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Risk Assessment in Geotechnical Engineering

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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.

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Risk Assessment in Geotechnical Engineering

  1. 1. Risk Assessment in Geotechnical Engineering Presented by : Soumaya Addou スマイヤ アッドー 東北大学 1
  2. 2. References This presentation is made based on the information provided by mainly these two books 2
  3. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 21. Thank you for your attention ご清聴ありがとうございました 21

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