SLOPE RELIABILITY   

 

 

MBTC FR 1014   

 

Sam I. Thornton   

 

 

 

 

 

 

 

 

 

 

 

 

DISCLAIMER

 

 

The contents of this report reflect the views of the authors, who are responsible for the facts and accuracy of the information presented herein.  This document is disseminated under the sponsorship of the Department of Transportation, University Transportation Centers Program, in the interest of information exchange.  The U.S. Government assumes no liability for the contents or use thereof.

 

 

 

 

FINAL REPORT

 

 

 

 

 

Slope Reliability

 

 

 

 

 

Submitted

 

to

 

Mack-Blackwell National Rural Transportation Study Center

University of Arkansas, Fayetteville

 

and

 

Arkansas Highway and Transportation Department

Little Rock, Arkansas

 

 

 

 

 

Prepared

 

by

 

Sam I. Thornton

 

 

 

 

Civil Engineering Department

University of Arkansas

 

June 1995

 

 

 

 

 

 

 

 

 

 

 

 

                                      SLOPE RELIABILITY

 

 

SUMMARY

 

Analysis methods for slope stability are routinely applied by geotechnical engineers. Slope designs, however, are usually based on a "safety factor" which does not account for soil variability (soil variability is due to actual in-place conditions and not due to sampling procedures and/or testing methods).  As a result, the true safety of a slope is unknown.

 

A reliability approach, using probability calculations which account for the variability in soil strength, is superior to the factor of safety approach.  The method is based on the point estimate method and allows engineers to calculate a probability of failure for the slope.  Knowing the probability of failure improves engineering judgement by providing a rational basis for making a safe and economical slope design.

 

Examples show how soil variability affects slope reliability and how the method is applied.   The factor of safety is 1.30 in the first two examples.  In the first example, the soil deposits are uniform and the probability of failure is acceptable; In the second example, the soils have more soil strength variation and the probability of failure is higher than recommended.  

 

 

1  INTRODUCTION

 

Geotechnical engineers routinely calculate a factor of safety (FS) to evaluate the stability of earth slopes.  The Simplified Bishop method (Wright, et al, 1973) is a popular basis for computer analysis programs.  A minimum FS of 1.3 is commonly considered as the design basis for most slopes.  Failure is assumed to occur when the FS is less than 1.0. 

 

Because the FS analysis does not have a way to consider the variability of the soil strength, the true safety of a slope is unknown.  A reliability approach, where a probability of failure is calculated,  is a better method for slope design because it accounts for variability in soil strengths.  Other factors, like an inadequate field investigation, missing a critical geologic detail (Christian, et, al., 1994) or progressive slope failures (Chowdhury, R,. N., Sept. 1994) are not included in the method described in this report.

 

The probability of slope failure method is based on the "Point Estimate Method" (PEM) which was developed by Rosenblueth (1975 and 1981) and described by Harr (1987).  In the PEM method, a distribution of the variable must be found or assumed.  If a normal distribution is assumed, the problem is simplified.  Details of the PEM method and a discussion of other distributions are contained in a thesis by Garrett (1989) and a paper by McGuffy, Iori, Kyfor and Grivas (1981). 

 

 

2  APPLICATION OF THE POINT ESTIMATE METHOD TO

    SLOPE STABILITY

 

2.1 MEAN AND STANDARD DEVIATION

 

To apply the PEM, the mean and standard deviation of the soil strength in each layer must be found.  Soil strength may be cohesion, C, and/or internal friction, φ.  Between layers, strength parameters are considered independent.  Within a soil layer, however, the cohesion may be correlated to the internal friction. 

 

                      

 

                                                                                                                        (Eqn. 1)

 

 

                      

 

 

                                                                                                                        (Eqn. 2)

 

 

                        where, x = the C or φ values in the layer

                                                n = the number of C or φ (tests performed) values in the                                                layer

 

2.2  CORRELATION COEFFICIENT

 

For a soil layer with C and  φ, the correlation between C and φ, must be found.  Correlations vary with the type of strength test.  For the consolidated undrained triaxial test, Harr (1987) reports a correlation, r, of about +0.25.  A positive correlation means the internal friction increases when the cohesion increases.  The undrained triaxial test is the best predictor for quick failures caused by earthquakes or the sudden drawdown of water at a levee or dam.  Drained triaxial tests often have negative correlations and are usually the best predictor of field performance.  Wolff reported a drained triaxial correlation of -0.47 (Harr, 1987).  The correlation coefficient, r, is calculated by the following:

 

                      

 

                                                                                                                        (Eqn. 3)

 

 

            where, N = the number of strength tests

 

2.3  HIGH AND LOW STRENGTH VALUES

 

Variation in C and φ is accounted for by adding or subtracting the standard deviation.  For example, a high cohesion, C+, is obtained by adding the standard deviation of the cohesion to the mean.  A low cohesion, C-, is the mean less the standard deviation.  In turn, φ+ and φ- is the mean internal friction + or - the standard deviation of internal friction.

                      

 

                      

 
                                                                                                                        (Eqn. 4a)

                                                                                                                        (Eqn. 4b)

            

 

                                    where,

 

                                             

 

                                                                                                                        (Eqn. 5a)

                                                                                                                        (Eqn. 5b)

            

 

                                    where,

 

2.4 SLOPE SAFETY FACTORS

 

Safety Factors must be found for all combinations of soil strength.  The number of combinations is 2n, where n is the number of variables (soil strengths).  A slope with two layers, each layer with a C and φ, has 24 or 16 combinations of soil strength.  The set of safety factors reflects the variation of soil strength.  The symbol FS++++ is used for a slope containing two soil layers with C+ and φ+ used for strength values in both layers.  FS-+++ is the symbol for the FS when C- and φ+ are used for the first layer and C+ and φ+ are used for the second layer.

 

2.5  WEIGHING FUNCTIONS

 

Weighing functions must be applied to the FS's.  The weighing functions are point estimates, p, of the distribution of the FS's.  The symbol p++++ is used for the point applied to FS++++ as described in section 2.4.  The sum of the p's is equal to 1.

 

2.51 Independent Layers

 

For the case where each soil layer has only a C (a clay) or φ (a sand) the soil strengths are not correlated. If normal distribution is assumed, the point estimates are:

 

                                                p = 1/2n                                                           (Eqn. 6)

 

                        where, n = the number of variables (layers when each layer has                                                 only a C or φ). 

 

The points for two soil layers with C or φ are:

 

                                    p++ = p+- = p-+ = p-- = 1/4

 

For three soil layers with C or φ, the points are:

 

            p+++ = p++- = p+-+ = p-++ = p+-- = p-+- = p--+ = p--- = 1/8

 

The points for four soil layers are:

 

            p++++ = p+++- = p++-+ = p+-++ = p-+++ = p++-- =

            p+--+ = p--++ = p+--- = p-+-- = p--+- = p---+ = p-++- =

            p+-+- = p---- = p-+-+ = 1/16

           

2.52  Correlated Layers         

 

When a slope has a single layer with both C and φ (two variables), the points are:

 

                                    p++ = p-- = (1 + r)/4                                     (Eqn. 7a)

 

                                    p+- = p-+ = (1 - r)/4                                      (Eqn. 7b)

 

A slope that has two soil layers, each with C and φ (four variables), will have the following points:

 

            p++++ = p---- = p++-- = p--++ = (1+r1+r2)/16    (Eqn.8a)

 

            p+++- = p---+ = p++-+ = p--+- = (1+r1-r2)/16      (Eqn.8b)

 

            p+--- = p-+++ = p+-++ = p--+- = (1-r1+r2)/16      (Eqn.8c)

 

            p+-+- = p-+-+ = p+--+ = p-++- = (1-r1-r2)/16                    (Eqn.8d)

 

2.53 Mixed Layers

 

For the case where there are two layers of soil, one layer contains either C or φ and the other contains both C and φ, the points are:

 

                        p+++ = p+-- = p--- = p-++ = (1+r2)/8                        (Eqn.9a)

 

                        p++- = p+-+ = p--+ = p-+- = (1-r2)/8             (Eqn.9b)

 

 

2.6 STANDARD DEVIATION OF THE FS'S

 

The expected value of the factor of safety, E[FS], and the expected value of the squared FS's, must be found in order to calculate the standard deviation of the FS, σ[FS]. 

 

2.61  Two Variables

 

For a slope with two variables (either two layers with C or φ, or one layer with C and φ):         

 

                        E[FS] = p++(FS++) + p+-(FS+-)

                                      + p-+(FS-+) + p--(FS--)                                           (Eqn.10a)

 

                        E[FS2] = p++(FS++)2 + p+-(FS+-)2

                                       + p-+(FS-+)2 + p--(FS--)2                           (Eqn.10b)

 

                        σ[FS] = (E[FS2] - E[FS]2).5                                                  (Eqn.11)

 

 

 

 

2.62  Three Variables

 

For a slope with three variables (either three layers with C or φ, or two layers; one layer with C or φ, and one layer with C and φ):

 

                        E[FS] = p+++(FS+++) + p++-(FS++-)

                                      + p+--(FS+--) + p---(FS---)

                                      + p--+(FS--+) + p-++(FS--+)

                                      + p-+-(FS-+-) + p+-+(FS+-+)                    (Eqn.12a)

 

                        E[FS2] = p+++(FS+++)2 + p++-(FS++-)2

                                      + p+--(FS+--)2 + p---(FS---)2

                                      + p--+(FS--+)2 + p-++(FS-++)2

                                      + p-+-(FS-+-)2 + p+-+(FS+-+)2                 (Eqn.12b)

 

                        σ[FS] = (E[FS2] - E[FS]2).5                                                  (Eqn.11)

 

 

2.63 Four or More Variables

 

For four or more variables, the expected FS, E[FS], is found by multiplying the points, p, by their respective FS's and summing the products (see equations 10a and 12a).

 

The E[FS2] is found by multiplying the points, p, by their respective squared FS's and summing the products (see equations 10b and 12b).

 

The standard deviation is found from equation 11.

 

2.7  PROBABILITY OF FAILURE

 

For normal distribution, the standardized variable Z is:

 

                                                Z = (FS - E[FS])/σ[FS]                                 (Eqn. 13)

 

                        where, FS = the cutoff value to be evaluated (FS = 1)

 

E[FS] and σ[FS] are found from section 2.6.

 

With Z, the probability that the FS will be less than 1 can be found from the normal distribution table in Appendix A.

 

 

 

3  ACCEPTABLE FAILURE PROBABILITIES

 

In order to evaluate a design, the calculated probability of failure should be compared to an acceptable probability.  A table of acceptable failure probabilities was proposed by Santamarina, et. al. (1992).  A partial listing of the table is contained in Table 1.

 

 

TABLE 1.  Slope Stability - Probability of Failure

CONDITIONS

Pf

Unacceptable in most cases

>0.1

Temporary structures with low repair cost

0.1

Low consequences of failure repairs when time permits

0.02

Existing large cut on interstate highway

0.01

Acceptable in most cases except if lives may be lost

0.001

Acceptable for all slopes

0.0001

Unnecessarily low

0.00001

 

 

 

4  EXAMPLES

 

 

4.1  CONVERSION FACTORS

 

                        SI to English                                         English to SI

                        1 m = 3.281 ft                          1 ft = 0.3048 m

                        1 kN/m2 = 20.885 lb/ft2                        1 lb/ft2 = 0.04788 kN/m2

                        1 kN/m3 = 6.361 lb/ft3              1 lb/ft3 = 0.1572 kN/m3

 

 

4.2  TWO LAYERS WITH EITHER C OR Φ

 

Two examples using the slope in Figure 1 will show how the method is applied.  The unit weight of both soil layers is 20 kN/m3

 

 

 

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

FIGURE 1

 

 

4.21  Example 1: Slope with Uniform Soils

 

The internal friction, φ, and cohesion, C, from tests for the soil in Figure 1 are:

 

                                            SAND                       CLAY

                                                  φE                     C (kN/m2)

 

                                                33.5                             60

                                                36.5                             63

                                                35.5                             64

                                                34.5                             58

                                                35.1                             62.5

                                                34.9

 

The mean (Eqn.1) and standard deviation (Eqn.2) are as follows:

 

                                         SAND                                CLAY

 

                                    mean φ = 35E  mean C = 61.5 kN/m2 

                                       σ (φ) = 1E                   σ (C) = 2.45

 

The high and low strength values (Eqn. 4a, 4b, 5a, and 5b) used to determine slope stability factors of safety are:

 

                                    φ+ = 36E                     C+ = 63.95 kN/m2

                                    φ-   = 34E                    C-   = 59.05 kN/m2

 

The strength combinations for slope stability analysis are:

 

                        ++ Sand φ  = 36E                    Clay C = 63.95 kN/m2

                        +- Sand φ    = 36E                   Clay C = 59.05 kN/m2

                        -+ Sand φ    = 34E                   Clay C = 63.95 kN/m2

                        -- Sand φ      = 34E                  Clay C = 59.05 kN/m2

 

The resulting factors of safety from the computer program PCSTABL5 (Bishop Method) are:

 

                                    ++ FS    = 1.350

                                    +- FS     = 1.248

                                    -+ FS     = 1.348

                                    -- FS                  = 1.246

 

The weighing functions (Eqn. 6) for two soil types with 1 strength parameter per layer is:

 

                                    p++ = p+- = p-+ = p-- = 0.25

 

The expected FS (Eqn. 10a) is:

 

                        E[FS] = 0.25(1.350) + 0.25(1.248) + 0.25(1.348) +0 .25(1.246)

                                 = 1.298

 

The expected FS2 (Eqn. 10b) is:

 

                        E[FS2] = 0.25(1.350)2 + 0.25(1.248)2 + 0.25(1.348)2 + 0.25(1.246)2

                                  = 1.6874

 

The standard deviation of the FS's (Eqn. 11) is:

 

                        σ [FS] = [(1.687) - (1.298)2)].5

                                  = 0.051

 

The standardized variable (Eqn. 13) is:

 

                        Z = (1-1.298)/0.051 = -5.84

 

For a FS = 1, where failure is assumed to occur, the probability of failure, Pf , is (Appendix A):

 

                                    Pf < 0.0000001

 

This probability of failure, according to Table 1, is unnecessarily low.

 

 

4.22  Example 2: Slope with Variable Clay

 

Strength test results for the soil in Figure 1 are as follows:

 

                                    Sand                            Clay

                                      φE                         C(kN/m2)

 

                                    36.5                                55

                                    34                                   50

                                    34.5                                71

                                    35.5                                82

                                    34.5                                53    

                                                                           58

 

The mean (Eqn. 1) and standard deviation (Eqn. 2) are as follows:

 

                                    Sand                                        Clay

                             mean φ = 35E                     mean C = 61.5 kN/m2

                                σ (φ) = 1E                          σ (C) = 12.4 kN/m2

 

 

The high and low strength values (Eqn. 4a, 4b, 5a, and 5b) used to determine slope stability factors of safety are:

 

                                    φ+ = 36E                     C+ = 73.9 kN/m2

                                    φ-   = 34E                    C-   = 49.1 kN/m2

 

The strength combinations for slope stability analysis are:

 

                        ++ Sand φ = 30E                     Clay C = 73.9 kN/m2

                        +- Sand φ   = 36E                    Clay C = 49.1 kN/m2

                        -+ Sand φ   = 34E                    Clay C = 73.9 kN/m2

                        --  Sand  φ   = 34E                   Clay C = 49.1 kN/m2

 

The factors of safety from the computer program PCSTABL5 (Bishop Method) are:

 

                                    ++ FS = 1.556

                                    +- FS     = 1.040

                                    -+ FS     = 1.554

                                    -- FS      = 1.039

 

 

The expected FS (Eqn. 10a) is:

 

                                                E[FS] = 1.297

 

The standard deviation of the FS's (Eqn. 11) is:

 

                                                σ [FS] = 0.2578

 

The standardized variable (Eqn. 13) and the probability of failure (Appendix A) are:

 

                                    Z = (1-1.297)/0.2578 = -1.1637

                                    Pf = 0.125

 

This probability of failure, according to Table 1 is too high, even for temporary structures with low repair costs.

 

 

4.23  Example Comparison

 

The probability that the slopes in the two examples would fail is greatly different;  less than 0.00001% for the first example vs. 12.5% for the second example.  This difference is surprising because the geometry unit weight, and average strength of the soil layers within the slopes are the same.  The reason for the difference in probability of failure is the variability in cohesion of the clay layer.  In the uniform clay layer (section 4.11) the standard deviation of the cohesion is 2.45 kN/m2 or 4% of the average cohesion.  The variable clay layer (section 4.12) has a standard deviation  of 12.4 kN/m2 or 20% of the average cohesion.

 

 

4.3  EXAMPLE 3: THREE LAYERS WITH EITHER C OR Φ

 

The figure below is a slope on Interstate 40 near Morrilton, Arkansas.  The slope has been divided into 3 layers.
 

 

 

 

 

 

 

 

 

 

FIGURE 2

In this example, the only strength parameter in each layer is cohesion.  From the strength tests, the mean and standard deviation of each layer obtained from Eqn. 1 and 2 are:

 

LAYER NO.

MEAN STRENGTH

STANDARD  DEVIATION

1

180 lb/ft2

16 lb/ft2

2

410 lb/ft2

54 lb/ft2

3

600 lb/ft2

138 lb/ft2

 

 

The high and low values (Eqn. 4a and 4b) for cohesion in lb/ft2 are:

 

                          Layer 1                        Layer 2                        Layer 3 

                        C1+ = 196                    C2+ = 464                    C3+ = 738

                        C1 -  = 164                  C2 -  = 356                  C3 -  = 462

 

The next step is putting together the strength combinations.  In this case, since there are 3 strength parameters, the are 23, or 8 strength combinations.

 

The strength combinations and factors of safety from the computer program PCSTABL5 (Bishop Method) for each combination are as follows:

 

COMBINATION

C1

C2

C3

FS

+++

196

464

738

1.466

++-

196

464

462

1.293

+-+

196

356

738

1.145

-++

164

464

738

 1.452 

+--

196

356

462

1.131

-+-

164

464

462

1.285

--+

164

356

738

1.145

---

164

356

462

1.131

 

 

The next step is the calculation of the expected FS (Eqn. 12a), expected value of the squared FS's (Eqn. 12b),  and standard deviation of the FS's (Eqn. 11):

 

 

                                                E[FS]  = 1.256

                                                E[FS2] = 1.595

                                                σ[FS]  = 0.1326

 

Then the standardized variable (Eqn. 13) is found for a FS = 1.

 

                                    Z = (1-1.256)/0.1326 = 1.93

 

By using this Z and the probability chart in Appendix A, the probability of failure for this slope is 2.68%.

 

 

4.4  EXAMPLE 4: FOUR LAYERS WITH C OR Φ

 

The example for four layers of soil is taken from the thesis at the University of Arkansas by Steven Garrett (1989).  Figure 3 contains the geometry of the slope.

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

FIGURE 3

 

 

In this example, the first and third layers are clay and the second and fourth layers are sand.  From the strength tests, the mean (Eqn.1) and standard deviation (Eqn. 2) of strengths are:

 

 

 

            LAYER            MEAN STRENGTH                STANDARD DEVIATION

                1                         3500 lb/ft2                                        200 lb/ft2

                2                         27E                                                  5E

                3                         2000 lb/ft2                                        300 lb/ft2

                4                         32E                                                  2.5E

 

The high and low strength values (Eqn. 4a, 4b, 5a, and 5b) are:

 

          LAYER  HIGH STRENGTH                  LOW STRENGTH

              1                           3700 lb/ft2                         3300 lb/ft2

              2                            32E                                               22E

              3                           2300 lb/ft2                         1700 lb/ft2

              4                           34.5E                                            29.5E

 

The strength combinations and factors of safety from the computer program PCSTABL5 (Bishop Method) for each combination are:

 

COMBINATION

C1  lb/ft2

φ2 E

C3  lb/ft2

φ4 E

FS

++++

3700

32

2300

34.5

1.4024

+++-

3700

32

2300

29.5

1.1966

++--

3700

32

1700

29.5

1.1428

+---

3700

22

1700

29.5

1.1239

----

3300

22

1700

29.5

1.1235

-+--

3300

32

1700

29.5

1.1424

-++-

3300

32

2300

29.5

1.1966

-+++

3300

32

2300

34.5

1.4021

--+-

3300

22

2300

29.5

1.1798

--++

3300

22

2300

34.5

1.3786

---+

3300

22

1700

34.5

1.1798

-+-+

3300

32

1700

34.5

1.3352

+-+-

3700

22

2300

29.5

1.1798

+--+

3700

22

1700

34.5

1.3130

+-++

3700

22

2300

34.5

1.3790

++-+

3700

32

1700

34.5

1.3356

The expected FS, expected value of the squared FS's, and standard deviation of the FS's are found per article 2.63:

 

          E[FS] = 1/16 [l.4024 + 1.1966 + 1.1428 + 1.1239 + 1.1235 + 1.1424 +                                                1.1966 + 1.4021 + 1.1798 + 1.3786 + 1.3126 + 1.3352 +

                         1.1798 +1.3130 + 1.3790 + 1.3356]

                   = 1.2590

 

          E[FS2] = 1/16 [l.40242 + 1.19662 + 1.14282 + 1.12392 + 1.12352 + 1.14242+

                          1.19662+ 1.40212 + 1.17982 + 1.37862 + 1.31262 + 1.33522 +                                               1.17972 + 1.31302 + 1.37902 + 1.33562]

                    = 1.5958

 

          σ[FS] = (l.5958 - 1.25902).5

                   = 0.1035

 

Then the standardized variable is found for a FS = 1.

 

                                    Z = (1.0 - 1.2590)/0.1305

                            = 2.50

 

Using the probability chart in Appendix A, the probability of failure is 0.62%.

 

 

4.5  EXAMPLE 5: ONE SOIL WITH TWO VARIABLES

 

This example is taken from a paper by Verduin and Lovell (1988).  The embankment is 40 feet high and is built on a slope of two horizontal to one vertical (Figure 4).  The soil has a unit weight of 140 lb/ft3.

 
 

 

 

 

 

 

 

 

 

 

 

 

 

FIGURE 4

The mean and standard deviation of the soil strength are:

 

                                    mean C = 200 lb/ft2      σ(C) = 80  lb/ft2

                                    mean φ = 25E              σ(φ) = 2.5°

 

The correlation coefficient (Eqn. 3) as determined from laboratory tests is +0.25.

 

The high and low strength values (Eqn 4a, 4b, 5a, and 5b) used to determine slope stability factors of safety are:

 

                        φ+ = 25 + 2.5 = 27.5E             C+ = 200 + 80 = 280 lb/ft2

                        φ-   = 25 - 2.5  = 22.5E                       C-   = 200 - 80  = 120 lb/ft2

 

The slope factors of safety from the computer program PCSTABL5 (Bishop Method) are:

 

                                    FS++ = 1.685

                                    FS+-    = 1.454

                                    FS-+    = 1.373

                                    FS--     = 1.140

 

The weighing functions (Eqn. 7a and 7b) are:

 

                        p++ = p-- = 0.25(1+0.25) = 0.3125

                        p+-  = p-+ = 0.25(1-0.25)  = 0.1875

 

The expected FS (Eqn. 10a), expected value of the squared FS's (Eqn.10b), and standard deviation of the FS's (Eqn. 11) are:

 

                        E[FS] = 0.3125(1.685) + 0.1875(1.454) +

                                       0.1875(1.373) + 0.3125(1.140)

                                 = 1.413

 

                        E[FS2] = 0.3125(1.685)2 + 0.1875(1.454)2 +

                                        0.1875(1.373)2 + 0.3125(1.140)2

                                  = 2.043

 

                        σ[FS] = (2.043 - (1.413)2).5

                     = 0.216

 

The standardized variable (Eqn. 13) and probability of failure (Appendix A) are:

 

                                    Z = (1.0 - 1.413)/0.216 = -1.91

                                    Pf = 2.8%

 

4.6  TWO SOIL LAYERS WITH TWO VARIABLES EACH

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

                                                               FIGURE 5

 

                                                Unit Weight Layer 1 = 110 lb/ft3

                                                Unit Weight Layer 2 = 120 lb/ft3

 

 

The mean (Eqn. 1) and standard deviation (Eqn. 2) of the soil strength are:

 

                              First Layer                   Second Layer  

                        C (lb/ft2)           φE                    C (lb/ft2)             φE

 

                          200               31                       150                  27

                          180               33                       110                  30

                          210                28                      240                   24

                          230                27                      220                   25

                          160                34                      120                   32

 

            Layer 1            mean C = 196 lb/ft2                  σ(C) = 27 lb/ft2           

                                    mean φ = 30.6E                       σ(φ) = 3.05°

 

            Layer 2            mean C = 168 lb/ft2                  σ(C) = 58.9 lb/ft2        

                                    mean φ = 27.6E                       σ(φ) = 3.36°

 

The correlation coefficients (Eqn. 3) are -0.964 for layer 1 and -0.927 for layer 2.

 

The high and low strength values (Eqn. 4a, 4b, 5a, and 5b) are:

 

                        C1+ = 223 lb/ft2                        φ1+ = 33.65E

                        C1-   = 169 lb/ft2                      φ1-   = 27.55E

                        C2+ = 226.9 lb/ft2                     φ2+ = 30.96E

                        C2-   = 109.1 lb/ft2                   φ2-   = 24.24E

 

The slope factors of safety from the computer program PCSTABL5 (Bishop Method) are:

 

                        FS++++ = 1.6235

                        FS+++-       = 1.3798

                        FS++--       = 1.2123

                        FS+--- = 1.1714

                        FS----         = 1.1413

                        FS---+ = 1.3573

                        FS--++       = 1.5226

                        FS-+++       = 1.5897

                        FS-+-+       = 1.4579

                        FS+-+-       = 1.3295

                        FS+--+       = 1.3977

                        FS-++-       = 1.3527

                        FS++-+       = 1.4560

                        FS--+- = 1.3014

                        FS+-++       = 1.5595

                        FS-+-- = 1.1873

 

The weighing functions (Eqn. 8a, 8b, 8c, and 8d) are:

 

          p++++ = p---- = p++-- = p--++ = (1-0.964-0.927)/16  = -0.05569

          p+++- = p---+ = p++-+ = p--+- = (1-0.964+0.927)/16 = 0.0602

          p+--- = p-+++ = p+-++ = p-+-- = (1+0.964-0.927)/16 = 0.0648

          p+-+- = p-+-+ = p+--+ = p-++- = (1+0.964+0.927)/16 = 0.1807

 

The expected FS, expected value of the squared FS's, and standard deviation of the FS's are:

 

                   E[FS] = -0.05569(1.6235) + 0.0602(1.333798) + -0.05569(1.2123)

                               + 0.0648(1.1714) + -0.05569(1.1413) + 0.0602(1.3573)

                               + -0.05569(1.5226) + 0.0648(1.5897) + 0.1807(1.4579)

                               + 0.1807(1.3527) + 0.1807(1.3977) + 0.0602(1.4560)

                               + 0.0602(1.3014) + 0.1807 (1.3295) + 0.0648(1.5595)

                               + 0.0648(1.1873)

                              =  1.3821

 

                    E[FS2]  = -0.05569(1.62352) + 0.0602(1.37982) + -0.05569(1.21232)

                                    + 0.0648(1.17142) + -0.05569(1.14132) + 0.0602(1.35732)

                                    + -0.05569(1.52262) + 0.0648(1.58972) + 0.1807(1.45782)

                                    + 0.1807(1.35272) + 0.1807(1.39772) + 0.0602(1.45602)

                                    + 0.0602(1.30142) + 0.1807(1.32952) + 0.0648(1.55952)

                                    + 0.0648(1.18732)

                                 =  1.9136

 

                    σ[FS]   = (1.9136 - 1.38212).5 = 0.05798

 

The standardized variable (Eqn. 13) is:

 

                        Z = (1.0 - 1.382)/0.058

                                       = -6.59

 

The probability of failure (Appendix A) is less than .003%.

 

 

5 ARKANSAS SITES

 

Two sites were selected by the Arkansas Highway and Transportation Department for possible application of the method to existing slopes.  The sites were at Highway 67 at State Highway 14 near Newport and I-40 at mile 103.79 near Morrilton.

 

5.1 NEWPORT

 

The embankment at Newport was constructed on a 3 horizontal to 1 vertical slope rising 28 feet.  Two soil layers were contained in the slope.  The first soil layer was a fill 28 feet in height.  The second soil layer was the subsoil which was at ground level.  Soil characteristics for the layers are:

 

                                    SOIL 1                        SOIL 2

                                      φE                    φE

 

          Mean                  37.9                              30.5

     Std. Deviation            2.84                              1.26

          φ+                      40.74                           31.76

          φ-                       35.06                           29.24

 

The resulting factors of safety from the computer program PCSTABL5 are:

 

                                    FS++ = 2.763

                                    FS+-    = 2.658

                                    FS-+    = 2.384

                                    FS--     = 2.315

 

The functions (Eqn. 6) for two soil types with 1 strength parameter per layer is:

 

            p++ = p+- = p-+ = p-- = 0.25

 

The expected FS (Eqn.10a) is:

    

          E[FS] = 0.25(2.763) + 0.25(2.658) + 0.25(2.384) +0 .25(2.315)

                   = 2.53

 

The expected FS2 (Eqn. 10b) is:

 

          E[FS2] = 0.25(2.763)2 + 0.25(2.658)2 + 0.25(2.384)2 + 0.25(2.315)2

                    = 6.43

 

The standard deviation of the FS's (Eqn. 11) is:

 

          σ [FS] = [(6.34) - (2.53)2)].5

                    = 0.1859

 

The standardized variable (Eqn. 13) is:

 

          Z = (1-2.53)/0.1859 = -8.23

 

For a FS = 1, where failure is assumed to occur, the probability of failure, Pf , is (Appendix A):

 

            Pf < 0.0002

 

This probability of failure, according to Table 1, is acceptable for this slope.

 

5.2 MORRILTON

 

The Morrilton site, based on the strength data supplied by the Arkansas Highway and Transportation Department, had an expected FS of 3.58.  Because the FS is so high, the slope was not analyzed for a probability of failure.

 

6  CONCLUSION

 

The reliability approach to slope stability is superior to the safety factor approach because it accounts for variability in soil strength.

 

7  REFERENCES

 

Chowdhury, R. N., Sept. 1994, "Decisions On Landslides Based on Risk Assessment", International Conference on Landslides, Slope Stability and the Safety of Infrastructures, Kuala Lumpur, Malaysia (ISBN:981-00-5813-6).

 

Christian, John T., Charles C. Ladd, and Gregory B. Baecher, Dec. 1994, "Reliability Applied to Slope Stability Analysis", Jour. of Geotechnical Engineers, American Society of Civil Engineers, p 2181.

 

Garrett, Steven Ray, 1988, Slope Failure Probability In Layered Soils, Master's Thesis, University of Arkansas, Fayetteville, AR 72701.

 

Harr, M. E., 1987, Reliability Based Design in Civil Engineering, McGraw-Hill, Inc., pp. 205-220.

 

McGuffey, V., Z. Iori, Z. Kyfor, and D. Athanasoiu-Grivas, 1981, "Use of Point Estimates for Probability Moments in Geotechnical Engineering", Transportation Research Record 809, TRB, National Research Council, Washington D. C.

 

Rosenblueth, E. Milid, Oct. 1975, "Point Estimates for Probability Moments", Proc. Nat. Acad. Scie.,vol. 72, no. 10, pp 3812-3814.

 

Rosenblueth, E. Milid, Oct. 1981, "Two Point Estimates in Probabilities", Appl. Math Modelling, vol. 5, pp 324-334.

 

Santamarina, J.C., A.G. Altschaeffl, and J.L. Chameau, Jan 1992, Reliability of Slopes, Transportation Research Board, Paper #920569, Washington D. C.

 

Wright, Stephen G., Fred H. Kulhaway, and James M. Duncan, Oct 1973, "Accuracy of Equilibrium Slope Stability Analysis", Jour. Soil Mechanics and Foundations, Amer. Soc. of Civil Engineers, New York.

 

 

 

APPENDIX A

 

NORMAL DISTRIBUTION CURVE AREAS