frequ

ency

[%]

effective friction angle φ´0 5 10 15 20 25 30 35 40

40

35

30

25

20

15

10

5

0

074.574.20

≈ν

°=σ

°=µ

ϕ′

ϕ′

ϕ′

ϕ′µ

ϕ′σ

Gaussian distribution

ϕ′σ

frequ

ency

[%]

effective friction angle φ´0 5 10 15 20 25 30 35 40

40

35

30

25

20

15

10

5

0

074.574.20

≈ν

°=σ

°=µ

ϕ′

ϕ′

ϕ′

ϕ′µ

ϕ′σ

Gaussian distribution

ϕ′σ frequ

ency

[%]

effective cohesion c´0 10 20 30 40 50 60 70 80 90

25

20

15

10

5

0

213.0kPa47.20

kPa66.39

c

c

c

=ν=σ=µ

′

′

′

c′µ

ϕ′σ ϕ′σ

Lognormal distribution

frequ

ency

[%]

effective cohesion c´0 10 20 30 40 50 60 70 80 90

25

20

15

10

5

0

213.0kPa47.20

kPa66.39

c

c

c

=ν=σ=µ

′

′

′

c′µ

ϕ′σ ϕ′σ

Lognormal distribution

1.7

PROBABILISTIC METHODS APPLIED TO GEOTECHNICAL ENGINEERING

Consolata Russelli (russelli@igs.uni-stuttgart.de), Pieter A. Vermeer

Institute of Geotechnical Engineering, University of Stuttgart, S-Vaihingen, Germany

ABSTRACT Geotechnical problems are often dominated by uncertainty, such as inherent spatial variability of soil properties or scarcity of representative data. Engineers try to solve these problems by deterministic calculation, but there maybe so many uncertainties that even advanced deterministic methods become useless. Therefore probabilistic methods are applied. This paper describes the probabilistic Point Estimate Method and shows its application in the field of geotechnical engineering. The bearing capacity study of a shallow foundation is considered.

1. Introduction

Geotechnical variability results from different sources of uncertainties. The three primary sources are inherent variability, measurement error and model uncertainty. Inherent variability results primarily from natural geologic processes that created in situ soil layers. Measurement error is caused by sampling and lab testing. This error is increased by statistical uncertainty that arises from limited amount of information. Finally the model uncertainty is introduced when field or laboratory measurements are transformed into input parameters for design models involving simplifications and idealisations.

For the already mentioned geotechnical problem homogeneous soil layers are considered, whose shear strength parameters are described by given probability distribution functions. The Gaussian (normal) function is the simplest and best known distribution function (see figure 1a). However in geotechnical engineering the distribution of parameters around the mean value would not always seem to be symmetrical as suggested by the Gaussian distribution. Especially the soil cohesion seems to be better approximated by the asymmetrical lognormal function, as indicated in figure 1b.

The normal and lognormal distributions are characterised by the mean value µX, standard deviation σX and skewness coefficient νX. Considering a certain random variable x, as the soil cohesion or friction angle, and its corresponding distribution function f(x), then it yields:

∫+∞

∞−

⋅=µ dx)x(fxX dx)x(f)x( 2XX

2 ∫+∞

∞−

⋅µ−=σ ( )

dx)x(fx3X

3X

X ⋅σµ−

=ν ∫∞+

∞−

Instead of specifying the standard deviation, one often gives the coefficient of variation, which is

the ratio of the standard deviation over the mean value, i.e. COV = σX / µX. Phoon and Kulhawy (1999) presented data for a sand and a clay layer to find values between 5-15% for the effective friction angle. This range was also put forward by Harr (1987). Occasionally higher COV-values can be found, as in figure 1a, where its value is about 30%. Considering the effective cohesion, Harr suggests a value of about 20%, Li and Lumb (1987) report a particular clay layer with 40% and Moormann and Katzenbach (2000) find 50% for Frankfurt clay as indicated in figure 1b. It can thus be concluded that the available data on the effective cohesion show much more variation then the data of the effective friction angle.

Figure 1. Probability density functions of strength parameters of Frankfurt Clay (Moormann & Katzenbach, 2000)

(a) (b)

The skewness coefficient describes the degree of asymmetry of a distribution function. When the skewness coefficient is equal to zero then a function is symmetric, as in the case of the Gaussian approximation of the friction angle of Frankfurt clay, as shown in figure 1a. Figure 1b shows the case of a positively skewed lognormal distribution which is steep for low values of the cohesion and flat for large values. A negatively skewed distribution will be flat for low values of the soil parameter and steep for large values. However negative skewnesses would seem to be unrealistic for the distribution of soil parameters. Considering data as reported by Moormann & Katzenbach and El Ramly et al. (2005), it would seem that the skewness coefficient of the effective friction angle could be disregarded and the choice of a Gaussian distribution would seem suitable. On the other hand, the above authors find skewness of νc ≈1.7 (Moormann & Katzenbach) and 4 (El Ramly et al.) for effective cohesion, thus a lognormal distribution would seem to be appropriate. Anyway very little data about the skewness coefficient are available in literature and the skewness values would need further testing. Some authors have based their probabilistic studies on the assumption that soil cohesion and friction angle are uncorrelated, but this is often done to simplify calculations. Lumb (1969) was probably the first to study the correlation between soil cohesion and friction angle on the basis of experimental data. He found a negative correlation implying that low values of cohesion are associated with high values of the tangent of friction angle, and vice versa. However in some cases the correlation was found to be insignificant. He concluded that the assumption of independence of the strength parameters simplifies strength interpretation considerably, and also leads to conservative results if the correlation is in fact negative. In contrast the results of Cherubini (1998) indicate a significant negative correlation between effective cohesion and friction angle. Similarly Schad (1985) reported a strong negative correlation. In order to incorporate a dependence between the strength parameters one needs input data on the so-called correlation coefficient, denoted as ρ. Both Cherubini and Schad find ρ=-0.6, whereas Lump reported values in the range –0.3<ρ<-0.7. Hence it would seem that a value of about -0.6 is realistic. In this presentation the strength parameters are assumed to be negatively correlated. It will be shown that negative correlation coefficient decreases the standard deviation of computational results, such as bearing capacity of a footing or safety factor of a slope. It can be concluded that it seems to be interesting to incorporate a negative correlation for strength parameters in the analysis.

2. Bearing capacity of a shallow foundation

The bearing capacity problem refers to a shallow foundation with a width of 2m on top of a homogeneous soil layer. The unit soil weight is 15 kN/m3 and an initial surcharge 10 kPa is considered. The effective friction angle and cohesion are assumed respectively as normally and lognormally distributed variables with their corresponding mean value, standard deviation and skewness coefficient, i.e.: µc´=4 kPa, σc´=3.3 kPa, νc´=3.1 and µϕ´=25°, σϕ´=2.6°, νϕ´=0. The bearing capacity, denoted as qf, from Brinch-Hansen´s formula gives a deterministic value of 287.6 kPa.

Initially independent input parameters are considered and Monte Carlo simulations are applied. In Monte Carlo simulations the accuracy of the computed function depends on the number of realizations as carried out for each input parameters. 1000 realizations were carried out in order to obtain a nearly exact solution. Table 1 shows the results of the Monte Carlo method with a mean bearing capacity of 304.72 kPa, which differs only 6% from the deterministic value. For practice Monte Carlo simulations are computationally too time consuming, therefore alternative methods have to be considered, such as the Point Estimate Method (Rosenblueth, 1981).

With the Point Estimate Method the distribution function of the shear strength parameters are discretised by choosing some sampling points and the mean value, standard deviation and skewness of the bearing capacity can be computed. The determination of these values is realized through the weights sum of every discrete realizations (sampling points). The results of the Point Estimate Method considering uncorrelated strength parameters are listed in table 1. When these results are compared to those of Monte Carlo simulations, it can be observed that the mean values and standard deviation are very similar; on the other hand there is a significant difference between the skewness coefficients. The Point Estimate Method doesn’t give the entirely probability density function of the bearing capacity as the Monte Carlo method does. Thus assuming a lognormal distribution, the probability density function can be plotted, as figure 2 shows. Despite the different skewness, the curves are practically coincident. Hence the Point Estimate Method apparently seems to be insensitive to the skewness and virtually gives the same curve as the computationally no-attractive Monte Carlo simulations.

Additionally in table 1 the results from the Point Estimate Method are listed for a correlation coefficient of –0.6. For this case the mean value of the bearing capacity hardly changes, but the standard deviation strongly decreases, down to 69.97 kPa. Thus reducing the uncertainty in qf.

0

0,001

0,002

0,003

0,004

0,005

0,006

0,007

0,001 55 11

016

522

027

532

538

043

549

054

560

510

0015

50

Bearing capacity qf (kPa)

Prob

abili

ty d

ensi

ty

FS=2

Brinch-Hansen

PEM ρ=0

PEM ρ=- 0.6

Monte Carlo ρ=0

0

0,001

0,002

0,003

0,004

0,005

0,006

0,007

0,001 55 11

016

522

027

532

538

043

549

054

560

510

0015

50

Bearing capacity qf (kPa)

Prob

abili

ty d

ensi

ty

0

0,001

0,002

0,003

0,004

0,005

0,006

0,007

0,001 55 11

016

522

027

532

538

043

549

054

560

510

0015

50

Bearing capacity qf (kPa)

0

0,001

0,002

0,003

0,004

0,005

0,006

0,007

0,001 55 11

016

522

027

532

538

043

549

054

560

510

0015

50

Bearing capacity qf (kPa)

Prob

abili

ty d

ensi

ty

FS=2

Brinch-Hansen

PEM ρ=0

PEM ρ=- 0.6

Monte Carlo ρ=0

As a consequence also the skewness coefficient and the COV decrease. Hence the bearing

capacity density function becomes narrower, as shown in figure 2. In many countries a safety factor, denoted as FS, of two on the bearing capacity used to be

applied. In figure 2 the line for FS=2 is also plotted. On neglecting the correlation coefficient for the strength parameters the probabilistic analysis, i.e. for ρ=0, the results show that a safety factor of two still involves a considerable risk of failure. However on introducing a realistic negative correlation of –0.6 the variation of the bearing capacity decreases considerably and this matches well with the use of a safety factor equal two.

3. Conclusions

From the probabilistic analysis of the bearing capacity problem it is possible to compare advantages and disadvantages of well-known probabilistic methods. In spite of some limitations, the Point Estimate Method shows to be a simple, but powerful technique as it matches Monte Carlo simulations extremely well. The Point Estimate Method will be used for a further study as it is much faster than Monte Carlo simulations. As an alternative to the Point Estimate Method, the First Order Second Moment method was applied, which gave results similar to those of the Point Estimate Method. However it doesn’t allow the evaluation of the skewness, thus being substantially less accurate.

From figure 2 an enormous variation for the bearing capacity can be observed, when neglecting the correlation between the strength parameters. These results would seem representative for many soil problems as typical input data for the strength parameters, including the coefficients of variation, have been applied. It would seem very important to do probabilistic analysis and to use a negative correlation. A slope stability analysis was also executed, which confirmed the advantages of the Point Estmate Method. However additional case studies would be required for a more definitive judgement.

4. References

Cherubini, C. (1998). Reliability of shallow foundation bearing capacity on c´, ϕ´ soils. Canadian Geotechnical Journal, 37: 264-269.

El-Ramly, H., Morgenstern, N.R., Cruden, D.M. (2005). Probabilistic assessment of stability of a cut slope in residual soil. Géotechnique, 55: 77-84.

Harr, M.E. (1987). Reliability-based design in civil engineering. McGraw-Hill, New York. Li, K.S., Lumb, P. (1987). Probabilistic design of slopes. Canadian Geotechnical Journal, 24: 520-535. Lumb, P. (1969). Safety Factors and the Probability Distribution of Soil Strength. Canadian Geotechnical Journal,

7(3): 225-242. Moormann, C., Katzenbach, R. (2003). Überlegungen zu stochastischen Methoden in der Bodenmechanik am

Peispiel des Frankfurter Tons. Heft 16 der Gruppe Geotechnik Graz. Technische Universität Graz. Phoon, K.-K., Kulhawy, F. H. (1999). Characterization of geotechnical variability. Can. Geotech. J., 36: 612-624.

Figure 2. Probability density functions of the bearingcapacity from the Monte Carlo and the Point Estimate

Table 1. Bearing capacity results predicted bythe Monte Carlo and the Point Estimate

.

. 0=ρ

Method (kPa) (kPa) COVqf

Monte Carlo 304.72 115.94 1 9 0.38PEM 305.39 110.48 1 2 0.36PEM 298.79 69.97 0.59 0.23

fqσfqµ fqν

0=ρ6.0−=ρ

306.92 113.65 0.371.3

Rosenblueth, E. (1981). Two-point estimates in probabilities. Appl. Math. Modelling, 5(2): 329-335. Schad, H., Wallrauch, E. (1985). Rutschungen im Keuper. Otto-Graf Institute, Stuttgart University (MPA).