International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Bayesian methods and liquefaction Christian, John T.; Baecher, Gregory B. Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1 Bayesian Methods and Liquefaction John T. Christian Consulting Engineer, 36E Seven Springs Lane, Burlington, MA 01803, USA Gregory B. Baecher Prof., Dept. of Civil and Env. Engineering, University of Maryland, College Park, Maryland, USA ABSTRACT: Since the beginning of serious study of liquefaction in the 1960s, uncertainties in both observed data and in processing field and laboratory tests have been major concerns. Assigning reason-able coefficients of variation to the parameters and correction factors in the conventional deterministic analyses indicates that a site with SPT N values between 12 and 22 and deterministic factors of safety of 1.5 can actually have liquefaction probability above 20%. About a third of the variance in the factor of safety comes from uncertainty in the load (magnitude scaling, stress reduction factor, etc.), which is independent of the method used to estimate the resistance. Researchers have traditionally presented the results of case studies in the form of charts showing instances in which liquefaction did and did not oc-cur and have developed relations to separate the two. Although the original researchers developed the separation lines informally, recent work has applied statistical methods, such as discriminant analysis and logistic regression or combinations of them. In their original form, these methods give the sam-pling distributions of the observed data (i.e., the probability of observing the data given the hypothesis) rather than the probability of the hypothesis given the data, but the engineer needs the latter, that is, the probability of liquefaction given a set of observations. Researchers have addressed this issue using Bayesian methods, adopting non-informative priors to develop the results. Published curves of lique-faction probabilities can thus be interpreted as likelihood ratios. Other, independent work demonstrates that geological, meteorological, and historical data can be used to develop prior liquefaction probabili-ties that are not non-informative, so it may not be necessary to assume a non-informative prior. The ac-tual prior can then be combined with the previously developed likelihood ratios to provide rational probabilities of liquefaction.   Although the earthquake literature (e. g., Richter 1958) describes many historical instances of liq-uefaction of soils during earthquakes, the scien-tific study of the phenomenon can be said to begin with the work of H. B. Seed and his col-leagues (Seed and Lee 1966, Seed and Idriss 1971). Since the 1960s liquefaction has been a major concern of the geotechnical engineering community, and the literature on the subject has become vast. There have been at least two pro-fessional workshops attempting to elucidate the state-of-the-art (NRC/NAE 1985, Youd et al. 2001) and two editions of an EERI monograph on the subject (Seed and Idriss 1982, Idriss and Boulanger 2008). These four publications con-tain extensive bibliographies on the subject, which, in the interest of saving space, will not be repeated here. As this is being written, a new Na-tional Research Council study of the subject is under way. Most engineering practice employs a form of the simplified method proposed by Seed and Idriss (1971), modified many times since, and presented in the most widely available modern form by Idriss and Boulanger (2008). The ap-proach has two parts. In one, the soil’s resistance to liquefaction is evaluated from in situ meas-urements such as the Standard Penetration Test (SPT), Cone Penetration Test (CPT), or shear wave velocity (vs), modified to account for ef-12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2 fects that cannot be controlled during the test, and expressed as a Cyclic Resistance Ratio (CRR). In the other, the cyclic loading is de-scribed by a Cyclic Stress Ratio (CSR), which is calculated from parameters describing the earth-quake acceleration, patterns of dynamic stress distribution, magnitude, and other features of the imposed loading. In practice the required values of CSR are usually found by estimating from ob-served field cases of liquefaction and non-liquefaction the CRR values required to prevent liquefaction. For a particular location and a par-ticular expected earthquake loading, the ratio be-tween the estimated values of CRR and CSR is defined as the Factor of Safety (FS), or FS = CRR / CSR. There is a great deal of uncertainty in this process. This papr addresses two aspects of the uncertainty: (a) If the published critical relations between CSR and CRR are accepted as valid, how much uncertainty is introduced into FS by the uncertainty in the parameters that enter into the estimates of CSR and CRR? (b) When rela-tions between CSR and CRR are stated in proba-bilistic terms, what do the results mean and how can they be used in practice? 1. PARAMETRIC UNCERTAINTY The CSR at some location in the potentially liq-uefiable soil is conventionally defined by:   max0.65 vc dvcσ aCSR rσ g= ¢   (1) The terms are as follows: • 0.65 is a factor to account for the average ef-fective acceleration in an accelerogram being less than the peak value. The value 0.65 has been generally accepted as a reasonable esti-mate of the combined effects of multiple cy-cles, and it is usually simply stated as the numerical value without any symbol. How-ever, there is clearly some uncertainty in this value. • vcσ  is total vertical stress on the horizontal plane.  • vcσʹ′  is effective vertical stress on the horizon-tal plane.  • amax is peak ground acceleration. • g is the acceleration of gravity.  • rd is a stress reduction factor to account for the flexibility of the soil profile, which caus-es the shear stress transmitted through the profile to decrease with depth.   Different methods of evaluating the in situ conditions lead to different formulas for estimat-ing CRR. If the SPT is used, the expression is 𝐶𝑅𝑅 = (𝑓 𝑁? ™ ™ ×𝑁𝑆𝐹×𝐾?  𝑁? ™ ™ = 𝐶?𝐶?𝐶?𝐶?𝐶?𝑁? + ∆𝑁? ™ ™  (2) In these equations: • f 𝑁? ™ ™  is a function based on observed field behavior.  • MSF is a magnitude-scaling factor account-ing for the fact that earthquakes of larger magnitude have more cycles of loading. • Kσ brings all the empirical observations to the same overburden stress. • CN accounts for the effect of depth on SPT. • CE accounts for the effect of the transmitted energy. • CB accounts for the effects of the diameter of the borehole. • CR accounts for the effects of the length of the drill rod. • CS accounts for the effects of the configura-tion of the sampler. • ( )1 60Δ N  accounts for the presence of fines.  If the cone penetration test (CPT) is used, the corresponding relations are 𝐶𝑅𝑅 = 𝑓(𝑞?? ™? )×𝑀𝑆𝐹×𝐾?  𝑞?? ™? = 𝐶?𝑞™ + Δ𝑞??? (3) For the CPT tests: • f((qc1Ncs) is a function of the corrected CPT based on observed field behavior. • qcN is the measured tip resistance of the cone corrected for various factors such as layer 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3 thickness, near surface values, and cone fric-tion effects. • 1Δ c Nq  is an additional term to account for the presence of fines.    Figure 1: Adjusted N and FS from example of Idriss and Boulanger (2008): (a) Revised values of N, (b) Factors of Safety from revised N values  The corrections to the measured tip re-sistance involve a number of empirical relations that are not simple multiplication factors. The SPT and CPT cannot be used in coarse-grained soils, so correlations have been proposed for tests such as the Becker hammer test in those materi-als. The measured in situ shear wave velocity has also been used as an alternative to the SPT and CPT, and corrections similar to those for the SPT and CPT have been proposed. Because the values of the coefficients and input parameters are not known precisely, there is uncertainty in applying current procedures to estimate liquefaction potential. A numerical study was carried out using the spreadsheet pro-vided in Appendix A of Idriss’s and Boulanger’s (2008) monograph. The raw values of N in the example provided in that report are so low for most of the samples that the computed factors of safety are much less than 1.0. Thefore, values of N were raised so that all the factors of safety for the sand samples were close to 1.5. Figure 1 shows the revised values of N and the new fac-tors of safety.  It is reasonable to assume that the parame-ters and the resulting values of CSR, CRR, and FS are log-normally distributed. If the mean and standard deviation of the values of a log-normally distributed variable are designated µ and σ, respectively, and the mean and standard deviation of the logarithms of those values are designated λ and ζ, respectively, and COV stands for the coefficient of variation of the data, then the well-known relations are 𝐶𝑂𝑉 = 𝜎𝜇 𝜍? = 𝑙𝑛1+ 𝐶𝑂𝑉?  𝜆 = 𝑙𝑛𝜇 + ?? 𝜍? (4) The variance of the logarithm of the product or ratio of several independent log-normally dis-tributed variables can be found by simply adding the variances of the input. Since equations (1), (2), and (3) are largely multiplications and divi-sions, the calculations are straightforward. The major exception is the evaluation of the vari-ances of the f () functions in equations (2) and (3). These are exponentials of polynomial ex-pressions whose variances can be approximated by a first-order Taylor series. The values of the COVs for the parameters were estimated from their published descriptions. The values selected for the present analysis are listed in Table 1. In order not to exaggerate the uncertainty, the values selected for most of the COVs are small. The two exceptions are the COVs of 0.20 for MSF and rd, but the range of proposed values for these parameters suggests that these parameters really are poorly known and incorporating large uncertainties is appropri-ate.  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4  Table 1. COVs and values of ζ2 for parameters and coefficients Parameter COV ζ2 N meas. 0.10 0.009950 CE 0.02 0.000400 CE 0.02 0.000400 CE 0.02 0.000400 CE 0.02 0.000400 CE 0.02 0.000400 MSF 0.20 0.039221 Kσ 0.07 0.004888 0.65 factor 0.05 0.002497 ʹ′vc vcσ σ  0.05 0.002497 amax 0.10 0.009950 rd 0.20 0.039221  The spreadsheet was modified to incorporate the variances and to compute the variances of CRR and CSR, the reliability index, and the probabil-ity of liquefaction. For example, at a depth of 4.1m, N=18, and 𝜍 ™?? = 0.054165   𝜍 ™?? = 0.084555 𝜍 ™? = 0.138720   𝜍 ™ = 0.372541 𝜆™ = 0.305101 𝛽? ™ /? ™ = 0.819170 𝑝™?? = Φ− 𝛽 = 0.206345 ≈ 20% (5) Results for the other samples are comparable. Figure 2 gives the results for the entire profile. In this example the values of N and the computed deterministic factors of safety of 1.5 are so large that most practicing geotechnical en-gineers would consider the site safe against liq-uefaction. Nevertheless, there is significant prob-ability of liquefaction, in part because multiply-ing several uncertain parameters has a strong im-pact on overall uncertainty. Even when the calcu-lated factor of safety is 1.5, there is 20% proba-bility of liquefaction simply because of paramet-ric uncertainty.  Figure 2. Probabilities of liquefaction computed from uncertainties in coefficients and input parameters, from the example of Idriss and Boulanger (2008)  Because there are fewer uncertain coeffi-cients in the CPT-based approach, the estimated uncertainty is less. However, the uncertainty in CSR, which contributes at least one third of the overall uncertainty, is unchanged from the SPT-based analysis, so large uncertainty exists in the results, even when the resistance is based on in situ tests such as CPT or vs. 2. QUESTION (B) – PROBABILISTIC CRR – CSR RELATIONS Almost all the criteria for relating CRR and CSR to in situ measurements are derived from obser-vations at sites where liquefaction either did or did not occur during earthquakes. Such studies are inevitably beset the scarcity of data taken be-fore the earthquake occurred, relatively fewer cases in which liquefaction did not occur, and difficulties distinguishing values representative of the site as a whole from a mass of data from individual borings.  The data are usually presented in a plot against two axes. The horizontal axis consists of units such as (N1)60, qc1, vs, or some other pa-rameter representing the normalized measure-ment of soil strength. The vertical axis is ex-pressed in terms of the estimated load imposed by the earthquake. Cases in which liquefaction occurred are plotted as filled-in circles, and cases without liquefaction are represented as open cir-cles. Figure 3 contains two typical versions, as 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5 developed by Idriss and Boulanger (2008), one for SPT results the other for CPT results. The task is then to develop a function or line or sur-face that separates the closed from the open cir-cles and that can be expressed as a function of the corrected strength. The lines drawn in Figure 3 are based largely on judgment, and it is not surprising that different researchers have come to different conclusions about where the division between the closed and open circles should be placed. There are essentially two reasons for the discrepancies: (a) argument over how to catego-rize particular case histories and what values to assign to the parameters and (b) disagreement over how to separate the two regions even after the locations of the pints have been established. The problem of separating two regions de-scribed by empirical observations arises in many fields of science and engineering, so it is not sur-prising that, since Fisher (1936) addressed it, many approaches have been proposed, each hav-ing advantages and disadvantages, proponents and disparagers. There are essentially two ways to address the problem: discriminant analysis and logistic regression analysis. Discriminant analy-sis essentially rotaties the axes to find the orien-tation that maximizes the separation between the two sets of points. Logistic regression is an itera-tive procedure that adjusts the parameters in a function of the data (often a polynomial) so that the likelihood of observing the data is maxim-ized. It can be shown that, under some not-too-unreasonable conditions, the two approaches give the same result. Although the first attempt to apply these methods to the liquefaction prob-lem by Christian and Swiger (1975) employed linear discriminant analysis, Liao et al. (1988) used logistic regression, and that has become the preferred approach (Jha et al. 2009, Juang et al. 2002, 2006, Moss et al. 2006).  Regardless of the mathematical tool, the re-sults depend strongly on choices made by the an-alyst, especially on the way the data are normal-ized and on the form of the function that is to separate the two classes of observations. It is re-markable that the published discrimination func-tions developed by different researchers differ by only about 30% in the principal region of inter-est. There are, however, several issues that re-main to be addressed or even to be understood by the users of these plots.  Based on SPT Results (b) Based on CPT Results Figure 3. Typical Plots of CSR versus Field Data, from Idriss and Boulanger (2008)  One of the most important is the meaning of the probability of misclassification in classic dis-criminant or logistic regression analysis. The probability in those analyses is the probability that, if a case belongs to one class, the data will indicate that it belongs in the other. In other words, it is the probability that, if the site actual-ly liquefies, the data indicate that the site should be safe. What the engineer wants is the reverse of this. The engineer wants to know, given a set of data that indicate a safe site, the probability that the site will liquefy. Originally (e. g., Christian and Swiger 1975) plotted the probabilities of misclassification, that is, without Bayesian up-dating. More recently Juang et al. (2002, 2006), 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6 Cetin et al. (2002, 2004), Moss et al. (2006), Kayen et al. (2013), and others have applied Bayesian updating. This requires a prior estimate of the probability of liquefaction and an estimate of the probability that a site that does not liquefy has data indicating that it will. Since researchers cannot have an estimate of the prior probability of liquefaction for the abstract exercise of creat-ing general plots, a common procedure is to as-sume a non-informative prior. This is, in fact, what was done in most of the published analyses, and it means that the analysts assumed equal pri-or probability of liquefaction and non-liquefaction, or 0.50. Figure 4 is a generic figure, similar to those presented by the above research-ers, showing the shape of the typical probability curves, without specifying the parameters for ei-ther axis, but, in the more recent work, represent-ing Bayesian updating.   Figure 4. Generic Plot of Probabilities of Liquefac-tion Reflecting Bayesian Updating.  If Bayes’s theorem is expressed in terms of odds, it becomes a statement that the posterior odds of an event are equal to the prior odds mul-tiplied by the Likelihood Ratio, which is the probability of observing the data in the case of liquefaction divided by the probability of observ-ing the data in the case of no liquefaction:  ?[ ™???????∪? | ™?? ]?[ ™    ™???????∪? | ™?? ]= ??[ ™???????∪? ]??[ ™    ™???????∪? ]× ?[ ™?? | ™??????℩? ]?[ ™?? | ™    ™???????∪? ] =  ??[ ™???????∪? ]??[ ™    ™???????∪? ]×𝐿𝑅 (6) For a non-informative prior, the prior odds are 0.5/0.5 = 1.0, and equation (6) reduces to  1 PLR P= -­‐   (7) in which P is the probability for any of the lines in Figure 4. Let us suppose that the data for a particular site fall on the heavy line with 20% probability. The odds are 0.2 / (1-0.2) = 0.25., and this is the value of LR as well, and plots like Figure 4 can be converted into plots of LR. It might be more convenient if reports of Bayesian analyses of liquefaction data presented the results as LR plots instead of probabilities with assumed non-informative priors. This is of more than academic interest when dealing with a case for which there is an in-formed prior. For example, Prof. Baise and her colleagues have developed methods for estimat-ing the probability of liquefaction on the basis of geological, meteorological, and historical data (Brankman and Baise 2008, Zhu et al. 2014). Consider what happens for a ssite where their procedure identified an overall probability of liq-uefaction of 0.8. This is then a prior probability of liquefaction. If the field data for this site falls on the 20% line in Figure 4, corresponding to LR = 0.25, the posterior probability of liquefaction is found from ?[ ™???????∪? | ™?? ]???[ ™    ™???????∪? | ™?? ] =  ??[ ™???????∪? ]????[ ™    ™???????∪? ]×LR = ?.??.? 0.25 = 1.0  𝑃 𝑙𝑖𝑞𝑢𝑒𝑓𝑎𝑐𝑡𝑖𝑜𝑛 𝑑𝑎𝑡𝑎 = 0.5 (8) This compares with the values of 0.20 for the non-informative prior. The prior probability can have a big effect. 3. COMBINED EFFECTS Idriss and Boulanger (2010) give an explicit form to the CRR curve and give the following equation for the probability of liquefaction, that 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  7 is, the probability of misclassifying the data for a particular site: 𝑃?𝑁?, ™ ™ 𝐶𝑆𝑅???.?,????? ™? =  ??, ™ ™™ .? ? ??, ™ ™™? ?? ??, ™ ™™ .? ?? ??, ™ ™™? .? ???. ™ ? ™? ℡ ??.?,????? ™? ??.?,? ™?  (9) The numerator of equation (9) is the function embedded in the Idriss and Boulanger (2008) spreadsheet, and R is shortened notation for CRR. Idriss and Boulanger (2010) recommend that ( )ln 0.13.Rσ =    Figure 5. Probabilities of liquefaction from uncer-tainty in location of CSR/CRR curves (red circles) and data uncertainty (blue squares)  The same parametric values as in Figure 1 are input to equation (9), and the results are plot-ted in a modified version of Figure 2, shown as Figure 5.  The red points represent the probabili-ties of liquefaction from equation (9), given that the values of the input data for a specific point are known precisely. In other words, they repre-sent the uncertainty in the location, with respect to the observed case study data, of the CSR/CRR curves. The blue dots represent the probabilities of failure assuming that equation (9) is absolute-ly true but the values of the input data are uncer-tain.  The simplest way to combine the two re-sults is to assume they are independent. For most of the points in Figure 5 the uncertainty in equa-tion (9) is so small that the scatter dominates, but in the case of the lowest point the probabilities are approximately 0.07 and 0.21, so the com-bined probability is 0.265.  4. CONCLUSIONS  This paper has examined only two aspects of the uncertainties in liquefaction evaluations. Howev-er, even this limited exploration reveals some points: The uncertainties in the parameters in the conventional computations of FS are large enough that there is significant uncertainty in the result. A case with N values of 15 to 20 and es-timated FS = 1.5 may have 20% probability of liquefaction even though the probability associ-ated with the location of the CSR/CRR curves is small.  One of the principal reasons for the large uncertainty in FS is the number of multiplied correction factors, each contributing to the over-all uncertainty. Attempts to refine the precision of the analyses may have increased their uncer-tainty. People presenting plots of probabilities from empirical studies of liquefaction should state whether these are probabilities of liquefaction, of observing the data, or of something else. The ex-planation should be on the figure, or at least in its caption. It should not be buried in the text. Many papers describe liquefaction probabil-ities computed by Bayesian updating using non-informative priors, which imply that the prior odds are 1.0 and that the likelihood ratio can be computed from the reported posterior probabili-ties. However, plots of the actual likelihood rati-os may be more useful. Although non-informative priors may be necessary in generic studies, prior probabilities often do exist for actual sites, and they have a major effect on the computed posterior probabil-ity of liquefaction. 5. REFERENCES Brankman, C. M., and Baise, L. G. (2008) “Liq-uefaction susceptibility mapping in Boston, Mas-sachusetts,” Environmental and Engineering Ge-oscience, 14, 1–16. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  8 Cetin, K. O., Der Kiureghian, A., and Seed, R. B. (2002) “Probabilistic models for the initiation of seismic soil liquefaction,” Structural Safe-ty, 24(1): 67-82. Cetin, K. O., Seed, R. B., Der Kiureghian, A., Tokimatsu, K., Harder, L. F., Jr., Kayen, R. E., and Moss, R. E. S. (2004) “SPT-based probabilistic and deterministic assessment of seismic soil liquefaction potential, Journal of Geotechnical and Geoenvironmental Engi-neering, ASCE, 130(12): 1314-1340. Christian, J. T., and Swiger, W. F. (1975) "Statis-tics of Liquefaction and SPT Results," Jour-nal of the Geotechnical Engineering Divi-sion, ASCE, 101(GT11): 1135-1150, also closure (1976) 102(GT12):1279-1281. Fisher, R. A. (1936) “The use of multiple meas-urements in taxonomic problems,” Annals of Eugenics 7: 179-188. Idriss, I. M., and Boulanger, R. W. (2008) Soil Liquefaction during Earthquakes, Earthquake Engineering Research Institute, MNO-12, Oakland, CA, 243 pp. Idriss, I. M., and Boulanger, R. W. (2010) “SPT-Based Liquefaction Triggering Procedures,” Report UCD/CGM-10/02, Department of Civil and Environmental Engineering, Uni-versity of California, Davis. Jha, S. K., and Suzuki, K. (2009) "Liquefaction Potential Index Considering Parameter Un-certainty," Engineering Geology, 107:55-60. Juang, H. C., Jiang, T., and Andrus, R. D. (2002) "Assessing Probability-based Methods for Liquefaction Potential Evaluation," Journal of Geotechnical and Geoenvironmental En-gineering, ASCE, 128(7): 580-589. Juang, H. C., Fang, S. Y., and Khor, E. H. (2006) "First-Order Reliability Method for Probabil-istic Liquefaction Triggering Analysis," Journal of Geotechnical and Geoenvironmen-tal Engineering, ASCE, 132(3): 337-350. Kayen, R. E., Moss, R. E. S., Thompson, E. M., Seed, Cetin, K. O., Der Keureghian, A., Tanaka, Y., and Tokimatsu, K. (2013) "Shear wave velocity-based probabilistic and deter-ministic assessment of seismic soil liquefac-tion potential," Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 139(3): 407-419. Liao, S. C. C., Veneziano, D., and Whitman, R. V. (1988) "Regression Models for Evaluating Liquefaction Probability," Journal of Ge-otechnical Engineering, ASCE, 114 (4): 389-411. Moss, R. E. S., Seed, R. B., Kayen, R. E., Stew-art, J. P., Der Keureghian, A., and Cetin, K. O. (2006) "CPT-based probabilistic and de-terministic assessment of in situ seismic soil liquefaction potential," Journal of Geotech-nical and Geoenvironmental Engineering, ASCE, 132 (8): 1032-1051. National Research Council/National Academy of Engineering (NRC/NAE) (1985) Liquefac-tion of Soils during Earthquakes, National Academy Press, Washington, DC. Richter, C. F. (1958) Elementary Seismology, W. H. Freeman & Co., San Francisco, CA, 768 pp. Seed, H. B., and Lee, K. L. (1966) “Liquefaction of saturated sands during cyclic loading,” Journal of the Soil Mechanics and Founda-tions Division, ASCE, 92(SM6): 105-134. Seed, H. B., and Idriss, I. M. (1971) “Simplified procedure for calculating soil liquefaction potential,” Journal of the Soil Mechanics and Foundations Division, ASCE, 97(SM9): 1249-1273. Seed, H. B., and Idriss, I. M. (1982) Ground Mo-tions and Soil Liquefaction during Earth-quakes, Earthquake Engineering Research Institute, MNO-5, Berkeley, CA, 134 pp. Youd, T. L., and 20 others (2001) “Liquefaction resistance of soils: summary report from the 1996 NCEER and 1998 NCEER/NSF work-shops on evaluation of liquefaction resistance of soils,” Journal of Geotechnical and Ge-oenvironmental Engineering, ASCE, 127(10): 817-833. Zhu, J., Daley, D., Baise, L. G., Thompson, E. M., Wald, D. J., Knudsen, K. L. (2014) “A geospatial liquefaction model for rapid re-sponse and loss estimation,” accepted 

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