# Probabilistic evaluation of liquefaction potential.

Typically, a deterministic method is used to determine the potential for liquefaction. In a deterministic method the uncertainty of many factors that may contribute to the occurrence of liquefaction are not incorporated. As a result, liquefaction potential is evaluated only by the factor of safety. In this paper, probabilistic concepts are used in determination of liquefaction potential. In order to determine the conditional probability of liquefaction, the Taylor's Series reliability method and Monte Carlo simulation method are used. The "simplified procedure", as embodied in the "NCEER report" will be used as the basis for the probabilistic procedure. The methods which are described in this paper are applied to an actual site after Manjil-Rudbar earthquake of June 21, 1990.

INTRODUCTION

Evaluation of liquefaction potential of soils is an important aspect in geotechnical investigations in earthquake-liable region. The most common deterministic procedure around the world for evaluating liquefaction resistance is "simplified procedure" developed originally by Seed and Idriss (1971) using blow counts from the Standard Penetration Test correlated with a parameter called the cyclic stress ratio which represents the cyclic loading on the soil. Since 1971, this procedure has been modified and updated.

In deterministic methods, the values of parameters that may contribute to the occurrence of liquefaction are considered constant. But there is uncertainty in many parameters, so deterministic methods aren't reliable methods. The probabilistic methods differ in incorporating the uncertainty of parameters compared with deterministic methods. Some probabilistic methods include Haldar and Tang (1979) and the Taylor's Series reliability method, Wolf (1994); USACE (1997, 1998); Duncan (2000).

In this paper probabilistic concepts are used in determination of liquefaction potential. In order to determine the conditional probability of liquefaction, the Taylor's Series reliability method and Monte Carlo simulation method are used.

SIMPLIFIED PROCEDURES FOR DETERMINISTIC LIQUEFACTION EVALUATION

The most common deterministic method to evaluate liquefaction potential of a site, is the simplified procedure" originally developed by Seed and Idriss (1971). The method has been modified and improved since it was developed. A review of the current state-of-practice in liquefaction evaluation using the simplified procedure is given in the "1996 NCEER and 1998 NCEER/NSF Workshops on Evaluation of Liquefaction Resistance", Youd et al. (2001), referred as "NCEER Report" in this paper. The current version of the simplified procedure calculates the factor of safety FS against liquefaction of a level ground in terms of the cyclic stress ratio CSR (the demand), and the cyclic resistance ratio CRR (the capacity) according to:

FS = CR[R.sub.7.5]/CSR MSF.[K.sub.[sigma]] (1)

Where CR[R.sub.7.5]=Cyclic resistance ratio for magnitude 7.5 earthquakes, CSR=cyclic stress ratio, MSF = magnitude scaling factor, and [K.sub.[sigma]] = correction for non-linear effects of confining stress. The complete equations are given in NCEER Report.

PROBABILITY OF LIQUEFACTION

The Taylor's Series reliability method

In the Taylor's Series reliability method, Wolff (1994); USACE (1997, 1998); Duncan (2000), conditional probability of liquefaction [P.sub.L] is determined, assuming a log-normal distribution of the factor of safety. This requires the calculation of a reliability lognormal reliability index [[beta].sub.LN] and using the formula:

[P.sub.L] = 1 - [PHI] ([[beta].sub.LN] (2)

Where [PHI]()= the standard normal cumulative distribution function. To calculate [[beta].sub.LN], the moments of the function of the safety factor must be calculated from the moments of the parameters.

Monte Carlo simulation method

Generation of random numbers with predefined probability distribution functions is one of the most important applications of Monte Carlo simulation method. In Monte Carlo simulation, values of the random variables are generated consistent with their probability distribution, and the function of the safety factor is calculated for each generated set of variables. The process is repeated numerous times, typically thousands, and the expected value, standard deviation, and probability distribution of the function of the safety factor are taken to match that of the calculated values. Monte Carlo method permits one to estimate the shape of distribution of the function of the safety factor, permitting more accurate estimation of probability of liquefaction.

LIQUEFACTION POTENTIAL ANALYSIS IN ASTANEH AFTER MANJILRUDBAR EARTHQUAKE OF JUNE 21, 1990

The Manjil, Iran earthquake, [M.sub.s]=7.7, Berberian et al. (1992), occurred on June 21, 1990. Official estimates indicate that more than 35,000 people lost their lives. During the earthquake, the liquefaction of soils played a significant role in the destruction of many houses and commercial and public buildings. There was dramatic evidence of this widespread liquefaction-induced damage as far away as 80 km from the fault.

Astaneh-Ashrafieh region is located to the east of the major city of Rasht. Liquefaction on the east of Rasht occurred almost entirely in the levee deposits along the banks of the Sefid and Heshmat rivers.

To evaluate the in situ properties of the liquefied soils in the town of Astaneh-Ashrafieh, standard penetration tests (SPT) were conducted in 16 boreholes. It is noted that these field borings were made in the summer of 1991 almost a year after the earthquake. In all cases, the SPT tests were performed using a standard 2-inch split spoon sampler without a liner. The borehole AS-3 located in the area that liquefaction was observed after Manjil earthquake. But in the borehole AS-14 liquefaction was not observed, Yegian et al. (1995).

Figure 1 shows SPT, N-values in boreholes AS-3 and AS-14 from Astaneh-Ashrafieh. The sands from this region are characterized as gray poorly graded fine sands with less than 10 % silts. The density of soil is about 19.5 (KN/m3). At the time of the earthquake the ground water level was about 2 meters below the ground surface.

[FIGURE 1 OMITTED]

According to the peak ground acceleration obtained from this earthquake, Naderzadeh (1991), the peak ground acceleration in Astaneh-Ashrafieh was 0.18g.

To evaluate the conditional probability of liquefaction, the average values and estimated coefficient of variation (COV) of the uncertainty involved parameters have been considered according to table 1.

The standard penetration test (SPT) results show that the distribution function of [N.sub.spt] is lognormal. In this paper the distribution function is assumed normal for the other uncertainty involved parameters.

Figure 1 shows the liquefaction safety factor in different depths of boreholes AS-3 and AS-14. The safety factor has been determined according to the deterministic method, Youd et al. (2001).

Figure 2 shows the conditional probability of liquefaction in the different depths of boreholes AS-3 and AS-14. The conditional probability of liquefaction has been determined according to the probabilistic methods, Monte Carlo simulation method and the Taylor's Series Reliability Method, Wolff (1994); USACE (1997, 1998); Duncan (2000). Monte Carlo simulation was carried with 5000 iterations.

[FIGURE 2 OMITTED]

Figure 3 shows the relation between safety factor and probability of liquefaction. According to figure 3, if safety factor is 1.5 and if coefficient of variation of parameters is as indicated in this paper, then the predicted probability of liquefaction is 0.17 in Monte Carlo simulation method and 0.18 in the Taylor's Series reliability method. Also the value of safety factor required to have the probability of liquefaction be less than 0.1 is about 1.6. According to the figure 3 the Taylor's Series reliability method is more conservative than Monte Carlo simulation method.

[FIGURE 3 OMITTED]

REFERENCES

Barghamadi, H. (2001). "Liquefaction potential analysis by using Monte Carlo simulation method", MSC thesis, Tehran University, Iran.

Berberian, M., Qorashi, M., Jackson, J.A., Priestley, K., and Wallace, T. (1992). "The Rudbar-Tarom 20 June 1990 earthquake in northwest Pesia: Preliminary field and seismological observations, and its tectonic significance", Bulletin of the Seismological Society of America, 82(4), 1726-1755.

Castro, G. (1995). "Empirical methods in liquefaction evaluation", Primer Ciclo de Conferencias Internationales, Leonardo Zeevart, Universidad Nacional Autonoma de Mexico, Mexico City.

Duncan, L. M. (2000). "Factors of safety and reliability in geotechnical engineering", J. Geotech. Envir. Eng., ASCE, vol. 126, No. 4, 307-316.

Haldar, A. and Tang, W.H. (1979). "Probabilistic Evaluation of Liquefaction Potential", Journal of the Geotechnical Engineering Division, ASCE, Vol. 105, No. GT2, 145-163.

Harr, M.E. (1984). "Reliability-based design in civil engineering", 1984 Henry M. Shaw Lecture, Dept. of Civil Eng. North Carolina State Univ., Raleigh, NC.

Kulhawy, F.H. (1992). "On the evaluation of soil properties", J. Geotech. Spec. Publ., ASCE, 95-115.

Naderzadeh, A. (1991). "Personal Communications".

Seed, H.B. and Idriss, I.M. (1971). "Simplified procedure for evaluating soil liquefaction potential", J. Geotech. Engr. Div., ASCE, vol. 97, No. 9, 1249-1273.

Skempton, A.W. (1986). "Standard penetration test procedures and the effects in sands of overburden pressure, relative density, particle size, aging and over consolidation", Geotechnique, vol. 36, No. 3, 425-447.

USACE (1998). "Risk-based analysis in geotechnical engineering for support of planning studies", Engr. Tech. Ltr. No. 1110-2-554, U.S. Army Corps of Engrs., Dept. of the Army, Washington, D.C. (27 Feb. 1998).

USACE (1997). "Engineering and design introduction to probability and reliability methods for geotechnical engineering", Engr. Tech. Ltr. No. 1110-2-547, U.S. Army Corps of Engrs., Dept. of the Army, Washington, D.C. (30 Sept. 1997).

Wolff, T.F. (1994). "Evaluating the reliability of existing levees", Rep. Res. Reliability of existing levees, prepared for the U.S. Army Engineers Waterways Experiment Station Geotech. Lab., Vicksburg, Miss.

Yegian, M.K., Ghahraman, V.G., Nogole-Sadat, M.A.A., and Daraie, H. (1995). "Liquefaction During 1990 Manjil, Iran, Earthquake, I- Case Histories Data", Bulletin of the Seismological Society of America, February.

Youd, T.L. and 20 others (2001). "Liquefaction resistance of soils: Summary report from the 1996 NCEER and 1998 NCEER/NSF Workshop on Evaluation of Liquefaction Resistance of Soi1s", J. Geotech. Envir. Eng., ASCE, vol. 127, No. 10, 817-833.

HADI BARGHAMADI

Soil and Rock Mechanics Group, Mahab Ghodss Consulting Engineer Tehran, Iran

MANOUCHEHR LATIFI NAMIN

University of Tehran, Faculty of Engineering Tehran, Iran
```Table 1. The average and estimated coefficient of variation (COV).

Parameter Average Estimated COV (%)
value

Magnitude scaling NCEER COV(|MSF-1|)-22
factor, MSF Report

Stress-reduction NCEER
coefficient, [r.sub.d] Report COV(1 - [r.sub.d])=22

moist unit weights, Actual
[[gamma].sub.m] data COV([[gamma].sub.m])=
9 (SP-SW)

SPT blow count, Actual
[N.sub.spt] data In this paper COV=30

Overburden NCEER COV(I-[C.sub.N])23.1
correction factor, [C.sub.N] Report

Energy correction 0.95 9
factor. [C.sub.E]

Borehole diameter 1.07 1
correction factor, [C.sub.B]

Rod length 0.82 24
correction factor, [C.sub.R]

Sample correction 1.2 3
factor, [C.sub.s]

Parameter Reference

Magnitude scaling from three-sigma
factor, MSF rule

Stress-reduction from three-sigma
coefficient, [r.sub.d] rule

moist unit weights, Based on in-place
[[gamma].sub.m] density data from
several USBR dams

SPT blow count, COV=15-40%
[N.sub.spt] (from Harr (1984),
Kulhawv (1992))

Overburden Based on data from
correction factor, [C.sub.N] Castro (1995)

Energy correction Based on data for
factor. [C.sub.E] Safety hammer in
NCEER Report

Borehole diameter Based on data for
correction factor, [C.sub.B] NCEER Report

Rod length Based on data from
correction factor, [C.sub.R] Skempton (1986)

Sample correction Based on data for
factor, [C.sub.s] NCEER Report
```
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