1 Slope Stability Santiago Chile, November 2009Quantif the reliability of Rock Slopes In the introduction to Guidelines for Open Pit Slope Design (Guidelines) (1), it is stated, “It is often no longer sufficient to present slope design in deterministic (factor of safety) terms…. ” It is also noted, “ A measure is required so that mine executives have sufficient information and understanding to be able to establish acceptable levels of risk. ” To these assertions, as is well documented in the Guidelines, are the considerable uncertainties in geological and hydrological parameters (properties) and spatial variations and correlati ons that must be accommodated to obtain meaningful quantitative assessments of the adequacy of rock slopes. A me thod of ana lysis is off ered to meet these needs. Its o utputs are the probabilit y of failure (PoF) a nd reli ability (R = 1 - PoF), the necessary ingredients for quantifying risk. The procedur e can account for the contributi ons of many uncertain (random) parameters , whethe r correlated or n ot. What is m ore, its versa tility can provide reliab ility me asures for s tate of the art eq uations, graphs and char ts or computer gene rated studies. Its flexibili ty permit s the evaluation an d comparison of many po ssible sce narios. Many exam ples are presented to provide b ackground a nd to illus trate the m ethodo logy at the lev el of the practitio ner. It will be assumed that the reader is familiar with Appendix 2 of the Guidelines. Reference to equations and figur es in the Guidelines will use the chapter and number designation: for example, Figur e 6.43 will refer to the 43 rd figure in Chapter 6; likewise, eqn. 10.2 will designate the second equation in Chapter 10. Figures and equations in the present paper will use integer notation; Figure 9 and Table 9 will refer to the 9 th figure and table in the present pape r . AbstractM.E. Harr Professor Emeritus of Civil Engineering Purdue University SOME BASIC CONCEPTS Shown in Figure 1 are N derived factors of safety (FoS, see Tables 9.1 to 9.7) in selected ranges of values and their frequency of occurrence (fN). Note that the system is similar to a system of vertical forces on a beam: except, in the probabilistic case, the frequencies must sum to unity. Example 1 Given the following 10 tabular realizations for the factors of safety, obtain their expected value, variance, standard deviation, and coefficient of variation. FoS Occurrence Frequency , f 1.3 3 0.3 1.5 4 0.4 1.8 3 0.3 Sum 10 1.0 Figure 1– Example 1
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1Slope Stability Santiago Chile, November 2009
Quantif the reliability of RockSlopes
In the introduction to Guidelines for Open Pit Slope Design (Guidelines) (1), it is stated, “It is often no longer sufficient to present slopedesign in deterministic (factor of safety) terms….” It is also noted, “A measure is required so that mine executives have sufficientinformation and understanding to be able to establish acceptable levels of risk.” To these assertions, as is well documented in theGuidelines, are the considerable uncertainties in geological and hydrological parameters (properties) and spatial variations andcorrelations that must be accommodated to obtain meaningful quantitative assessments of the adequacy of rock slopes.
A method of analysis is offered to meet these needs. Its outputs are the probabilit y of failure (PoF) and reli ability (R = 1 - PoF),the necessary ingredients for quantifying risk. The procedure can account for the contributions of many uncertain (random)
parameters, whether correlated or not. What is more, its versatility can provide reliability me asures for s tate of the art equations, graphs and char ts or computer generated studies. Its flexibili ty permits the evaluation and comparison of many possible scenarios. Many exam ples are presented to provide b ackground and to illus trate the m ethodology at the level of the practitioner.It will be assumed that the reader is familiar with Appendix 2 of the Guidelines. Reference to equations and figures in the Guidelineswill use the chapter and number designation: for example, Figure 6.43 will refer to the 43rd figure in Chapter 6; likewise, eqn. 10.2will designate the second equation in Chapter 10. Figures and equations in the present paper will use integer notation; Figure 9and Table 9 will refer to the 9 th figure and table in the present pape r.
Abstract
M.E. Harr
Professor Emeritus of Civil
Engineering
Purdue University
SOME BASIC CONCEPTS
Shown in Figure 1 are N derived factors of safety (FoS, see Tables 9.1 to 9.7) in selected ranges of values and their frequency ofoccurrence (fN). Note that the system is similar to a system of vertical forces on a beam: except, in the probabilistic case, the frequenciesmust sum to unity.
Example 1Given the following 10 tabular realizations for the factors of safety, obtain their expected value, variance, standard deviation, and
coefficient of variation.
FoS Occurrence Frequency, f
1.3 3 0.3
1.5 4 0.4
1.8 3 0.3
Sum 10 1.0
Figure 1 – Example 1
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2Santiago Chile, November 2009 Slope Stability
Parameter Coefcient of variation, %
Friction angle Ø
Gravel 7Sand 12
c, strength parameter(cohesion) 40
Porosity 10
Specic gravity 2
Unit weight 3
Coefcient of permeability (240 at 80% saturation
to 90 at 100% saturation)
Example 2
Given the probabilistic parameters of the FoS in the previous example, obtain its probability distribution consistent to the principle of
maximum entropy.
In Table A2.2 it was shown that, given only the expected value and standard deviation of a distribution, the principle of maximum entropy
prescribes the symmetrical Normal (Gaussian) Distribution. Some examples were shown in Figure A2.4. Note that the transformation is made
from the discrete frequencies of Figure 1 to the continuous normal distribution, with probabilities measured as areas under the curve.
With the availability of computers today in the workplace, many probabilistic functions are readily and easily evaluated. Microsoft’s office
EXCEL functions will be used in the present paper. The EXCEL function for the normal distribution is:
+NORMDIST(x, expected value, standard deviation, true) (eqn. 1)
yielding the value for a normal distribution less than the variate x for the specified parameters. The word ‘true’ and the + sign
are required.
Example 3
Assuming a normal distribution and using the results of Example 1, obtain the probability that the factor of safety will lie between 1.30
+BETAINV(0.10, 4,4, 0.93,2.13) = 1.26. The normal distribution gave 1.27!
The variance of a distribution can also be expressed as the product of expectations
V[x] = E[(x - E[x])(x – E[x])]
Suppose x and y are two variables associated with the same event. Their relative dependency is given by their covariance:
cov[x , y] = E[(x – E[x])(y –E[y])] (eqn. 5)
The measure of their relationship is the correlation coefficient, defined as:
ρ = cov[x , y]/ σ[x]σ[y] (eqn. 6a)
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4Santiago Chile, November 2009 Slope Stability
The correlation coefficient satisfies the condition:
+1 ≤ ρ ≤ +1 (eqn. 6b)
Where ρ = ± 1 denote perfect correlations, all corresponding values will lie on the same straight line: ρ = -1 designates when onevariable decreases the other increases, ρ =0 indicates no linear relationship between the two. Some examples are shown in Figure 2.
Figure 2 - Some examples of the correlation coefficient.
The EXCEL function +CORREL(x1:x
N,y
1:y
N ) gives the correlation coefficient between columns of N corresponding x, y variates. The
subscripts indicate the spreadsheet ranges.
Example 8
Given the following data set arranged in an EXCEL spreadsheet as shown, obtain the correlation coefficient.
Row x y 2 1 56 3 3 28 4 5 20
5 7 16 6 9 14
+CORREL(x2:x6,y2:y6 ) = -0.883
As expected, the minus sign indicates that as x increases y decreases and that there is a strong negative correlation.
POINT ESTIMATE METHOD (PEM)
The expected value and the standard deviation supply information concerning the central tendency and scatter about that value of arandom (uncertain) variable. The beam analogy previously referred to in Figure 1, is shown again in Figure 3a. It is seen that the beam issupported by a unit force acting at E[x]. Rosenblueth (3), recommended the analogue beam, shown in Figure 3b, be supported by forcesof 1/2, acting at E[x] – σ[x] and E[x] + σ[x] ( the derivation can be found in Harr (2), p206).
Figure 3 - Continuously distributed vertical force system on a rigid beam (a), single support (b), two supports
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Example 9
The well-known coefficient of active earth pressure is K A = tan2(45 - Ø/2).
If E[Ø] =45o and V(Ø) = 12%, obtain the coefficient of variation of K A.
The standard deviation is σ[Ø] = 0.12(45°) = 5.4o. Hence, Ø- = 45 - 5.4 = 39.6o and Ø+ = 45 + 5.4 = 50.4o.
Substituting into the expression for K A yields K
A+ = 0.13 and K
A- = 0.22. It follows that:
E[K A] = 1/2 (0.13 + 0.22) =0.18 and
E[K 2 A] = 1/2 ( 0.017 + 0.048) = 0.033
From eqn. A2.8, V[K A] = 0.033 – 0.182 = 0.001 and σ[K
A] = 0.024.
Hence, V(K A ) = (0.024/0.18) 100 = 13.6%
Figure 4 shows the procedure followed to obtain the solution.
Figure 4 - Example 9.
Example 10
Using the results of the previous example, estimate P[KA ≤ 0.16 ].
Having only estimates of the expected value and standard deviation, the normal distribution will be employed with:
Given H = 14m, c= 15kPa, Ø = 30o , γ = 18kN/m3 , and FoS = 1.3, using Figure 6 find the effective bench face angle β. The solution
is indicated by the arrows on the figure to yield β = 40o.
Example 13
If the coefficients of variation of c and Ø are V(c) = 40% and V(Ø) = 10% and β = 40o , estimate the PoF and reliability R for the data
of Example 12 using the same Hoek and Bray chart in Figure 6, assume the variates are not correlated.
The spreadsheet solution is given in Figure 7, PoF = 5.4% and the reliability = 94.6%.
FoS(c,Ø) = FoS(±, ±)
Factors Units
E[c] 15 kPa
E[Ø] 30 degrees 0.52 radians
β 40 degrees 0.70 radians
V(c) = 40%, V(Ø) = 10%
γ 18 Mg/m3
H 14 m
σ [c] 6
σ [Ø] 3
E[c]+σ [c] 21
E[c]- σ [c] 9
E[Ø]+σ [Ø] 33 degrees 0.58 radians
E[Ø]-σ [Ø] 27 degrees 0.47 radians
Entering Figure 6 with the following values of c, Ø, β
c, Ø c/γHtanØ tanØ/FoS FoS FoS2
+ + 0.13 0.39 1.67 2.79
+ - 0.16 0.35 1.46 2.13
- + 0.05 0.52 1.25 1.56
- - 0.07 0.48 1.06 1.12
E[ ] = 1.36 1.90
V[FoS] = 0.050
σ [FoS] = 0.22
P[FoS ≤ 1] 5.4 %
Reliability, R 94.6 %
Figure 7 - Spreadsheet solution to Example 13
The point estimate method for more than two random variables is given in Chapter 4 of Harr (2, pp 218-220). For very many variables,
whether correlated or not, see Harr (5). Care should be taken in the latter publication (5), as a mathematical error was found that
telescopes through the paper.
A three point estimate solution can be used for functions of random variables that display minimum or maximum values within the range
of E[x] ± σ[x]. Instead of p- = 1/2 and p+ = 1/2, as was shown for two points, values are calculated at p- = 1/6, p
+= 1/6, acting at E[x]
- √ 3 σ[x] and E[x] + √ 3 σ[x], respectively, and po = 2/3, acting at E[x].
Example 14Obtain the three point solution for Example 9.
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9Slope Stability Santiago Chile, November 2009
From Example 9, K A = tan2(45 - Ø/2), with E[Ø] =45o and V(Ø) = 12%, σ[Ø]= 5.4, it follows:
E[Ø] + √ 3 σ[Ø ] = 54.35o; K A+ = 0.103 1/6 K
A+ = 0.017
E[Ø] - √ 3 σ[Ø ] = 35.65o; K A- = 0.264 1/6 K
A- = 0.044
E[Ø] = 45o K Ao
= 0.172 2/3 K Ao
= 0.115
E[K A] = 0.176
1/6 K 2 A = 0.002
1/6 K 2 A- = 0.012
2/3 K 2 Ao
= 0.020
E[K 2 A
] = 0. 034
Within round off accuracy, as the KA function is monotonic, the result is the same as in Example 9.
Test x1 x2 x3 dx1 dx2 dx3
1 2 1 2 -5.25 -6.67 -4.92
2 4 3 1 -3.25 -4.67 -5.92
3 6 7 5 -1.25 -0.67 -1.92
4 7 6 3 -0.25 -1.67 -3.92
5 5 4 7 -2.25 -3.67 0.08
6 8 9 6 0.75 1.33 -0.92
7 9 8 7 1.75 0.33 0.08
8 7 10 9 -0.25 2.33 2.08
9 10 11 11 2.75 3.33 4.08
10 8 9 8 0.75 1.33 1.08
11 12 11 10 4.75 3.33 3.08
12 9 13 14 1.75 5.33 7.08
E[xi] 7.25 7.67 6.92
Figure 8 - Tabular values of test results for three variates
Suppose that 12 tests are performed and sample measurements of variables x1, x2, and x3 yield the values in Figure 8. Also shown
are the sample expected values 7.25, 7.67, and 6.92. The columns labeled dx1, dx2, and dx3 are mean corrected; that is, the respective
expected values have been subtracted from the raw data (for example, for test 1, 2 – 7.25 = -5.25)
Designate a matrix D (2, pp 236-237 and Appendix A) composed of the 12 rows and 3 columns of dx1, dx2, and dx3. Performing the
matrix multiplication (DT is the transpose of D), produces the symmetrical covariance matrix C:
7.48 8.73 7.84
C = (1/11)DTD = 8.73 12.97 12.24
7.84 12.24 14.63
The elements on the principal diagonal are the respective variances of x1, x2, and x3 (7.48, 12.97, and 14.63). Hence the respectivestandard deviations are σ[x1] = 2.73, σ[x2] = 3.60, and σ[x3] = 3.82. The off-diagonal elements are the respective covariances (for
example, 8.73 = σ[x1]σ[x2]ρ1,2). Hence, the correlation coefficient between x1 and x2 is:
ρ1,2 = 8.73/(2.73)(3.60) = 0.89
Thus, it is seen that data such as in Figure 8 can readily be reduced to produce the expected values, standard deviations, and
correlation coefficients of random variables, the very fodder of probabilistic analysis. Computer software abound that yield the above
results directly.
CLOSURE
It was the intent of this paper to demonstrate how quantitative measures of reliability and probability can be obtained by a simple
yet very versatile methodology. It is believed that the wide use of examples in the paper has enhanced the reader’s ability to apply thedeveloped methodology to the design and evaluation of rock slopes.
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APPENDIX
E [x] = x ̄ (eqn. A2.6)
V [xi ] = E [xi2
] - ( E [xi ])2
(eqn. A2.8)
σ[xi] = (eqn. A2.9)
V(x) = x 100 (%) (eqn. A2.10)
α = (eqn. A2.23a)
Figure A2.4 - Examples of normal probability distribution funcions
Table IX.1 - Examples of acceptable FoS values (Pries & Brown 1983)
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11Slope Stability Santiago Chile, November 2009
H Chebyshev’s
inequality
Gauss’
inequality
Exponential
distribution
Normal
distribution
Uniform
distribution
0,5 0 0 0,780 0,380 0,29
1 0 0,56 0,860 0,680 0,58
2 0,75 0,89 0,950 0,960 1,00
3 0,89 0,95 0,982 0,997 1,00
4 0,94 0,97 0,993 0,999 1,00
Table A2.1 - Probabilities for range of expected values ± h sigma units
Table A2.2 - Maximum entropy probability distribution
REFERENCES
1. J.R.L. Read and P.F. Stacey, Guidelines for Open Pit Slope Design, CSIRO Publishing, Melbourne, Australia, 2009.
2. M.E. Harr, Reliability-based Design in Civil Engineering, Dover Publications, New York, 1987.3. E. Rosenblueth, “Point Estimates for Probability Moments”, Proc. Nat. Acad. Sci. USA, V72(2), 1975.
4. E. Hoek and J. Bray, Rock Slope Engineering (3rd edn.), IMM, London 1981.
5. M.E. Harr, “Probabilistic Estimate For Multivariate Analyses”, Appl. Math. Modelling, V13, May 1989.