LIQUEFACTION POTENTIAL MAPPING FOR SAN FRANCISCO

By EdwardjKavazanjian, Jr.,1 M. ASCE, Richard A.lRoth,2

anffHeriberto (Echezuria3

ABSTRACT: The potential of saturated deposits of cohesionless soil in down­town San Francisco to experience initial liquefaction due to seismically induced pore pressure is evaluated. Initial liquefaction, or the zero effective stress state, is used as the index of liquefaction potential because it provides the best avail­able index for damage due to seismically induced pore pressures. Liquefaction potential is evaluated by comparing the conditional probability of liquefaction, or liquefaction susceptibility, to the expected intensity of seismic loading, or liquefaction opportunity. The probabilistic evaluation is made using a liquefac­tion hazard model developed by Chameau. (Assuming the water table to be at the ground surface, results indicate that while no liquefaction is expected any­where for an intensity with an annual probability of exceedance of 0.05, only the most resistant deposits will survive an event with an annual probability of exceedance of 0.02. For an event with an annual probability of exceedance of 0.01, initial liquefaction is expected to occur within all saturated, cohesionless soil deposits in the downtown San Francisco area'L It must be emphasized that in dense soil deposits the consequences of initial.liquefaction may be minimal because of their limited shear strain potential, i •'

\ , V INTRODUCTION

The liquefaction potential of saturated, cohesionless soil deposits within the downtown area of San Francisco subject to seismic loading has been evaluated probabilistically as part of the U.S. Geological Survey Earth­quake Hazard Reduction Program (11). The probability of occurrence of initial liquefaction is used as the index of liquefaction potential. Initial liquefaction refers to a zero effective stress state due to seismically gen­erated pore pressures. Initial liquefaction is a necessary, but not suffi­

c i e n t condition for damaging ground movements to occur in horizontal deposits subjected fo| seismic shaking] Actual damage will also depend upon the shear strain potential of the soil.

Large portions of downtown San Francisco are underlain by saturated, cohesionless soil deposits. Historical evidence indicates that liquefaction-induced ground movements have occurred in these deposits three times in the last 120 yr (26). Knowledge of the liquefaction potential of these cohesionless deposits is useful in the design and location of structures and utilities, in evaluating the benefits of remedial measures, in lifeline engineering for estimating the extent of damage, and in disaster pre­paredness and response planning.

Liquefaction potential is determined by comparing the conditional !Asst. Prof, of Civ. Engrg., Terman Engrg. Center, Stanford Univ., Stanford,

Calif. 94305. 2Staff Engr., Dames and Moore, 455 S. Figueroa St., Suite 3500, Los Angeles,

Calif. 90071. 3Ingeniero Civil, INTEVEP, Caracas, Venezuela. Note,—Discussion open until June 1, 1985. To extend the closing date one month,

a written request must be filed with the ASCE Manager of Journals. The manu­script for this paper was submitted for review and possible publication on No­vember 15, 1983. This paper is part of the Journal of Geotechnical Engineering, Vol. I l l , No. 1, January, 1985. ©ASCE, ISSN 0733-9410/85/0001-0054/$01.00. Pa­per No. 19400.

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probability of liquefaction, or liquefaction susceptibility, to the expected intensity of seismic loading, or the opportunity for liquefaction-induced ground failure. Liquefaction susceptibility describes the resistance of a soil element to seismic loading. Liquefaction susceptibility is determined on the basis of grain size characteristics, relative density, and depth. A liquefaction susceptibility map composed of six zones, each of which contains materials of similar susceptibility, was developed for the down­town San Francisco area. The zone boundaries were determined by ref­erence to surficial geology, historical information, and bore log data. Rel­ative density values were derived from the results of dynamic penetration tests.

The expected intensities of ground shaking are determined on the ba­sis of an attenuation relationship for strong ground motions and a prob­abilistic seismic hazard analysis to estimate the contributions from all potential seismic sources. The attenuation relationship for this study was developed by statistical analysis of 83 strong motion records from 18 earthquakes. The results of the hazard analysis yields the annual prob­ability of exceedance for levels of ground shaking intensity for down­town San Francisco.

The information on liquefaction susceptibility and expected intensity of ground shaking are combined using a probabilistic model proposed by Chameau (2) to evaluate liquefaction potential. Results are presented as tables which give representative values of the probability of initial liquefaction for intensities with annual probabilities of exceedance of 0.05, 0.02, and 0.01. A method of calculating probabilities of liquefaction for any of the three predominant soil types found in the study area for any specified annual probability of exceedance or singular seismic event, is developed to facilitate site-specific studies of liquefaction hazard.

PROBABILISTIC LIQUEFACTION POTENTIAL MAPPING

Youd and Perkins (27) developed a methodology for mapping the po­tential for liquefaction-induced ground failure by associating the inci­dence of ground failure with achievement of the zero effective stress state in the ground. The justification for this association is that lique­faction and ground failure are likely to occur simultaneously at the more vulnerable sites even though significant strength loss and damage is possible prior to initial liquefaction in sloping terrain. Also, initial liq­uefaction may not result in damaging ground movements at very dense sites. The framework for regional liquefaction assessment developed by Youd and Perkins was adapted in this study to provide a probabilistic assessment of liquefaction potential.

The methodology of Youd and Perkins prescribes that liquefaction po­tential be evaluated in terms of a capacity-demand or resistance-loading problem. Capacity or resistance is represented by a regional map of liq­uefaction susceptibility. Liquefaction susceptibility is a measure of the relative likelihood of the subsurface materials to liquefy during seismic shaking, and is a function of the mechanical properties of the materials at a given site. Seismic demand or loading is depicted by a map of what is called liquefaction opportunity. It is a measure of the expected inten­sity of ground shaking at a given site and ideally should take into ac-

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count distance to the seismic source(s), attenuation of the seismic waves with distance from the source, and modifications of the seismic waves owing to travel path and local site conditions.

The Youd and Perkins procedure can be easily adapted to express liq­uefaction potential in probabilistic terms. The likelihood that a deposit will liquefy can be evaluated by

P(L) = j P(L|z)/Ki)dt (1)

in which P(L) = the probability of liquefaction; P(L\i) = the conditional probability of liquefaction given the occurrence of a level of earthquake intensity, z; and /j(z) = the probability density function of intensity.

The conditional probability of liquefaction expresses the likelihood of liquefaction given a particular level of seismic shaking. This is also re­ferred to as the seismic fragility of the soil. It provides a measure of material resistance and is analogous to the information that is contained in a map of liquefaction susceptibility. Several analytic procedures have been presented to probabilistically evaluate this term (2,7,10,24).

The probability density function of intensity, /;(z), can be evaluated in terms of a recurrence relationship and an attenuation relationship. The recurrence relationship gives the annual probabilities of exceedance for the range of earthquake magnitudes of interest for each potential seismic source. The attenuation formula describes the decay of the earthquake intensity as a function of distance from the seismic source and event magnitude. This combination of a recurrence relationship and an atten­uation relationship sometimes is referred to as a hazard model.

In the study described herein, the model of Chameau and Clough (2,3) was used to determine probabilistically the distribution of excess pore pressure as a function of the intensity and duration of earthquake (cyclic) shaking. The intensity is represented by the root mean square of the shear stress time history, rs, while the number of positive zero cross­ings, n, represents duration. The model assumes random arrivals of Ray-leigh-distributed shear stress peaks between positive zero crossings.

Chameau's pore pressure model requires two experimentally deter­mined relations as input; a cyclic strength curve and a pore pressure generation curve. The cyclic strength curve relates the number of uni­form cycles to liquefaction, Nt, to the ratio of the uniform cyclic shear stress to initial effective stress, T^/CT^ = S. The pore pressure generation curve expresses the amount of excess pore pressure development nor­malized by the initial effective confining stress, ue/a'0 = R, as a function of the number of cycles of uniform loading, N, normalized to the num­ber of cycles to liquefaction, N/Nt.

Chameau provided a methodology for incorporating this pore pres­sure model as a measure of liquefaction susceptibility into a compre­hensive assessment of liquefaction potential. By setting the pore-pres­sure ratio to unity, the probability of liquefaction, P(L), can be evaluated by

P(L) = P(R = 1) = p ( R = l\rs,n)frs,N(rs,n) drfin (2) JN Jr,

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in which P(R = 1) = the probability of achieving a pore pressure ratio of one; P(R = l\rs, n) = the conditional probability of liquefaction given n, the number of cycles of loading and rs, the root mean square accel­eration; and fh)N (rs,n) = the joint probability density function of rs and n.

Eq. 2 has the same form as the more general Eq. 1. The only difference is that two parameters, the root mean square of the shear stress time history and the number of positive zero crossings, are used to define the intensity. Chameau showed that probabilities of liquefaction calcu­lated with this approach have the potential to describe field behavior during past earthquakes better than when a single parameter such as peak ground acceleration (PGA) is used to define intensity.

Eq. 2 can be simplified if liquefaction potential is to be evaluated at a value of rs that corresponds to a specified annual probability of exceed-ance, or to a specified event. The probability of liquefaction then be­comes:

P(L) = P(R = 1) = P(R = l\rs,n)fN(n\rs)dn (3) JN

The conditional probability term in Eqs. 2 and 3, P(R = l\rs,n), com­pletely defines the liquefaction susceptibility, or seismic fragility, while the probability density terms, fra,N(rs,n) and fN(n\rs), define the expected intensity of ground shaking. The Chameau hazard model is therefore compatible with the mapping procedure of Youd and Perkins. Input to the probabilistic hazard model on liquefaction susceptibility can be ob­tained from a description of relative density with depth for each soil type in the study area, along with cyclic strength and pore pressure gener­ation curves for each combination of soil type and relative density. Input on seismic loading requires a seismic hazard analysis for the root mean square of acceleration, r„, to determine expected levels of the intensity and duration of seismic loading in the study area. The loading param­eter, rs, is directly related to r„ through depth and soil unit weights.

LIQUEFACTION SUSCEPTIBILITY MAPPING FOR DOWNTOWN SAN FRANCISCO

General Approach.—The susceptibility map was developed by divid­ing the study area into zones each of which is characterized by a typical soil profile. The profiles contain water table depth, soil type, and relative densities. Zone boundaries were chosen based upon the criterion that all the locations within a given zone should behave similarly during seis­mic shaking. Since the liquefaction susceptibility of the cohesionless soils was determined primarily by relative density and grain-size distribution, the zone boundaries enclose material that is similar in terms of these mechanical properties. Preliminary zone boundaries were defined on the basis of performance during the 1906 earthquake (4,25) (Fig. 1), a sub­jective assessment of liquefaction susceptibility based on geology follow­ing Youd and Perkins (27), and historical accounts of dates of filling and emplacement procedures along the waterfront (5).

The extent of the cohesionless soil deposits was determined by ref­erence to a geologic map of San Francisco (18) and to historical infor-

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Original 1853 shoreline

» Approximate boundaries of liquefied zone in 1906.

FIG. 1.—Areas of Liquefaction Induced Ground Failure During 1906 Earthquake

mation documenting the construction of the waterfront fills. For the nat­ural dune sand deposit, the available geologic information was sufficient for drawing preliminary zone boundaries. Determination of zone bound­aries within areas of artificial fill, however, was more complicated. The artificial fills tend to be far more nonhomogeneous than the natural de­posits. There is no reason to assume the lateral stratification that is found in a naturally deposited, sedimentary unit also persists throughout the artificial fills. They comprise a conglomeration of natural and man-made

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materials, such as wood, brick, refuse and debris. The geologic map des­ignates artificial fills as such, without further description of the constit­uents.

Historical documentation of the nature and sequence of the filling op­erations in San Francisco (5) proved to be a valuable alternative to geo­logic criteria. Inferences were drawn from this information that provided insight as to the composition of the several sites of artificial fill along the San Francisco waterfront. The major assumption inherent in this ap­proach was that locations of artificial fill emplaced at the same time or that had similar sources of material would be similar in their dynamic behavior. Roth (15) has compiled detailed historical documentation of the San Francisco waterfront development.

Once the preliminary zone boundaries were determined, the files of several consulting firms that have done extensive work in and around downtown San Francisco were examined for reports of geotechnical in­vestigations that were completed within the study area. Cross sections were drawn from these reports through each zone to confirm prelimi­nary zone boundaries, and representative profiles were developed. Grain-size distribution curves were extracted from the geotechnical reports where available. The relative densities of the subsurface materials were derived from results of dynamic penetration tests that were conducted as part of most of the geotechnical investigations.

Compilation of Bore Log Data.—Data for determining the mechanical properties of the liquefiable soils in the study area was compiled from over 350 bore logs representing 83 different sites (Fig. 2). The informa­tion for each site was obtained from a geotechnical investigation that was undertaken for a construction project at that location.

Representative profiles were constructed for each site. Each soil strata in the representative profile was described by visual characteristics, a classification according to the Unified Soils Classification System, and results of dynamic penetration tests performed within that soil strata from all borings at the site.

The mean and standard deviation of blow counts from the dynamic penetration tests within a particular soil strata were converted to mean and standard deviation relative density values by use of a Taylor series expansion of the correlation equation developed by Marcuson and Bie-ganousky (13). This correlation equation assumes that the blow counts were obtained with the equipment prescribed by ASTM D1586-67 for the Standard Penetration Test (SPT). Most of the penetration test results compiled for this report were not obtained with standard equipment. Twenty-three different combinations of hammer weight, height of fall, and sampler size were encountered in the 350 plus boring logs. Non­standard penetration test blow counts were converted to equivalent standard blow counts using a pile driving-wave equation type model of dynamic penetration tests developed by Salah-Mars (17) to correlate the nonstandard test results to SPT blow counts.

Once representative profiles were constructed for each site, cross sec­tions were developed through sites that were common to one of the preliminary zones (Fig. 2). This was done to verify whether or not the sites within a given preliminary zone were similar enough to justify the use of one idealized profile for that zone. No adjustment of the prelim-

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O Bore logs with penetration test data

D Bore logs without penetration test data

40 Site number

FIG. 2.—Location Map for Bore Log Data and Cross-Sections Through Map Zones

inary zone boundaries was deemed necessary. The resulting final map zones corresponded to the Yerba Buena Cove, Telegraph Hill, North Beach, South Beach and Mission Bay fills and the natural, dune sand deposits shown in Fig. 1. Information contained in the cross sections was then condensed into idealized profiles to represent the character­istics of each zone in the susceptibility map. Development of the cross sections and idealized profiles is described in detail elsewhere (11,15,16).

Conversion of Standard Blow Counts to Relative Densities.—The blow count data compiled from bore logs for this study were converted to relative density values using a correlation equation developed by Mar-

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cuson and Bieganousky on the basis of laboratory simulations of the Standard Penetration Test (13). Marcuson and Bieganousky conducted tests on Reid-Bedford Model sand and Ottawa sand; median grain sizes, D50 = 0.25 and 0.21 mm, respectively, and coefficients of uniformity, C„ = 1.6 and 1.5, respectively. The grain-size distribution curves of the two test materials are compared with that of dune sand from Yerba Buena Cove in Fig. 3. The similarity in grain-size characteristics is important since the work of Gibbs and Holtz (8) and Marcuson and Bieganousky indicates that the correlation between blow count and relative density is sensitive to grain-size distribution.

Marcuson and Bieganousky (13) derived the following equation by performing penetration tests on samples that had been constituted to different relative densities:

N = 6.5 + 0.3cr; + 0.0045(Dr)2 (4)

in which N = the blow count; Dr = the relative density in percent; and o'0 = the effective, overburden pressure. The correlation coefficient for this equation was computed as 0.85.

Relative density values were calculated from the mean and standard deviation values of blow counts for each soil strata within a profile. The effective overburden was considered to be deterministic. Equations to compute mean and standard deviation values of relative density were developed by the Taylor series expansion method (1) after rearranging Eq. 4. The following equations were obtained:

D„,, = VC ,'Var(N)

1 - 6,173' ' (5)

Var (N) Var (Dr) = 111 x — ^ (6)

in which Drm = mean relative density; Var (N) = the blow count vari­ance; Var (Dr) = the variance in the relative density value; and C = (N„, + 6.5 - 0.23a„)/0.0045 with Nm representing the mean blow count value.

Assignment of Material Properties.—After the information contained in the cross sections was condensed into idealized profiles to represent the characteristics of each zone in the susceptibility map, material prop­erties were assigned to each soil type and relative density encountered. Examination of the profiles indicated that for purposes of creating the susceptibility map, there are three different potentially liquefiable soil types that need to be considered within downtown San Francisco. Cyclic strength and pore pressure development curves had to be assigned to each of those soil types for a range of relative densities.

The first soil type is a uniform, fine-grained sand (D50 = 0.2 mm, C„ = 0.5) found in the Yerba Buena Cover, North Beach and Mission Bay fills and in the dune sand deposits, classified SP according to the Unified Soil Classification System. The second potentially liquefiable soil unit is a well-graded, silty sand with greater than 15% fines (D50 = 0.2 mm, C„ > 10), classified SM. This material can be found in parts of the Yerba Buena Cove and North Beach fills. The third type of material is a sandy gravel that shows a range in grain sizes similar to that of the silty sand

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" • " H COARSE I MEbluM m» I — T O T SILT OR CLAY

Raid Bedford model sand

\ \ ^Ottawa sand

sand (site « D

2.0 1.0 0.5 0.4 0.3 0.2 0.1

GRAIN SIZE IN MILLIMETERS

FIG. 3.—Grain Size Curves for Materials Used to Derive Relative Density Corre­lations

o

Lu

O _l

.6

5 .4

<

< |cvi.<i

w en LU K H

No te : Wet Tamping Compoction

D Or = 7 0 % A Dr = 6 0 % V Dr = 5 0 % o Dr = 4 0 %

10 100

NUMBER OF CYCLES

1000

FIG. 4.-—Cyclic Strength Curves for Dune Sand (from Clough and Chameau, 1979)

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just mentioned (D50 = 1 mm, C„ > 10), designated GW in the Unified System.

The source material for each strata in each profile was inferred from the historic information and bore log data. The mean and standard de­viation of blow count values were calculated for each soil strata in the profile from dynamic penetration tests performed within that strata at locations throughout the zone after correction was made for other than standard equipment. Mean and standard deviation values of relative density were calculated using Eqs. 5 and 6. It was assumed that the normalized pore pressure development characteristics were the same for each soil unit, and were given by the results of Clough and Chameau (4). These were the only experimental results available for the potentially liquefiable soils within the study area. The use of a single pore pressure generation curve is consistent with the work of Seed, Martin, and Lys-mer (20) which indicated that the normalized pore pressure develop­ment characteristics of a variety of cohesionless soils falls within a rel­atively narrow band. The cyclic strength characteristics of the SP sand were provided by laboratory cyclic shear tests performed on remolded samples of dune sand recovered from the Yerba Buena Cove fill (4). The tests were performed at four different relative densities (Fig. 4). Cyclic strength curves for any other values of relative density were derived by interpolation of these curves.

There are a variety of factors that influence the relationship between the cyclic shear strength of remolded laboratory samples and the in-situ material, including the method of sample preparation, age, and prior stress and strain histories. However, no quantitative assessment of these factors was feasible. Thus the cyclic strength of the remolded samples was assumed representative of the mean cyclic strength of the deposit, and a relatively large log-normal standard deviation of 0.5 was assigned to the cyclic strength in the probabilistic analysis. A sensitivity study showed results of the analysis to be insensitive to the value of the log normal standard deviation on cyclic strength over the range of 0.3-0.7.

Review of the available literature revealed little in the way of labora­tory test results that would be applicable to the other two potentially liquefiable soils. Most of the published data have been on uniform soils, but there has been work to suggest that the cyclic stress required to cause initial liquefaction in a well-graded soil could be about 60% less than that required for a uniformly graded soil (23).

Lee and Fitton (12) tested a well-graded SM sand (D50 = 0.37 mm, C„ = 32) with greater than 15% fines at a relative density of 50%. The grain size characteristics of this material are very similar to those of the SM sand found in some of the fills in downtown San Francisco, except that the mean grain size is slightly larger (Fig. 5).

Several workers have compiled data to show the effects of median grain size on the cyclic stress required to cause either initial liquefaction or some specified level of strain (12,19,21,23). Cyclic strength curves for the well-graded, sandy gravel and the well-graded, silty sand were de­rived by applying correction factors to account for median grain-size and relative density to the cyclic strength curve obtained by Lee and Fitton (12) for the well-graded SM sand. The grain size correction factors were derived from the work of Wong, Seed, and Chan (23). The cyclic strength

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GRAIN SIZE IN MILLIMETERS

FIG. 5.—Grain-Size Comparison of SM Fill Sand and SM Sand Tested (12) by Lee and Fltton (1969)

NUMBER OF UNIFORM CYCLES OF LOADING

FiG. 6.—Range of Cyclic Strength Curves Included in Analysis

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curve for the GW, sandy gravel (D^, = 1 mm) was obtained by shifting Lee and Fitton's cyclic strength curve upward by multiplying the cyclic stress ratio by a factor of 1.1. The cyclic strength curve obtained by Lee and Fitton was assigned without correction to the SM sand (D50 = 0.2 mm) since the correction factor was very close to 1.0.

The cyclic strength curves thus derived are only valid for a relative density of 50% since that is the value at which Lee and Fitton ran their tests. An additional correction must be made to model materials of the same grain-size distribution, but different relative densities.

Laboratory tests have shown that for relative densities less than about 80%, the cyclic shear stress required to cause initial liquefaction is ap­proximately proportional to the relative density (19). The cyclic strength curve obtained for a relative density of 50% was therefore corrected for relative density effects by applying a correction factor equal to Dr/50 to the cyclic shear strength, in which Dr = the relative density (as a per­centage) of the material that is being modeled. Fig. 6 shows the range of cyclic strength curves investigated for this project. Results allow the prediction of liquefaction potential for any parallel cyclic strength curve within these ranges.

LIQUEFACTION OPPORTUNITY MAPPING FOR DOWNTOWN

SAN FRANCISCO

General Approach.—To determine liquefaction opportunity for San Francisco expected values of the root mean square (RMS) of the shear stress time history and the duration of seismic loading (the number of positive zero crossings) must be determined. The RMS of the shear stress time history, rs, is proportional to the RMS of the acceleration time his­tory, ra, thus the expected value of rs was determined based upon a seismic hazard analysis for r„. The expected duration and its depen­dence on earthquake magnitude was determined by analyses of histo­grams generated from strong motion records.

The seismic hazard analysis for RMS of acceleration requires infor­mation on the location and activity of all potential seismic sources af­fecting the study area and an attenuation relationship for RMS of ac­celeration.

Most existing studies of the attenuation of earthquake ground motions are concerned with PGA. No adequate RMS attenuation relationship was available for use in this project. As part of this study, a comprehensive set of strong motion records obtained within a distance of 100 km of the seismic source for earthquakes of magnitude 5.5 or greater were ana­lyzed. Four different definitions of the duration of strong shaking were used to calculate RMS and three different functional forms for the at­tenuation equation to develop an RMS attenuation relationship. The re­sulting attenuation equation used a definition from Vanmarcke and Lai (22) and a functional form developed for this project by modifying ex­isting PGA attenuation equations. This duration definition was chosen on the basis of the best fit with the available data from strong motion records and of theoretical consistency. The statistical analyses performed to develop the attenuation relationship are described in detail by Ech-ezuria (6). Fig. 7 shows the resulting RMS attenuation relationship.

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0.20

0.10

0.02

0.01

\ M = 7 . 5

•vM-6 • 5 N .

N. M-5.5 N.

-

V \ \

\ \ \ 5.0 10. 50. 100.

Distance to Causative Fault (Km)

FIG. 7.—RMS Attenuation Relationship

NUMBER OF POSITIVE ZERO CROSSINGS

50 100

NUMBER OF POSITIVE ZERO CROSSINGS

FIG. 8.—Distributions of Number of Positive Zero Crossings: (a) M < 6.25; (b) M > 6.25

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FIG. 9.—Potential Seismic Sources Considered in Analysis and Haiard Grid

0.4

0.3

0.2

.1 .01 .001

ANNUAL PROBABILITY OF EXCEEDANCE

FIG. 10.—Results of RMS Hazard Analysis

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The RMS attenuation equation developed for this project was substi­tuted for the PGA attenuation equation within an existing seismic haz­ard program (9) to predict expected levels of RMS of acceleration at four locations surrounding the study area. Expected values of RMS were cal­culated in a Bayesian manner, using data obtained after 1906 to update historically derived magnitude recurrence relationships from 1800-1906 for each seismic source (14).

Histograms of the number of positive zero crossings using Vanmarcke and Lai's definition of RMS and duration were generated at the same time the attenuation relationship was being developed. Inspection of these histograms showed some dependence of the number of zero crossings on earthquake magnitude. Accordingly, two separate lognormal distri­butions were used for the number of zero crossings, depending upon whether the magnitude of the earthquake was greater or less than 6.25. A magnitude of 6.25 was chosen as the dividing line because it was close to the centroid of the data set. Fig. 8 shows the histograms and the lognormal distributions used to characterize them. The results of the seismic hazard analysis were also separated into these two magnitude-dependent subsets.

Assessment of Regional Seismicity.—Four faults were assumed ca­pable of generating seismic events of large enough magnitude to induce liquefaction in downtown San Francisco. These faults are the San An­dreas, Hay ward, Calaveras, and San Gregorio. The location of these four faults with respect to the study area is shown in Fig. 9 along with the four points that define the grid for the hazard analysis.

Table 1 presents the magnitude-recurrence rates assumed for each seismic source in this analysis. These recurrence rates are similar to those used in previous seismic hazard analyses of the Bay Area (15). Note that the San Andreas fault is split into two sections, the San Andreas North and the San Andreas creep section, for the purposes of the hazard anal­ysis (Fig. 9).

Results of RMS Hazard Analysis.—Because earthquakes of magni­tude greater than 6.25 were assigned a different duration distribution than events of magnitude less than 6.25, the results of the hazard anal­ysis were separated into two data sets. The results of the hazard analysis are presented in Fig. 10. This figure shows the mean expected annual probability of exceedance of acceleration RMS for the four grid points for both subsets of earthquakes, plus the overall mean annual proba­bility of exceedance.

TABLE 1.—Assumed Annual Recurrence Rate for Potential Seismic Sources

Fault (1)

San Andreas North San Andreas Creep Hayward Calaveras San Gregorio

Magnitude Range

(2)

0.054 0.041 0.014 0.019 0.022

5.5^6 (3)

0.029 0.021 0.006 0.005 0.013

6-6.5 (4)

0.018 0.012 0.004 0.003 0.007

6.5=7 (5)

0.011 0.005 0.003 0.002 0.004

7-7.5 (6)

0.007 0.003 0.002 0.002 0.001

~7 tz O

(7) 0.004 0.002 0.001 0.001

>8 (8)

0.001 0.001

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For annual probabilities of exceedance of less than 0.01, the range of interest for this study, the hazard analysis yielded the same results for all four points on the hazard grid shown in Fig. 9. For annual proba­bilities of exceedance of less than 0.01, the use of the mean curve can lead to erroneous values of expected RMS. However, for these low prob­abilities of exceedance and the associated loadings, the probability of initial liquefaction for typical San Francisco soils is likely to be 1.0 re­gardless of which point on the grid is used.

EVALUATION OF LIQUEFACTION POTENTIAL IN DOWNTOWN SAN FRANCISCO

Methodology.—The loading parameter used to describe earthquake intensity is the RMS of the shear stress time history normalized by the vertical effective stress. The RMS of shear stress, rs, is related to the RMS of acceleration at the ground surface, ra, through the total vertical stress, CT0 , and a flexibility factor, /:

r, = frs<y0 (7)

The normalized value of rs, r's, the RMS of the cyclic shear stress ratio, becomes:

r'.-fr,-, (8)

in which v'0 = the vertical effective stress. The flexibility factor, /, for depths of 10 m or less is given by Seed and Idriss (19) as:

/ = 1 - O.Olz (9)

in which z = the depth in meters. Eqs. 8 and 9 are used to convert the design r„ values developed in the

hazard analysis to the shear stress loading parameter for the pore pres­sure model.

To calculate the probability of liquefaction for a specific design ra, the probability of liquefaction given ra was calculated for 1-105 cycles of loading using Chameau's pore pressure model. It was then calculated for both distributions of zero crossings by numerically integrating Eq. 3.

The total probability of liquefaction for a specified ra, P(L\ra), is the weighted sum of the probabilities for each magnitude range:

P(L|r„) = W[P(L\ra ,M < 6.25)] + (1 - W)[P(L\ra ,M > 6.25)] (10)

in which the weighting term, W = the ratio of the probability of ra from earthquakes of magnitude less than 6.25 to the probability of ra from earthquakes of all magnitudes.

W = P(RMS > ra\M < 6.25)|P(RMS > r„) (11)

Values of W for any RMS value can be calculated from Fig. 10. The constituent and total probabilities of liquefaction were calculated

for a wide variety of relative densities and RMS values over the range of interest for all three soil types encountered within the study area.

Conditional Probability of Liquefaction.—To aid in the interpretation

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FIG. 11.—Normalized Probability of Liquefaction Curves for Poorly Graded Dune Sand (SP)

FIG. 12.—Normalized Probability of Liquefaction Curves for Well-Graded Soils (SM, GW)

of the results, the conditional probability of liquefaction, or liquefaction fragility, was plotted against the normalized RMS of the cyclic shear stress ratio normalized by the relative density, r's/Dr, for each soil type and for both distributions of zero crossings. Since the cyclic shear strength curves representing material resistance for this analysis were assumed directly proportional to D r , and since the intensity of loading is directly proportional to r's, the ratio r's/Dr should provide a unique measure of the probability of liquefaction for a series of parallel cyclic strength curves. Since the cyclic strength curves for the SM and GW materials are par­allel, results from these two soils were plotted together with the nor­malizing relative density for the GW soil equal to 1.1 times its in situ relative density to account for its increased strength with respect to the SM soil.

Figs. 11 and 12 present the normalized plots of the probability of liq­uefaction for the poorly graded (SP) and well-graded (SM,GW) soils, respectively. As these figures show, the hypothesis of a unique rela­tionship between the probability of liquefaction and the ratio r'JDr is valid. These two figures completely define the conditional probability of liquefaction for any relative density for any of the three characteristic soil types within the study area. These figures can be used in conjunc­tion with Fig. 10 to predict the probability of liquefaction for any spec­ified annual probability of exceedance or specified design event.

Interpretation of Results.—Fig. 13 shows the final liquefaction sus­ceptibility map. Typical profiles developed for each of the six map zones shown in Fig. 13 are summarized in Table 2, which gives the mean rel­ative density, vertical effective stress, and vertical total stress for each profile along with the probability of liquefaction for intensities with an­nual probabilities of exceedance of 0.05, 0.02, and 0.01. The water table was assumed to be at the surface for all profiles with the exception of the dune sand profile, where it was assumed to be at a depth of 12 ft. The following conclusions can be drawn about the liquefaction potential of the cohesionless soil deposits in downtown San Francisco from the data presented in this table.

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FIG. 13.—Liquefaction Potential Map for San Francisco

For an intensity with an annual probability of exceedance of 0.05, none of the typical profiles are expected to experience any significant pore pressure development. For an annual probability of exceedance of 0.02, only the densest well-graded sandy gravel deposits, found primarily around the base of Telegraph Hill, and the dune sand deposits are not expected to experience significant pore pressure generation. For an in­tensity with an annual probability of exceedance of 0.01, even the dense sandy gravels and dune sands may achieve the zero effective stress state,

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TABLE 2.—Probability of Initial Liquefaction for Representative Profiles

Location (1)

Yerba Buena Cove Yerba Buena Cove Telegraph Hill North Beach North Beach Mission Bay Mission Bay South Beach Dune Sand"

Soil type (2)

SP SM GW GW SM SP SM GW SP

Relative density, as a per­centage

(3)

81 60 93 60 70 81 85 64 90

Effective vertical stress (PSI) (4)

1.9 3.8 3.8 1.9 3.8 1.9 1.9 3.8

13.98

Total vertical stress (PSI) (5) 4.1 8.1 8.1 4.1 8.1 4.1 4.1 8.1

17.5

PROBABILITY OF LIQUEFACTION

Return Period, in Years

20 (6)

0.01 0.01

0.01

50 (7)

0.98 0.99 0.14 0.97 1.0 0.98 0.67 0.93

100 (8)

1.0 1.0 0.98 1.0 1.0 1.0 1.0 1.0 0.84

"Water table depth equal to 12 ft. Water table at ground surface for other lo­cations.

or initial liquefaction. It must be emphasized that the consequences of initial liquefaction in a dense or gravelly soil may be minimal, thus the achievement of the zero effective stress state is not synonomous with damage for such soils. Furthermore, significant pore pressure dissipa­tion may occur in the gravelly soil, retarding pore pressure development and preventing initial liquefaction.

Expected levels of RMS may be related to specific seismic events to aid in the interpretation of the results. For the 0.05 annual probability of exceedance, the design RMS corresponds to a magnitude 5.5 event at 12 km, or a magnitude 7.5 event at 50 km. The 0.02 probability level corresponds to M6.5 at 14 km and M7.5 at 30 km. The 0.01 probability level corresponds to M6.5 at 3 km, or M7.5 at 16 km from the study area. Note that the nearest active fault to the study area, the San An­dreas, comes within 7-10 km of the downtown area at its closest ap­proach.

These are only representative combinations of magnitude and distance that will yield the specified RMS. There are actually an infinite number of magnitude/distance combinations that will yield a given design RMS, but the smaller magnitude events have much higher annuaj probabilities of occurrence than the larger ones.

The method described in the previous section can be used for site spe­cific studies of the probability pf liquefaction when the relative density is known. Of particular interest is the annual probability of exceedance or design earthquake for which the probability of liquefaction is 0.5. If the resistance and capacity are symmetrically distributed, a probability of liquefaction of 0.5 corresponds to a factor of safety 1.0. A probability of liquefaction greater than 0.5 may be interpreted as meaning signifi­cant initial liquefaction is likely. Table 3 presents return periods and seismic events resulting in a probability of liquefaction of 0.5 for the range of relative densities expected in all three soil types examined in this report.

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TABLE 3.—Representative Conditions for Probability of initial Liquefaction of 0.50

Soil type

(1)

SP SP SM SM SM GW GW GW

Relative density

as a per­centage

(2)

75 90 60 70 85 60 75 90

Acceleration RMS, in

gravitational acceleration

(3)

0.052 0.060 0.054 0.062 0.076 0.058 0.074 0.096

Annual probability

exceedence (4)

0.032 0.028 0.030 0.026 0.022 0.029 0.023 0.014

Distance, in Kilometers to Event of Magnitude

5.5 (5)

5 3 5 3 1 4 2 —

6.5 (6)

19 15 18 16 10 17 11 7

7.5 (7)

39 33 38 34 27 36 29 22

When calculating these probabilities, the sensitivity of the probability of liquefaction to shear stress RMS became evident. For the well-graded soils, a 10% increase in RMS can increase the probability of liquefaction for a major event (M > 6.25) from 0.30-0.70. One of the major factors affecting the shear stress RMS is the location of the water table. In these analyses, the water table was assumed to be at the ground surface.

Return periods for events causing significant liquefaction would in­crease to the 50-60 yr range if the water table were placed 5 ft below the ground surface. Many of the boring logs indicated the water table at this depth. However, since most of the borings were made during the summer dry season, the water table was assumed to have the po­tential to be at the ground surface, and liquefaction potential was eval­uated on this basis. Assuming the water table fluctuates between the surface and a depth of 5 ft, the return period for a probability of liq-

TABLE 4.—Significant Earthquakes with Close Eplcentral Proximity to San Fran­cisco Waterfront"

Date of event

(1) 1865

1868 1898 1906 1933 1957 1969

Estimated epicenter

area (2)

Santa Cruz mountains

Hayward Mare Island San Francisco Fremont Daly City Santa Rosa

Approximate distance of

epicenter from waterfront,

in miles (kilometers)

(3)

47 (75)

13 (21) 20 (32) 30" (48) 17 (27) 10 (16) 50 (80)

Intensity (modified mercali)

(4)

VIII-IX

IX-X VIII XI VII VII

VII-VIII

Magnitude (richter)

(5)

—

— — 8.3 — 5.3 5.7

Ground failure in

San Francisco (6)

slight

substantial none substantial none none none

"From Clough and Chameau (4). bDistance to fault break was 10 m (16 km).

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uefaction of 0.5 would be between 40 and 50 yr for most of the water­front fill areas.

Comparison with Historical Evidence.—Youd and Hoose (25) have presented historical evidence on the performance of the cohesionless soil deposits within San Francisco under seismic loading. This evidence, summarized in Table 4, indicates that liquefaction occurred three times in the 41 yr period from 1865-1906, although liquefaction has not oc­curred since the 1906 earthquake. The Bayesian RMS hazard analysis weights expected values of acceleration RMS towards the relatively ac­tive seismic period from 1800-1906 and away from the current relatively quiet period from 1907 to the present (due to the longer duration of the active period). Keeping this in mind, the prediction of return periods of between 40 and 50 yr for intensities resulting in a probability of lique­faction of 0.5 seems reasonable.

CONCLUSIONS

Despite the large uncertainties associated with a variety of the com­ponents of the liquefaction hazard analysis described in this paper, the analysis presented herein clearly shows that there is a high potential for widespread liquefaction-induced ground failure within the Yerba Buena Cove, North Beach, Mission Bay, and South Beach areas of San Fran­cisco. This conclusion is not surprising considering the historical ac­counts of liquefaction-induced failure from past earthquakes. In a major seismic event, there is likely to be many incidents of disrupted utility and transportation lines and damage to structures on shallow founda­tions in these areas. Planning of emergency response measures must take this into account.

In the Dune Sand and Telegraph Hill map zones, initial liquefaction may occur in extreme seismic events, however, the consequences of ini­tial liquefaction in these relatively dense soil deposits may not be as se­vere as in the more susceptible zones. Liquefaction of isolated pockets of loose material in these areas could conceivably cause damage in mod­erate seismic events. Site-specific studies are recommended for critical lifelines and facilities in these areas.

ACKNOWLEDGMENTS

This work was performed under the auspices of the John A. Blume Earthquake Engineering Center at Stanford University. Funding for this research was provided by the United States Geological Survey, Depart­ment of the Interior as part of the Earthquake Hazard Reduction Pro­gram under contract No. 14-08-0001-2059. The assistance of Dr. Martin McCann was invaluable in developing the RMS attenuation relationship and hazard analysis.

Mr. Roy Bell and Dr. William Villet of Dames and Moore, Mr. James Bowers of Harding-Lawson Associates, and Dr. Rudolph Bonaparte of Woodward-Clyde Consultants generously provided the writer with ac­cess to their companies files during the data collection phase of this proj­ect.

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APPENDIX.—REFERENCES

1. Benjamin, J. R., and Cornell, A., Probability, Statistics, and Decision for Civil Engineers, McGraw-Hill, 1970.

2. Chameau, J. L., "Probabilistic and Hazard Analysis for Pore Pressure In­crease in Soils due to Seismic Loading," dissertation presented to Stanford University, Department of Civil Engineering, at Stanford, Calif., in 1980, in partial fulfillment of the requirements for the degree of Ph.D.

3. Chameau, J. L., and Clough, G. W., "Probabilistic Pore Pressure Analysis for Seismic Loading," Journal of Geotechnical Engineering, ASCE, Vol. 109, No. GT4, Apr., 1983, pp. 507-524.

4. Clough, G. W., and Chameau, J. L., "A Study of the Behavior of the San Francisco Waterfront Fills during Seismic Loading," Report No. 35, John A. Blume Earthquake Engineering Center, Stanford University, Feb., 1979.

5. Dow, G. R., "Bay Fill in San Francisco: A History of Change," dissertation presented to California State University, at San Francisco, Calif., in 1973, in partial fulfillment of the requirements for the degree of M.A.

6. Echezuria, H., "Determination of Liquefaction Opportunity for Downtown San Francisco, California," dissertation presented to Stanford University, De­partment of Civil Engineering, at Stanford, Calif., in 1983, in partial fulfill­ment of the requirements for the degree of Doctor of Engineering.

7. Fardis, M. N., and Veneziano, D., "Probabilistic Analysis of Deposit Liq­uefaction," Journal of the Geotechnical Engineering Division, ASCE, Vol. 108, No. GT3, Mar., 1982, pp. 395-418.

8. Gibbs, H. J., and Holtz, W. G., "Research on Determining the Density of Sands by Spoon Penetration Testing," Proceedings of the 4th International Con­ference on Soil Mechanics, London, Vol. 1, 1957, pp. 35-39.

9. Guidi, G. A., "Computer Programs for Seismic Hazard Analysis. A User Manual," Report No. 36, The John A. Blume Earthquake Engineering Center, Stanford University, Mar., 1979, pp. 132-188.

10. Haldar, A., and Tang, W. H., "Probabilistic Evaluation of Liquefaction Po­tential," Journal of the Geotechnical Engineering Division, ASCE, Vol. 105, No. GT2, Feb., 1979, pp. 145-164.

11. Kavazanjian, E., Roth, R. A., and Echezuria, H., "Probabilistic Evaluation of Liquefaction Potential for Downtown San Francisco," Report No. 60, John A. Blume Earthquake Engineering Center, Stanford University, Apr., 1983.

12. Lee, K. L., and Fitton, J. A., "Factors Affecting the Cyclic Loading Strength of Soil," Vibration Effects of Earthquakes on Soils and Foundations, American So­ciety of Testing and Materials, Special Technical Publication 450, 1969, pp. 71-95.

13. Marcuson, W. F., and Bieganousky, W. A., "Laboratory Standard Penetra­tion Tests on Fine Sands," Journal of the Geotechnical Engineering Division, ASCE, Vol. 103, No. GT6, June, 1977, pp. 565-588.

14. Mortgat, C. P., and Shah, H. C , "A Bayesian Approach to Seismic Hazard Mapping," Report No. 28, John A. Blume Earthquake Engineering Center, Stanford University, 1978.

15. Roth, R. A., "Mapping Liquefaction Susceptibility in Downtown San Fran­cisco, Calif., dissertation presented to Stanford University, Department of Civil Engineering, at Stanford, Calif., in 1983, in partial fulfillment of the requirements for the degree of Doctor of Engineering.

16. Roth, R. A., and Kavazanjian, E., Jr., "Liquefaction Susceptibility Mapping for San Francisco, California," accepted for publication, Bulletin of the Asso­ciation of Engineering Geologists, Nov., 1984.

17. Salah-Mars, S., "Wave Equation Analysis Applied to the Standard Penetra­tion Test," dissertation presented to Stanford University, Department of Civil Engineering, at Stanford, Calif., in 1980, in partial fulfillment of the require­ment for the degree of Doctor of Engineering.

18. Schlocker, J., "Geology of the San Francisco North Quadrangle, California," U.S. Geological Survey Professional Paper 782, 1974.

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19. Seed, H. B., and Idriss, I. M., "Simplified Procedure for Evaluating Soil Liq­uefaction Potential," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 97, No. SM9, Sept., 1971, pp. 1249-1273.

20. Seed, H. B., Martin, P. P., and Lysmer, J., "Pore-Pressure Changes During Soil Liquefaction," Journal of the Geotechnical Engineering Division, ASCE, Vol. 102, No. GT4, Apr., 1976, pp. 323-346.

21. Seed, H. B., and Peacock, W. H., "Test Procedures for Measuring Soil Liq­uefaction Characteristics," Journal of the Soil Mechanics and Foundations Divi­sion, ASCE, Vol. 97, No. SM8, Aug., 1971.

22. Vanmarcke, E. H., and Lai, S. P., "Strong Motion Duration and RMS Am­plitude of Earthquake Records," Bulletin of the Seismological Society of America, Vol. 70, No. 4, Aug., 1980, pp. 1193-1307.

23. Wong, R. T., Seed, H. B., and Chan, C. K., "Cyclic Loading Liquefaction of Gravelly Soils," Journal of the Geotechnical Engineering Division, ASCE, Vol. 101, No. GT6, June, 1975, pp. 571-586.

24. Yegian, M. K., and Whitman, R. V., "Risk Analysis for Ground Failure by Liquefaction," Journal of the Geotechnical Engineering Division, ASCE, Vol. 104, No. GT7, July, 1978, pp. 921-938.

25. Youd, T. L., and Hoose, S. N., "Liquefaction during the 1906 San Francisco Earthquake," Journal of the Geotechnical Engineering Division, ASCE, Vol. 102, No. GT5, May, 1976, pp. 425-439.

26. Youd, T. L., and Hoose, S. N., "Historic Ground Failures in Northern Cal­ifornia Triggered by Earthquakes," U.S. Geological Survey Professional Paper 993, 1978.

27. Youd, T. L., and Perkins, D. M., "Mapping Liquefaction-Induced Ground Failure Potential," Journal of the Geotechnical Engineering Division, ASCE, Vol. 104, No. GT4, Apr., 1978, pp. 433-446.

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