): 717721.

Duncan, J.M. and Buchignani, A.L. (1973). Failure of underwater slope in San Francisco Bay. J. Soil Mechanics and Foundation Division, ASCE, 99(9): 687-703.

Duncan, J.M. and Wright, S.G. (2005). Soil strength and slope stability. Wiley & Sons, 297 p.

Low, B.K. (2003). Practical probabilistic slope stability analysis. Proceedings, Soil and Rock America 2003, Cambridge, Massachusetts, USA, June 22-26, 2003, Verlag Glckauf GmbH Essen, Vol. 2, 2777-2784.

Low, B.K. (2008). Practical reliability approach using spreadsheet, Chapter 3 (pages 134-168) of Reliability-Based Design in Geotechnical Engineering-Computations and Applications, Taylor and Francis, ed. K.K.Phoon.

Low, B.K. and Wilson H. Tang (1997). "Reliability analysis of reinforced embankments on soft ground." Canadian Geotechnical Journal, 34(5): 672-685.

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Low, B.K. and Wilson H. Tang (2007). Efficient spreadsheet algorithm for first-order reliability method. J. Engineering Mechanics, ASCE, 133(12): 1378-1387.

Low B.K., Gilbert, R.B. and Wright, S.G. (1998). "Slope reliability analysis using generalized method of slices." J. Geotech. and Geoenviron. Engrg. 124(4): 350-362.

Low, B.K., Lacasse, S. and Nadim, F. (2007). Slope reliability analysis accounting for spatial variation. Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, Taylor & Francis, London, 1(4): 177-189.

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Nash, D., A comparative review of limit equilibrium methods of stability analysis. In M. G. Anderson and K. S. Richards (Eds), Slope Stability, 1987, 11-75, Wiley: New York.

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