Probabilistic methods applied to Geotechnical Engineering

W(H)YDOC 05 2nd International Workshop of Young Doctors in Geomechanics Paris, November 23-25, 2005 Ecole Nationale des Ponts et Chaussées

Probabilistic methods applied to Geotechnical Engineering

Hong Kong

Dipl. Ing. Consolata Russelli 1

Supervisor: Prof. Pieter A. Vermeer Co-Advisor: Prof. A. Bárdossy

Contents

1. Research motivation 2. Overview of the probabilistic analysis 3. The Point Estimate Method (PEM) 4. PEM application to geotechnical problems 5. Conclusions

2

1. Research motivation

1. Research motivation

2. Overview of the probabilistic analysis

3. The Point Estimate Method

4. PEM application to geotechnical problems

5. Conclusions

3

100% Uncertainty

1. Research motivation

1. Research motivation

2. Overview of the probabilistic analysis

3. The Point Estimate Method

4. PEM application to geotechnical problems

5. Conclusions

3

100% Uncertainty 95% Uncertainty

1. Research motivation

1. Research motivation

2. Overview of the probabilistic analysis

3. The Point Estimate Method

4. PEM application to geotechnical problems

5. Conclusions

3

100% Uncertainty 80% 95% Uncertainty Uncertainty

1. Research motivation

1. Research motivation

2. Overview of the probabilistic analysis

3. The Point Estimate Method

4. PEM application to geotechnical problems

5. Conclusions

3

100% Uncertainty 50% 80% 95% Uncertainty Uncertainty

1. Research motivation

1. Research motivation

2. Overview of the probabilistic analysis

3. The Point Estimate Method

4. PEM application to geotechnical problems

5. Conclusions

3

100% 0% Uncertainty 50% 80% 95% Uncertainty Uncertainty Uncertainty

Geotechnical uncertainties ¾ Geological anomalies ¾ Inherent spatial variability of soil properties 1. Research motivation

2. Overview of the probabilistic analysis

¾ Scarcity of representative data ¾ Changing environmental conditions ¾ Unexpected failure mechanisms ¾ Simplifications and approximations adopted in geotechnical models ¾ Human mistakes in design and construction

3. The Point Estimate Method

Deterministic analysis leads to extremely conservative design with significant failure probability :

4. PEM application to geotechnical problems

unable to account for uncertainties in material and load properties.

5. Conclusions

4

Colorado

How to deal with uncertainties?

“Uncertainty is inevitable” 1. Research motivation

2. Overview of the probabilistic analysis

3. The Point Estimate Method

4. PEM application to geotechnical problems

5. Conclusions

5

Lack of perfect knowledge concerning phenomena and processes involved in problem definition and resolution.

Implementation of probabilistic analysis required - uncertainties rationally quantified and systematically incorporated into the design process,

- means to evaluate uncertainties influence on the likelihood of satisfactory performance for an engineering system.

La Conchita, California

Reluctance in adopting probabilistic analysis

1. Research motivation

1. Engineers´ training in probability theory often limited to basic information during their early years of education.

2. Overview of the probabilistic analysis

3. The Point Estimate Method

4. PEM application to geotechnical problems

5. Conclusions

6

2. Less comfortable dealing with probabilities than with deterministic analysis. 3. Common misconception that it requires significantly more data, time and effort. 4. Few published studies illustrate its implementation and benefits.

Deterministic analysis and probabilistic approach as complementary measures of acceptable design !!!

2. Overview of the probabilistic analyses

Choice of geotechnical problems 1. Research motivation

3. The Point Estimate Method

4. PEM application to geotechnical problems

5. Conclusions

7

Application of probabilistic methods

Results in terms of statistics values and probability distribution function probability density function of performance function g(xi)

2. Overview of the probabilistic analysis

Comparison with Monte Carlo Method

µg(x)

f[g(xi)]

? σg ( x ) σg ( x )

νg(x)

g(xi)

Probabilistic methods analysed

1. Research motivation

2. Overview of the probabilistic analysis

3. The Point Estimate Method

4. PEM application to geotechnical problems

5. Conclusions

8

¾ The First Order Second Moment Method (FOSM) ¾ The Second Order Second Moment Method (SOSM)

¾ The Point Estimate Method (PEM) ¾ Monte Carlo Simulations (MC)

3. The Point Estimate Method (Rosenblueth, 1975)

Computationally straightforward technique for uncertainty analysis: 1. Research motivation

2. Overview of the probabilistic analysis

3. The Point Estimate Method

4. PEM application to geotechnical problems

5. Conclusions

9

capable of estimating statistical values of a model output involving several stochastic variables, correlated or uncorrelated, symmetric or non-symmetric.

Weighted average method similar to numerical integration formulas involving “sampling points” and “weighting parameters”.

¾ Requires little knowledge of probability concepts and applies for any probabilistic distribution. ¾ Widely applied for reliability analysis and evaluation of failure probability.

Aim:

replace probability distributions for continuous random variables with discrete equivalent functions having the same mean value, standard deviation and skewness coefficient!

Procedure for implementing the PEM

2. Overview of the probabilistic analysis

3. The Point Estimate Method

4. PEM application to geotechnical problems

5. Conclusions

10

Consider a relationship between performance function f(Xi) and input random variables.

2.

Compute locations of sampling points (2n calculations) :

ξXi +

ν X i  = + 1+  2 

 ν Xi   2

ξ Xi − = ξ Xi + − ν Xi

x i − = µ Xi − ξ X i − ⋅ σ Xi x i + = µ Xi + ξ Xi + ⋅ σ Xi

  

2

   

1

2

probability density function

1. Research motivation

1.

f(tanφ´)

Ptanφ´-

µ tan ϕ′ − ξtan ϕ′− ⋅ σtan ϕ′

Ptanφ´+

tanφ´ µtanφ´

µtan ϕ′ + ξtan ϕ′+ ⋅ σtan ϕ′

Procedure for implementing the PEM

1. Research motivation

1.

Consider a relationship between performance function f(Xi) and input random variables.

2.

Compute locations of sampling points (2n calculations) :

ξXi +

2. Overview of the probabilistic analysis

 ν Xi   2

  

2

   

1

2

x i − = µ Xi − ξ X i − ⋅ σ Xi x i + = µ Xi + ξ Xi + ⋅ σ Xi (e.g. cohesion)

3.

Determine the weights Pi (probability concentrations) to obtain all the point estimates. single random variable :

5. Conclusions

10

ν X i  = + 1+  2 

ξ Xi − = ξ Xi + − ν Xi

3. The Point Estimate Method

4. PEM application to geotechnical problems

(e.g. friction angle)

f(X,Y)

PX i + =

ξxi − ξxi + + ξxi −

PX i − = 1 − PXi +

associated weights:

Ps1s 2 = PXs1 ⋅ PXs 2

Procedure for implementing the PEM 4. Determine the performance function value f(Xi) at each sampling point locations. 1. Research motivation

2. Overview of the probabilistic analysis

Sign

Pi 0.043 0.457 0.043 0.457

++ +-+ --

tan( ϕ′ ) 0.523 0.409 0.409 0.523

c´ 14.647 kPa 3.003 kPa 14.647 kPa 3.003 kPa

qf 653.548 kPa 194.171 kPa 391.703 kPa 365.685 kPa

5. Determine the first three moments of the performance function: 3. The Point Estimate Method

4. PEM application to geotechnical problems

5. Conclusions

11

µ f (X i ) =

σ

2

f (X i )

=

i

i

i =1

∑ P ⋅ (f (X ) − µ ( ) )

1 σ

∑ P ⋅ f (X )

2n

2

i =1

ν f (Xi ) =

2n

i

i

f Xi

2n

∑ P ⋅ (f (X ) − µ ( ) )

3 f (Xi ) i = 1

3

i

i

f Xi

4. PEM application to geotechnical problems

1. Research motivation

2. Overview of the probabilistic analysis

3. The Point Estimate Method

4. PEM application to geotechnical problems

5. Conclusions

12

Terzaghi´s Bearing Capacity: shallow foundation on a cohesive homogeneous soil - PEM with correlated input random variables. - PEM with uncorrelated input random variables. - Results comparison.

Slope Stability Analysis: fill embankment on undrained clay - PEM with uncorrelated input random variables. - Results comparison.

Terzaghi´s bearing capacity B=2m

qf

q

1. Research motivation

q f = c′ ⋅ N c + q ⋅ N q +

1 ⋅ γ ⋅ B ⋅ Nγ 2

q = 10 kPa

2. Overview of the probabilistic analysis

5. Conclusions

13

Gaussian (normal) distribution

µ tan ϕ′ = 0.47 σ tan ϕ′ = 0.06

f(tanφ´)

ν tan ϕ′ = 0 5% fractile 0.36

0.47

0.58

tan φ´

Lognormal distribution

probability density function

4. PEM application to geotechnical problems

γ = 15 kN/m3

probability density function

3. The Point Estimate Method

q

µ c′ = 4 kPa σ c′ = 3.3 kPa

f(c´)

ν c′ = 3.1

5% fractile 0

4

10

Cohesion c´ (kPa)

PEM results with uncorrelated ( ρ tan ϕ′ c′ = 0 ) input variables µ q f (kPa ) 305.39 1. Research motivation

0,0035

0,003

0,0025

0,002

0,0015

0,0005

14

σqf

50

75 10 0 12 5 15 0 17 5 20 0 22 5 25 0 27 5 30 0 32 0 34 5 37 0 39 5 42 0 44 5 47 0 49 5 52 0 54 5 57 5 60 0 65 0 90 0 11 50 14 00 16 50 19 00

µqf

0 0, 00

5. Conclusions

σqf

0,001

1

4. PEM application to geotechnical problems

COVq f 0.36

0,004

25

3. The Point Estimate Method

ν qf 1.2

0,0045

Probability density function of qf

2. Overview of the probabilistic analysis

σ q f (kPa ) 110.48

Bearing capacity qf (kPa)

PEM results with correlated input variables Rosenblueth (1981) – Two random variables

1. Research motivation

2. Overview of the probabilistic analysis

3. The Point Estimate Method

4. PEM application to geotechnical problems

5. Conclusions

15

Ps1s 2 = PXs1 ⋅ PXs 2

  + s1 ⋅ s 2 ⋅  ρ X1X 2   

ρ (tanϕ′,c´)

Mean value (qf)

-1,0 -0,9 -0,8 -0,7 -0,6 -0,5 -0,4 -0,3 -0,2 -0,1 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

294.110 295.279 296.449 297.618 298.788 299.957 301.127 302.297 303.466 304.636 305.805 306.975 308.144 309.314 310.484 311.653 312.823 313.992 315.162 316.331 317.501

 /  1 +  

 ν X1   2

Standard deviation (qf) 34.755 46.193 55.290 63.067 69.966 76.225 81.991 87.361 92.406 97.174 101.706 106.032 110.175 114.157 117.992 121.695 125.278 128.751 132.122 135.399 138.589

  

3

   ⋅ 1 +    

 ν X2   2

Skewness (qf) 0.000 0.614 0.678 0.647 0.594 0.537 0.483 0.432 0.385 0.341 0.300 0.262 0.226 0.193 0.161 0.131 0.102 0.075 0.049 0.024 0.000

  

3

   

1 2

     

Influence of the correlation coefficient on PEM results of the bearing capacity

5. Conclusions

16

and probability values sligthly decrease. If ρ tan ϕ′c′ decreases then p.d.f . is narrower

0,008

ρ tan ϕ′c′ = 0

and probability values increase.

0,006

0,004

ρ tan ϕ′c′ = 1.0

0,002

90 12 0 15 0 18 0 21 0 24 0 27 0 30 0 32 5 35 5 38 5 41 5 44 5 47 5 50 5 53 5 57 0 60 0 70 0 10 00 13 00 16 00 19 00

0 60

4. PEM application to geotechnical problems

If ρ tan ϕ′c′ increseas then p.d.f . is wider

0,01

30

3. The Point Estimate Method

ρ tan ϕ′c′ = − 1.0

0, 00 1

2. Overview of the probabilistic analysis

0,012

Probability density function

1. Research motivation

Bearing capacity (kPa) c.c.= 0

c.c. = 0.1

c.c. = 0.2

c.c. = 0.3

c.c. = 0.4

c.c. = 0.5

c.c. = 0.6

c.c. = 0.7

c.c. = 0.8

c.c. = 0.9

c.c. = 1.0

c.c. = -0.1

c.c. = -0.2

c.c. = -0.3

c.c. = -0.4

c.c. = -0.5

c.c. = -0.6

c.c. = -0.7

c.c. = -0.8

c.c. = -0.9

c.c. -1.0

Comparison of PEM and MC results Method Monte Carlo ρ = 0 PEM ρ=0 PEM ρ = − 0.6

1. Research motivation

17

FS=2 0,005

Monte Carlo ρ = 0

0,004

PEM ρ = 0 0,003 0,002

deterministic mean value 0,001

Bearing capacity qf (kPa)

00

00

19

00

16

13

0

0 00 10

70

0

60

5

57

53

90 12 0 15 0 18 0 21 0 24 0 27 0 30 0 32 5 35 5 38 5 41 5 44 5 47 5 50 5

287.6 kPa

0 60

5. Conclusions

69.97

COVqf 0.38 0.36 0.59 0.23

PEM ρ = − 0.6

30

4. PEM application to geotechnical problems

σqf (kPa) νqf 115.94 1. 9 110.48 1. 2

0,006

0, 00 1

3. The Point Estimate Method

(kPa) 304.72 305.39 298.79

0,007

Probability density function

2. Overview of the probabilistic analysis

µqf

Slope stability analysis Bishop´s simplified method of slices n

1. Research motivation

FS =

∑ [ci ⋅ bi i =1

+ Wi ⋅ tan ϕi ]⋅ n

∑ W ⋅ sin α i =1

2. Overview of the probabilistic analysis

1 m α (i )

i

i

O (37.5 m, 19 m)

3. The Point Estimate Method

R = 37.5 m

6m FILL 4. PEM application to geotechnical problems

5. Conclusions

18

4m

CRUST

8m

MARINE CLAY

6,5 m

LACUSTRINE CLAY

Source: J.T. Christian, C.C. Ladd, G.B. Baecher; „Realiability and Probability in Slope Stability Analysis“

Input uncorrelated soil parameters

Crust

5. Conclusions

27°

30°

40

γ f(φ)

19

33° Fill friction angle (°)

cu f(φ)

50 Crust cohesion (kPa)

probability density function of marine clay unit weight

γ = 20 . 31 kN / m 3 c u = 31.2 kPa

probability density function of fill unit weight

Lacustrine Clay

f(φ)

30

19

γ = 18 . 81 kN / m 3 c u = 34.5 kPa

probability density function of crust unit weight

4. PEM application to geotechnical problems

probability density function of crust cohesion

3. The Point Estimate Method

probability density function of fill friction angle

2. Overview of the probabilistic analysis

Marine Clay

γ f(φ)

17.9

19.8 18.81 Crust unit weight (kN/m3)

Coefficient of Variation 0.071 0.1 0.049 0.25 0.049 0.2 0.049 0.32

γ f(φ)

17.9

20 21 Fill unit weight (kN/m3)

18.81

19.8

cu f(φ)

27.5

Marine clay unit weight (kN/m3)

probability density function of lacustrine clay cohesion

1. Research motivation

Standard deviation 1 3 0.94 10 0.94 6 89 0.99 9 98

probability density function of marine clay cohesion

Fill

Mean Value γ = 20 kN / m 3 ϕ´u = 30 ° γ = 18 . 81 kN / m 3 c u = 40 kPa

probability density function of lacustrine clay unit weight

Layer

cu f(φ)

21.5

31.2

40.9

Lacustrine clay cohesion (kPa)

34.5 41.5 Marine clay cohesion (kPa)

γ f(φ)

19.3 20.31 21.3 Lacustrine clay unit weight (kN/m3)

Comparison of PEM and MC results

1. Research motivation

5. Conclusions

20

COVFS

ν FS

Monte Carlo PEM

1.462 1.535

0.279 0.286

0.191 0.186

0.012 0.00004

1,2

deterministic mean value 1 0,8 0,6

Gaussian fit

0,4 0,2

1.532

Factor of safety Monte Carlo

PEM (Rosenblueth, 1975)

2,97

2,86

2,75

2,64

2,53

2,42

2,31

2,2

2,09

1,98

1,87

1,76

1,65

1,54

1,43

1,32

1,1

1,21

0,99

0,88

0,77

0,66

0,55

0,44

0,33

0,22

0 0,11

4. PEM application to geotechnical problems

σ FS

1,4

0

3. The Point Estimate Method

µ FS

1,6

Probability density function

2. Overview of the probabilistic analysis

Method applied

5. Conclusions PEM advantages vs. MC simulations

1. Research motivation

2. Overview of the probabilistic analysis

3. The Point Estimate Method

4. PEM application to geotechnical problems

5. Conclusions

21

Reasonably robust and satisfactorily accurate for a wide range of practical problems !

¾ Results as reliable and accurate as MC simulations. ¾ Smaller computational effort for a comparable degree of accuracy ¾ No need of knowledge of p.d.f. shape of input random variables. ¾ Behaviour of non-linear function well captured.

5. Conclusions How to cope with PEM drawbacks ¾ Performance function p.d.f. to be assumed, thus introducing uncertainty. In Soil Mechanics normal and lognormal distributions frequently result as 1. Research motivation

2. Overview of the probabilistic analysis

output of probabilistic analysis.

¾ If more accuracy required than larger number of input variables necessary, i.e. number of required evaluations too high to be implemented practically. Rosenblueth approximation method (1981) for Gaussian distributed uncorrelated input

3. The Point Estimate Method

4. PEM application to geotechnical problems

5. Conclusions

22

variables.

¾ Results poor and not accurate for discontinuous functions or functions having discontinuous first derivatives and for large COVs of input variables. Typical geotechnical problems described by continuous functions, whose “non-linearity” not difficult to be treated by PEM. Small COV values in geotechnical literature, frequently lower than the unity.

5. Conclusions Final observations ¾ Different probabilistic methods applied to geotechnical problems to find out the most suitable one for the geotechnical field and for a further analysis. 1. Research motivation

2. Overview of the probabilistic analysis

3. The Point Estimate Method

4. PEM application to geotechnical problems

5. Conclusions

¾ PEM gave as reliable and accurate results as MC with less computational effort. ¾ It was easily applied to a multivariate problem.

¾ Output as a lower bound for the evaluation of failure probability, because effects of other factors not included in the analysis. ¾ Spatial variability ignored to simplify the autocorrelation, obtaining more conservative results.

calculations

assuming

perfect

PEM: simple, but powerful technique for uncertainty analysis. Its use in geotechnical reliability analysis justified by experience and theory.

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Thanks for your attention ! For further clarifications and discussion, please contact [email protected] Bainbridge Island, WA USA

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