Probabilistic Methods in Geotechnical Engineering (1995) / Chapter Skim
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Appendix C: Basic Concepts of Probability and Reliability
Appendix C: Basic Concepts of Probability and Reliability
Pages 57-80
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From page 57... ...
The actual capacity of a test pile ranges from as low as 40 percent of the value predicted to as great as twice that predicted value. The shape of the histogram is skewed toward higher values of the ratio ~ The pile tests were all in clay, and their capacities were predicted according to the procedures in the 16th edition of Recommended Practice 2A of the American Petroleum Institute (19861.
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From page 58... ...
It is a measure of the central tendency of the random variable. For the above case of pile capacity, if the prediction model does not contain any systematic bias, the mean value of the observed ratios will be approximately I.0.
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From page 59... ...
In a case where the engineer believes that a given set of measured data does represent a set of realistic sample values of the engineering variable and no other information is available, a PDF can be fitted over the frequency diagram, which is a modified histogram whose ordinate has been scaled, so that the area under the histogram is unity. For instance, a normal distribution is a common probability distribution model used to fit a symmetrical bellshaped histogram.
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From page 60... ...
Once the PDF of a random variable is established, it can be used to calculate the probability of an event associated with a range of values of the variable. For instance, suppose that the undrained shear strength of the soil at a site is modeled by a normal distribution with parameters ~ and ~ equal to 1.2 and 0.17 ksf, respectively.
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From page 61... ...
However, the measured data represent only the undrained shear strengths of discrete soil specimens, which are determined using a given test procedure. This is not necessarily the strength that governs the performance at the site.
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From page 62... ...
This phenomenon is a result of the increasing likelihood that unusually high property values at some points will be balanced by low values at other points; therefore, the average property is less likely to take on exceptionally high or low values. Second, the in situ soil property at incipient failure is not necessarily duplicated by the sampling and testing procedure performed on the soil specimen.
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From page 63... ...
As shown in Figure C-3 for a case where the test strength has a conservative bias, the o · _ c' ;^ ._ v, ;^ · ct o 4 , B t / ' a- -me A: average strength over slip surface `~ A B: strength of / / \ / \ ~ / \ / \ ~ \ \ / \ ~ -" 1 1 At'' 2.0 0.8 1.0 1.2 1.4 1.6 1.8 Undrained Shear Strength (ksf) test specimen //,: probability for Case B : probability for Case A Figure C-3 Discrepancy between distribution of in situ property and those of specimen property.
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From page 64... ...
Example The average density of a LO-m-thick soil layer is estimated based on the following information: Nine soil samples taken at widely scattered locations have been tested for their densities, which yielded a mean of 1,800 kg/m3 and a standard deviation of 200 kg/m3. Assume random test error is negligible compared with spatial variability.
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From page 65... ...
An example has been shown in Figure C-2 for the case of axial pile capacity based on a large number of pile tests. In principle, the probability distribution of this ratio, especially that established on the basis of high-quality field tests, can be used to determine the uncertainty associated with a given performance prediction model, which in turn may be used to estimate the probability of failure of the given geotechnical system.
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From page 66... ...
This implies that the pile capacity is expected to be 2.15 times the predicted value. On the other hand, if unconfined, unconsolidated strength tests on pushed samples were used for determining the soil strengths, the corresponding overall bias and c.o.v.
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From page 67... ...
Slope failure occurs if the total sliding resistance along a potential slip surface is less than the driving force caused by the soil weight and other loads. Hence, in the simplest case, a safety factor can be defined as the ratio of the available resistance, R
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From page 68... ...
(a) PDF o Probability ~ of Failure / \ Probabilistic Methods in Geotechnical Engineering 13=2.07 \ 1 _ Load Resistance o~_0.3 aR=0.3 PDF F=2.5 ~Probability _ / \ / \ of Failure ~ 0 1 2 3 4 0 1 2 3 4 S 6 (b)
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From page 69... ...
Example In dam design, structural engineers designing concrete gravity dams use F=3.0 for foundation design with respect to sliding failure, while geotechnical engineers designing earth dams use F=~.5 for similar foundation design. Does this mean that concrete gravity dams are twice as safe as earth dams in regard to sliding?
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From page 70... ...
On the other hand, if the geotechnical engineer had adopted a very conservative undrained soil strength equal to 40 percent of the average value measured, the design height of the earth dam would be 71 ft and the corresponding probability of sliding failure of the earth dam would be 0.00002. in spite of its smaller factor of safety, the earth dam would be only about onetwentieth as likely to fail as the concrete dam for this case.
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From page 71... ...
When the PDF of some (or even all) of the component variables are not prescribed but their mean values and c.o.v.s are available, the first-order reliability method may be used to determine the reliability index and the corresponding probability of failure approximately.
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From page 72... ...
to be multiplied by the mean resistance. By applying the first-order reliability method, one can determine the appropriate value of the resistance factor from the following equation hi = ~ - Al 13 Q (Eq.
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From page 73... ...
in this case, it represents the resistance factor for the undrained shear strength of soil. Alternatively, the design resistance can be obtained by dividing the mean resistance by a factor y (larger than one)
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From page 74... ...
Because most of the reliability analysis is based on the critical slip surface or the critical cross section of the tunnel, procedures are needed to extrapolate these critical component probabilities to those of the entire geotechnical system. Probabilistic methods can provide the necessary analytical framework.
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From page 75... ...
Waste-remediation decisions, which increasingly involve geotechnical engineers, can benefit greatly from the above-described probabilistic observational method; this is also true for many other geotechnical applications, ranging from dam safety to repair/maintenance decisions concerning deteriorating infrastructures. Hachich and Vanmarcke (]
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From page 76... ...
Before any of these wells is installed, the engineer believes that there is a 70 percent chance that the leakage has happened. Consider first the case that well A has been installed and no contaminants have been observed.
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From page 77... ...
Even if a given site exploration program has not encountered such geologic anomaly, the experience of the engineer with the geology of the region may suggest that it could still be present at the site. In this case, the engineer's judgment can be combined with the level of site exploration efforts spent (e.g., number of borings)
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From page 78... ...
The probability of a path is simply the product of the respective probabilities. The expected cost of each alternative is the summation of the path probability multiplied by the path consequence over all outcome scenarios for that alternative.
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From page 79... ...
Journal of Geotechnical Engineering, American Society of Civil Engineers Il9~24: 195-213.
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From page 80... ...
Journal of Geotechnical Engineering Division, American Society of Civil Engineers 107(GT12)
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