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.
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
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
7
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 .
8
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
12
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
16
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
17
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
Page 23 - 24
Natural : associated with inherent randomness of natural processes manifesting as variability over time and for phenomena that take time at a single location, or as variability over space for phenomena that take place at different locations but at single time or variability over both time and space
Knowledge : attributed to lack of data, lack of information about events and processes or lack of understanding of physical laws that limits our ability to model the real world.
- Site :subsurface geology resulting from data and exploration uncertainties
- Model: the degree to which a mathematical model accurately mimics reality
- Parameter: precision to which model parameters can be estimated
- Inability to know social objectives, values and time preferences.
Modeling site characterization involves the following steps:
-Develop hypotheses about site geology
Build a random process model based on the hypotheses
Make observations in the field or laboratory
Perform statistical analysis of the observations to draw inferences about the random process model
Apply des\cision analysis to optimize the type, number and location of observations
It is not wise to apply typical values of soil property variability from other sites in performing a reliability analysis.
Cc: compression index, Cr: recompression index, cv: coefficient of consolidation
So means and standard deviations are used to describe the variability in a set of soil property data. But they mask spatial information.
The trend is determined by an equation and the residuals are characterized statistically by a random variable.
Data are used to estimate a smooth trend, and remaining variations are described statistically, the variance of the residual reflects the uncertainty
Since changing the trend changes Rz, the autocorrelation function reflects a modelling decision too.
Both the loads and resistance may be uncertain.
It means finding a way to compute the margin of safety, factor of safety or other measure of performance like a simple equation or a computational procedure.
The parameters include the properties of geotechnical materials and also loads and geometry. Usually they are described by their means, variances and covariances.
This usually means calculating the mean and variance of the performance function.
Calculate
calculate
Provide insight into the functioning of a system and into the associated uncertainties about the way the system functions.
The analysis attempts to generate all the subsequent events. The event outcomes are represented as branches issuing from the chance node representing a particular event.
A conditional probability is associated with each event
𝜌 𝑋 𝑖 𝑋 𝑗 is the covariance between two variables
- The James Bay project required the construction of 50 Km of dikes on soft sensitive clays. The method was used to evaluate the single or multi-stage construction of a typical dike whose cross section is the following.
The goal of the analysis is to understand the relative safety of different designs, obtain insights about the influence of different parameters and establish consistent criteria for preliminary designs.
The random experimental variations represented by 𝑐 𝑒 due to error in measurements and small scale fluctuations in soil must be eliminated to find the shear strength. Only the spatial variance represent a real effect that occurs in the field and needs to be taken into account.
A limited number of tests is used and different set of measurements would yield a different estimate, the bias means that the experimental technique may not measure directly the quantity of interest
𝜎 𝐹 the variance. 𝐸 𝐹 the mean
The function can be differentiated formally or numerically by divided differences
Factor of safety : method of slices is used so numerical method is required to evaluate the variance
Soil profile and fill properties : thickness of the crust, the depth to the till, unit weight, friction angle
Shear strength : for single stage analyses, uncertainty in 𝑐 𝑢 is based on the field vane data. For multi stage case, shear strengths were established from a combination of undrained strength ratios and the in situ stress history
To extrapolate the results there are two way, one could assume that the variance of F is constant or assume that the coefficient of variation is constant
Conclusion : target probabilities are being selected based on reasons such as the relative contribution of different modes of failure. The target probability selection depends also on the costs of reconstruction ( The probability was reduced with the increase of the height of the embankment)
The FOSM method involve some approximations that may not be acceptable :
Suppose that the moments of the failure criterion can be estimated accurately enough by starting with the mean values of the variables and extrapolating linearly.
The from of the distribution of F is known and can be used to compute 𝑝 𝑓
The objective from transforming the variables 𝑥 𝑖 into 𝑥 ′ 𝑖 is to obtain a standardized space of Normal variables to aid in the computation of reliability Index.
Finally the reliability index is used to find the probability of failure
