A presentation Soumaya Addou a Master student in Tohoku University made about Risk Assessment in Geotechnical Engineering during meeting of Risk commission, that is part of the Japanese Geotechnical Society - Tohoku branch.

1. Risk Assessment in
Geotechnical Engineering
Presented by : Soumaya Addou スマイヤ アッドー
東北大学
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2. References
This presentation is made based on the information provided by mainly these two books
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3. 3
Outline
I. Introduction of concepts
1) Uncertainty and risk in Geotechnical engineering
2) Probability Theory and Random Variables
3) Random Process Models
4) Definition of Risk
II. Uncertainty in Geotechnical Context
1) Site Characterization
2) Soil Variability
3) Spatial Variability within homogeneous Deposits
III.Reliability analysis Methods
1) Introduction: Steps and Approximations
2) Event Tree Analysis
3) First Order Second Moment Method (FOSM)
4) First Order Reliability Method (FORM)
5) Monte Carle Simulation

4. I. Introduction of concepts
Most of the early pioneers in Geotechnical Engineering were aware of the limitations of
purely rational, deductive approaches to the uncertain conditions that prevail in the
Geological world. Their later writings are full of warnings not to take the results of
laboratory tests and analytical calculations too literally
Recently, there has been a trend to apply the results of reliability theory to Geotechnical
engineering. The offshore and nuclear power are at the forefront for the use of these
approaches.
The variability inherent in soils and rocks suggests that geotechnical systems are highly
amenable to a statistical interpretation.
① Uncertainty and risk in Geotechnical engineering
4

5. Risk analysis
Natural
variability
Temporal Spatial
Knowledge
uncertainty
Model
Site
characterization
Parameters
Decision model
uncertainty
Objectives Values
Time
Preferences
I. Introduction of concepts
The types of uncertainties that arise in Engineering practice :
Engineering data on soil or rock mass properties are usually scattered
Usage of statistics and graphical and simple probabilistic methods
① Uncertainty and risk in Geotechnical engineering
5

6. I. Introduction of concepts
The mathematical theory of probability deals
with:
- Experiments “random process generating
specific and a priori unknown results”
- Their outcomes “sample space”
In Geotechnical Engineering, we mostly deal with probability as a density function and
Probability is found by integrating the probability mass over a finite region.
𝑃 𝐴 =
𝐴
𝑓𝑋 𝑥 𝑑𝑥
It is convenient sometimes to represent probability by their moments
𝐸 𝑥 𝑛
=
−∞
+∞
𝑥 𝑛
𝑓𝑋 𝑥 𝑑𝑥
The most common is the second central moment , called the variance
𝜎2
= 𝐸 𝑥 − 𝐸(𝑥) 2
𝐸 𝑥 is the arithmetic average called the mean.
② Probability Theory and Random Variables
6

7. I. Introduction of concepts
For an uncertain quantity , various forms for the Probability Functions have been suggested :
- Probability Mass Function (pmf) :
Binomial (success and failures) : 𝐹 𝑥 𝑛 = 𝑥
𝑛
𝑝 𝑥
(1 − 𝑝) 𝑛−𝑥
Poisson distribution : 𝑓 𝑥 λ =
λ 𝑥 𝑒−λ
𝑥!
….etc
- Probability Distribution Function (pdf):
Exponential distribution : 𝑓 𝑠 λ = λ𝑒−λ𝑠
The Normal Probability Distribution
….etc
http://slideplayer.com/slide/5710846/
③ Random Process Models
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8. I. Introduction of concepts
The determinant of risk is the combination of uncertain event and the adverse
consequence
𝑅𝑖𝑠𝑘 = (𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦, 𝐶𝑜𝑛𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒)
④ Definition of Risk
Many approaches have been adopted to describe
risks.
They consist on plotting the exceedance
probability of risks against their associated
consequences.
Chart showing average annuals risks posed by a variety of
traditional civil facilities and other large structures .
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9. II. Uncertainty in Geotechnical Context
① Site Characterization
Concerns information about the geometry and material properties of local geological formations,
mainly :
 The geological nature of deposits and formations
 Location, thickness and material composition
 Engineering properties of formations
 Ground water level and its fluctuations
The random process models
used are usually models of
“Spatial variation”
Probability is in the model not the ground 9

10. II. Uncertainty in Geotechnical Context
② Soil Variability
The variability of elementary soil properties concerns various categories of physical properties :
- Index and classification properties : bulk properties, classification properties, …etc
- Consolidation properties: 𝐶𝑐, 𝐶𝑟, 𝑐 𝑣…etc
- Permeability: hydraulic conductivity
- Strength Properties: CPT, SPT parameters, effective friction angle, …etc
Variability in soil properties is inextricably related to the particular site
and to a specific regional geology
Parameter Soil Recorded COV (%) Source
𝐶𝑐, 𝐶𝑟
𝑐 𝑣
Bangkok Clay
Various
Dredge Spoils
Gulf of Mexico Clay
Ariake Clay
Singapore Clay
Bangkok clay
20
25-50
35
25-28
10
17
16
Zhu et al. (2001)
Lumb (1974)
Thevanayagam et al (1996)
Baecher and Ladd (1997)
Tanaka et al. (2001)
Tanaka et al. (2001)
Tanaka et al. (2001)
Values of the variability in consolidation parameter, expressed as Coefficient of Variation
10

11. II. Uncertainty in Geotechnical Context
3) Spatial Variability within homogeneous Deposits
 Describing the variation of soil properties in space requires additional tools
 In order to characterize the spatial variation of a soil deposit, a large number of tests is required
Use of a model 𝑧(𝑥) = 𝑡 𝑥 + 𝑢(𝑥)
Soil property
at location x
Trend at x
deterministic
residual variation at x
“random variable”
Estimate the trend by fitting well-
defined mathematical functions to
data points
Use of methods like “Regression
analysis”
Fitting the same data with a line versus a curve changes the residual variance11

12. II. Uncertainty in Geotechnical Context
3) Spatial Variability within homogeneous Deposits
The spatial association of residuals off the trend is expressed by a mathematical function that
describes the correlation of two residuals separated by a distance 𝛿, this description is called the
autocorrelation function.
𝑅 𝑧(𝛿) =
𝐶𝑜𝑣(𝑢(𝑥𝑖), 𝑢(𝑥𝑗))
𝑉𝑎𝑟 𝑢(𝑥)
𝑉𝑎𝑟 𝑢(𝑥) : The variance of the residuals across the site
Autocorrelation of rock fracture density in a copper porphyry deposit
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13. III. Reliability analysis Methods
① Introduction: Steps and Approximations
Reliability analysis deals with the relation between the loads “Q” a system carry, and its ability to
carry those loads “R”.
The goal of the analysis is to estimate the probability of failure 𝒑 𝒇, the steps are :
1. Establish an analytical model
2. Estimate statistical descriptions of the parameters
3. Calculate statistical moments of the performance function
4. Calculate the reliability index
5. Compute the probability of failure
I. First Order Second Moment Method (FOSM)
II. First Order Reliability Method (FORM)
III. Monte Carle Simulation
…..etc 13

14. III. Reliability analysis Methods
② Event Tree Analysis
A graphical representation of the many chains of events that might result from some initiating event.
Its objective is to provide the Probability of system failure.
Example of event tree of the probability of embankment breach of a dam due to liquefaction
The event tree
begins with an
accident initiating
event : Earthquake,
flood,….etc
A joint probability is obtained by multiplying the conditional event probabilities along the chain
14

15. III. Reliability analysis Methods
③ First Order Second Moment Method (FOSM)
It uses the first terms of a Taylor series expansion of the performance function “F” to estimate the
expected value and variance of the performance function. When the variables are uncorrelated
Example : The James Bay Dikes
“Reliability Applied to Slope Stability Analysis” John T. Christian; Charles C. Ladd, and Gregory B. Baecher, 1994.
Uncertainties
in soil
properties
Scatter
- Spatial Variability
- noise
𝛼𝑐 𝑢 𝐹𝑉 = 𝑐 𝑢 + 𝑐 𝑒
Systematic
error
- Limited number of tests
- Bias :
Ex : The factor α is a
function of the plasticity
index. It is taken 𝛼 = 1
𝑐 𝑒 is a random experimental error.
Should not be included in stability analysis
to be found by “Autocovariance function” 15

16. Identify all
the variables
Determine the best
estimate of each
variable (The mean)
and the best estimate
of the factor of Safety
Estimate the
uncertainty
(the
variance)
Calculate the
partial
derivatives
∆𝐹
∆𝑋𝑖
Obtain
𝑉𝑎𝑟 𝐹
Calculate 𝛽
then
Probability
of failure 𝑝 𝑓
III. Reliability analysis Methods
③ First Order Second Moment Method (FOSM)
FOSM Calculations
The variance 𝜎 𝐹
2
= 𝑖=1
𝑛
𝑗=1
𝑛 𝜕𝐹
𝜕𝑋 𝑖
𝜕𝐹
𝜕𝑋 𝑗
𝜌 𝑋 𝑖 𝑋 𝑗
𝜎 𝑋𝑖
𝜎 𝑋 𝑗
Reliability index 𝛽 =
𝐸 𝐹 −1
𝜎 𝐹
• Factor of Safety
• Soil Profile and fill Properties
• Shear strength of foundation
clay
𝑝 𝑓 were computed on the assumption that F is normally distributed
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 The selected 𝑝 𝑓 was selected smaller
for higher embankments
 Based on the revised target probabilities,
one obtains the consistent, desired
factors of safety.

17. III. Reliability analysis Methods
④ First Order Reliability Method (FORM)
This method, developed by Hasofer and Lind (1974) addressed some concerns about some
assumptions involved in the FOSM method.
For each variable 𝑥𝑖, we define 𝑥′
𝑖 having a mean value of zero and unit standard deviation.
𝑥′
𝑖 =
𝑥𝑖 − 𝜇 𝑥𝑖
𝜎𝑥𝑖
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Limit state function
𝑔 𝑥′
1, 𝑥′
2, … , 𝑥′
𝑛 = 0
Safe and unsafe regions (Du. 2005)
 Reliability index is interpreted geometrically as
the distance between the point defined by the
expected values of the variables and the closest
point on the failure criterion.
 The probability of failure is the volume of the
hill on the failure side.

18. III. Reliability analysis Methods
④ First Order Reliability Method (FORM)
Lagrange’s multipliers is used to
find the minimum distance as :
𝛽 = 𝑑 𝑚𝑖𝑛 = −
𝑥′∗
𝑖
𝜕𝑔
𝜕𝑥′ 𝑖 ∗
𝜕𝑔
𝜕𝑥′ 𝑖 ∗
2
The design point in the reduced
coordinate is :
𝑥′∗
𝑖 = −𝛼𝑖 𝛽
With 𝛼𝑖=
𝜕𝑔
𝜕𝑥′ 𝑖
𝜕𝑔
𝜕𝑥′ 𝑖 ∗
2
18
1. Define the limit state equation
2. Assume initial values of 𝑥′𝑖 and obtain reduced variables
𝑥′
𝑖 =
𝑥 𝑖−𝜇 𝑥 𝑖
𝜎 𝑥 𝑖
3.Evaluate 𝜕𝑔
𝜕𝑥′𝑖
and 𝛼𝑖 at 𝑥′
𝑖∗
4.Obtain the new design point 𝑥′
𝑖∗ in terms of 𝛽
5. Substitute the new 𝑥′
𝑖∗ in the limit state equation 𝑔(𝑥′
𝑖∗)=0
and solve for 𝛽
6. Using the 𝛽 value obtained in step 5, re-evaluate
𝑥′∗
𝑖 = −𝛼𝑖 𝛽
7.Repeat steps 3 through 6 until 𝛽 converges
Rackwitz algorithm

19. 19
III. Reliability analysis Methods
⑤ Monte Carlo Simulation Methods
Example :
A system has 2 random inputs 𝑍1 and 𝑍2, the response is a random function 𝑔(𝑍1, 𝑍2)
System failure occurs if 𝑔(𝑍1, 𝑍2) > 𝑔 𝑐𝑟𝑖𝑡
We want to find 𝑝 𝑓 = 𝑃 𝑔(𝑍1, 𝑍2) > 𝑔 𝑐𝑟𝑖𝑡
𝑍1 and 𝑍2 follow a certain probability distribution, so the 𝑝 𝑓 can be expressed in terms of the
joint probability density function
𝑝 𝑓 =
𝑧2∈𝐹 𝑧1∈𝐹
𝑓𝑧1 𝑧2
𝑧1, 𝑧2 𝑑𝑧1 𝑑𝑧
F: the failure region
This kind of integrals can be evaluated in most cases numerically
Monte Carlo Simulation

20. 20
III. Reliability analysis Methods
⑤ Monte Carlo Simulation Methods
After simulating the random realizations of 𝑍1 and 𝑍2, 𝑔(𝑍1, 𝑍2) is evaluated for each.
we check if 𝑔(𝑍1, 𝑍2) > 𝑔 𝑐𝑟𝑖𝑡
𝐼𝑖 =
1 if 𝑔(𝑧𝑖1, 𝑧𝑖2) > 𝑔 𝑐𝑟𝑖𝑡
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
The estimate of the probability is 𝑝 𝑓 =
1
𝑛 𝑖=1
𝑛
𝐼𝑖

21. Thank you for your attention
ご清聴ありがとうございました
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