Probabilistic correlation between laboratory and field liquefaction potentials using relative state parameter index (ξR)

Soil Dynamics and Earthquake Engineering 30 (2010) 1061–1072

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Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn

Probabilistic correlation between laboratory and field liquefaction potentials using relative state parameter index (xR) Y. Jafarian a, A. Sadeghi Abdollahi b, R. Vakili b, M.H. Baziar b,n a b

College of Civil Engineering, Semnan University, Semnan, Iran School of Civil Engineering, Iran University of Science and Technology, Tehran 16846-13114, Iran

a r t i c l e in fo

abstract

Article history: Received 4 March 2009 Received in revised form 10 April 2010 Accepted 17 April 2010

Cyclic triaxial tests on reconstituted sand samples are broadly employed in practice while they ignore the inherent characteristics of soil in field condition such as aging, fabric, and prior strain history. Relative state parameter index, xR, is utilized in a probabilistic framework to adjust the cyclic triaxial resistance ratio of sands at 15 uniform cycles (CRRtx,15) to field condition. A wide-ranging database containing the results of cyclic triaxial tests conducted on reconstituted samples has been compiled to derive a correlation between relative state parameter index (xR) and triaxial cyclic resistance ratio. The adjustment coefficients proposed by researchers are employed to correct CRRtx,15  xR relationship for actual field condition. The adjusted CRRtx,15  xR relationships are applied to a database of field liquefaction case histories composed of both SPT and CPT based data and their performances in field condition are evaluated. It is demonstrated that constant triaxial-to-field adjusting coefficients cannot ever predict conservative results. Logistic regression method is employed to derive a field probabilistic criterion that obtains the likelihood of liquefaction initiation in terms of xR. The xR-based boundary curve standing for 20% likelihood of liquefaction initiation is found to be the most conservative limit state boundary to be used in field conditions. Finally, the triaxial and field CRR  xR relationships are composed and a probabilistic triaxial-to-field adjustment coefficient is proposed in terms of xR and a given liquefaction probability. It is anticipated that the proposed relationship could reasonably correct the results of cyclic triaxial testing on freshly reconstituted sand samples. & 2010 Elsevier Ltd. All rights reserved.

1. Introduction The undrained cyclic behavior of saturated and silty sands in field and laboratory conditions is a concept of interest for geotechnical earthquake engineers. Cyclic response of granular soils in undrained condition leads to liquefaction when the earthquake induced excess pore water pressure reaches initial effective stress. Liquefaction has caused catastrophic failures during past earthquakes. Thus, proper estimation of liquefaction potential in projects that are located in places encountered this problematic response of granular soils is an essential step of the design procedure. In spite of the numerous studies performed around this subject, several issues have remained unclear due to the complicated and stochastic nature of soils and earthquakes. Two common approaches have been recommended for evaluating liquefaction potential in level and mildly sloping sites: (1) using laboratory cyclic tests and (2) using empirical correlations derived from the field performance of liquefiable soils.

n

Corresponding author. Tel.: + 98 21 77240450; fax: + 98 21 77240451. E-mail address: [email protected] (M.H. Baziar).

0267-7261/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2010.04.017

To avoid the difficulties associated with sampling and laboratory testing, field tests have become state-of-the-practice for routine liquefaction investigations [1]. The results of in situ tests such as standard penetration test (SPT), cone penetration test (CPT), and shear wave velocity have been employed in simplified frameworks to achieve empirical limit state boundary curves for the quick deterministic and probabilistic assessment of cyclic resistance ratio in field condition, CRRfield, [1–8]. Although the empirical limit state boundary curves have been updated and modified due to increase in number of field observations, a few laboratory cyclic tests are ever recommended for high risk and important projects. In addition, cyclic laboratory tests have provided valuable insights into the liquefaction behavior of soils in order to support and modify field empirical criteria. Cyclic simple shear and triaxial devices have been employed by researchers to investigate liquefaction resistance of reconstituted and frozen sand samples [9–14]. High quality cyclic testing by expensive frozen sampling techniques is beyond the budget of most engineering works. Meanwhile, permeability conditions in silty soils can make it impossible to obtain frozen samples without complete disturbance of soil samples due to ice expansion between particles [5]. On the other hand, the results of cyclic tests conducted on reconstituted samples are

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Nomenclature a coefficient to relate corrected standard penetration resistance and relative density CPT cone penetration test Cr correction factor to adjust for the difference between stress conditions of cyclic triaxial and simple shear tests CRR cyclic resistance ratio CRRfield field cyclic resistance ratio CRRfield,7.5 field cyclic resistance ratio at the earthquake magnitude of 7.5 CRRss cyclic resistance ratio in simple shear test condition CRRtx triaxial cyclic resistance ratio CRRtx,15 triaxial cyclic resistance ratio at 15 uniform cycles CRRxR xR-based probabilistic field cyclic resistance ratio CRRxR ,20% xR-based probabilistic field cyclic resistance ratio at 20% liquefaction probability Cr(VS) xR-based Cr derived from Vaid and Sivathayalan’s [24] data CSReq equivalent uniform cyclic stress ratio CSReq field cyclic stress ratio corrected for earthquake duration Ct  f xR-based triaxial-to-field adjustment coefficient at 20% liquefaction probability Ct  f, PL xR-based probabilistic triaxial-to-field adjustment coefficient Dr relative density Dr,cs relative density on critical state line DWFm duration weighting factor e void ratio ecs void ratio on critical state line Cd

questionable as they cannot often simulate the inherent characteristics of soils in field condition such as aging (e.g., [15]), fabric (e.g., [16]), prior strain history (e.g., [17]), overconsolidation (e.g., [18]), and cementation. Nevertheless, cyclic tests on reconstituted samples are further common than that on frozen ones in both practical and research works due to the difficult and expensive procedure of frozen sampling technique. Researchers (e.g., [6,15]) indicated that cyclic simple shear test can better simulate the behavior of a soil element in a level site under horizontal earthquake loading compared to cyclic triaxial test. In order to consider the influence of multi-directional excitation during earthquake in actual field condition, an adjustment coefficient of 0.9 was recommended to be applied to the CRR values obtained from simple shear test, CRRss, (e.g., [15,19]). Also, another correction factor, known as Cr, has been proposed to adjust CRR values for the difference between the stress conditions of triaxial and simple shear tests (e.g., [20–22]). Tokimatsu and Uchida [23] used the correlation between maximum shear modulus and shear wave velocity (Vs) to apply triaxial cyclic resistance to field case histories. They adjusted triaxial-based cyclic resistance of various reconstituted and frozen sand samples by using a constant coefficient and compared the triaxial-based results with Vs-based field data. However, Vaid and Sivathayalan [24] showed that Cr cannot be constant and depends on relative density and initial effective confining pressure of soil. In spite of the mentioned efforts, it seems that there is not a rigorous framework to bridge the gap between laboratory and field cyclic resistance ratios. Furthermore, a constant triaxial-tofield adjustment coefficient cannot introduce a uniform and

emax emin FC Fs IR K0 Ks MSF Mw n Nliq NSPT N1,60 Pa PL p’ Q

maximum void ratio minimum void ratio percentage of non-plastic fines content factor of safety relative density index lateral earth pressure at rest adjustment factor for overburden pressure magnitude scaling factor earthquake moment magnitude total number of explanatory variables number of cycles required for liquefaction initiation SPT blow count corrected standard penetration resistance atmospheric pressure likelihood of liquefaction occurrence initial mean effective confining pressure an empirical constant dependent to the mineralogy and breakage of soil qC1N normalized cone penetration resistance ru excess pore water pressure ratio SPT standard penetration test Vs shear wave velocity X¼[x1 ,x2,y,xn] vector of explanatory variables b ¼[b0 ,b1,y,bn] regression coefficients x state parameter xR relative state parameter index eDA double amplitude axial strain 0 s3 initial effective confining pressure sdev maximum cyclic deviatoric stress 0 sv0 initial vertical effective stress tcyc cyclic shear stress tmax maximum shear stress

specified level of conservatism or liquefaction risk for a given soil sample. On the other hand, cyclic triaxial tests on freshly reconstituted sand samples are broadly employed in practical and research works. Therefore, there should be a necessity for development of adjustment coefficient that improves the results of triaxial tests on freshly reconstituted specimens. Direct correlation between field and triaxial cyclic resistances seems to be rational to establish the basis for developing an advanced adjustment coefficient. Using a probabilistic approach, the current study develops an advanced triaxial-to-field adjustment coefficient that assures a uniform level of liquefaction risk for a given soil and site condition. In order to achieve this aim, triaxial and field cyclic resistances have to be expressed in terms of an identical parameter that can be conveniently determined in both laboratory and field conditions. Relative state parameter index (xR) that was gradually introduced by Been and Jeffries [25], Bolton [26], and Boulanger [27] has been found to be useful for this purpose. Two comprehensive databases of triaxial test results and field SPT and CPT based liquefaction case histories were gathered to derive laboratory and field based limit state boundaries. Logistic regression approach has been used to obtain the probabilistic adjustment coefficient. The proposed adjustment coefficient conservatively converts triaxial cyclic resistance ratio into field condition and improves some of the uncertainties involved in using cyclic tests on freshly reconstituted samples. It is demonstrated that the existing constant adjustment coefficients result in unsafe prediction of field cyclic resistances in contractive soils and over-conservative evaluation of that parameter in dilative soils.

Y. Jafarian et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 1061–1072

2. Correlating cyclic resistance ratio to nR for freshly reconstituted specimens in triaxial test condition 2.1. Relative state parameter index, xR From the earliest studies of soil behavior under cyclic loading condition, it has been concluded that the liquefaction inducing cyclic stress is profoundly influenced by the relative density (Dr) of the soil [15]. Convenient determination of relative density for granular soils in laboratory and also its correlations with field tests data (e.g., SPT and CPT) are the most important advantages of relative density that has been widely used to correlate laboratory and field studies (e.g., [28,29]). On the other hand, several researchers demonstrated the influence of initial effective confining pressure on liquefaction resistance (e.g., [9,27]). Been and Jeffries [25] indicated that the properties of sands cannot be expressed only in terms of relative density, but a description of effective stress level must also be included. As they showed, sands and silty sands behave similarly if test conditions assure an equal initial proximity to the steady state line. This proximity was identified by Been and Jeffries [25] as ‘‘state parameter’’ (x), which is the difference between initial and steady state void ratios at the same mean effective stress (Eq. (1)). This parameter appropriately reflects the combined effects of density and confining pressure in granular materials.

x ¼ eecs ¼ ðemax emin ÞðDr,cs Dr Þ

ð1Þ

Fig. 1. Relative state parameter index versus cyclic resistance ratio for Fraser Delta sand (after Vaid and Sivathayalan [24] data; adopted from Boulanger [27]).

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where e is the void ratio of the soil, ecs the void ratio of the soil on critical state line at the same effective stress, emax and emin are the maximum and minimum void ratios, Dr,cs the relative density on critical state line at the same effective stress, and Dr the relative density. Bolton [26] introduced relative density index, IR, as a measure to reflect the dilatancy potential of granular soils. Konrad [30] and Boulanger [27] used relative density index, IR, and normalized the state parameter (x) with respect to emax  emin to propose relative state parameter index (xR) as a parameter that is more useful and applicable than x in field condition:

xR ¼

1 Dr Q lnð100p0 =Pa Þ

where

0

p0 ¼ ð1 þ 2K0 Þsv0 =3

ð2Þ

where p0 is the initial mean effective confining pressure, Pa the 0 atmospheric pressure, K0 the lateral earth pressure at rest, sv0 the initial vertical effective stress, and Q an empirical constant dependent to the mineralogy and breakage of soil (e.g., Q¼10 for quartz sands). Bolton [26] estimated that Q is approximately 10 for quartz and feldspar, 8 for limestone, 7 for anthracite, and 5.5 for chalk mineral. Although the accurate estimation of Q is necessary for the soil under consideration, it can be approximately considered 10 for this study because most common sands and silty sands are composed of quartz and feldspar. According to Eq. (2), the variation of xR for Q¼10 is in the range of about 0.2 to  0.8 when the relative density and mean effective confining stress vary between 0–100% and 50–400 kPa, respectively. Pillai and Muhunthan [31] observed that cyclic resistance ratio (CRR) of clean sand is approximately constant for a specified value of x. Boulanger [27] employed the results of the cyclic triaxial and simple shear tests conducted by Vaid and Sivathayalan [24] on clean Frazer Delta Sand to illustrate the correlation between CRR and xR for triaxial and simple shear tests conditions (Fig. 1). Relative state parameter index, xR, has been found to be appropriate for the scope of the current study, i.e. developing a probabilistic triaxial-to-field adjustment coefficient. Therefore, triaxial and field cyclic resistances are correlated to xR since it inherently accounts for the influences of relative density and initial effective stress and can be directly calculated in both laboratory and field conditions. Furthermore, this parameter originates from dilatancy concept which is consistent with liquefaction phenomenon. The suitability of xR for this purpose is also confirmed by the conclusion of Vaid and Sivathayalan [24] indicating that the correlation between cyclic triaxial and simple shear resistances is a function of both relative density and initial

Table 1 Introducing the sands that their cyclic triaxial test results compiled into the database of laboratory tests, failure criterion is double amplitude axial strain of 5%.

Sand name Mai Liao Mai Liao Ottawa Ottawa Ottawa Ottawa Fraser Delta Ottawa Ottawa Fuzhou Monterey Ottawa 20–30

Sample preparation method Moisture tamping Moisture tamping Moisture tamping Moisture tamping Moisture tamping Moisture tamping Water sedimentation Moisture tamping Moisture tamping Saturated tamping Moisture tamping Moisture tamping and water sedimentation

Silts content (%)

emax

emin

Dr (%)

e

r0 3 (kPa)

Ref.

0 15 0 5 10 15 1 0 15 0 0

1.125 1.058 0.78 0.7 0.65 0.63 1 0.8 0.75 0.79 0.85

0.646 0.589 0.48 0.42 0.36 0.32 0.68 0.608 0.428 0.43 0.56

0.73–0.97 0.73–0.98 0.48–0.57 0.48–0.58 0.48–0.59 0.48–0.60 0.77–0.9 0.675 0.555 0.55–0.63 0.676

100 100 100 100 100 100 50–400 100 100 50–400 100

Huang et al. [13] Huang et al. [13] Carraro et al. [12] Carraro et al. [12] Carraro et al. [12] Carraro et al. [12] Vaid and Sivathayalan [24] Kanagalingam [38] Kanagalingam [38] Zhou and Chen [39]

10

0.87

0.49

32.4–76.1 32.4–76.2 19.4–80.4 19.4–80.5 19.4–80.6 19.4–80.7 31–72 65.10 60.56 45–65.3 60 40

0.68–0.75

100–250

Silver et al. [40] Amini and Qi [11]

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effective confining pressure rather than a single value related to initial stress condition (K0).

only cited in Table 1 and incorporated in the curve fitting procedure.

2.2. The CRRtx,15  xR relation A comprehensive database of cyclic test results is required to establish a correlation between triaxial based cyclic resistance (CRRtx) and relative state parameter index (xR). Numerous cyclic triaxial test results of freshly reconstituted samples reported by previous researchers were examined and those containing sufficient variations of initial relative density, void ratio, and effective confining pressure versus CRR were extracted. The results of the samples containing silts content more than 15% were excluded from the database because of the insufficient number of such data and difficulties in determining emax and emin for sands containing high fractions of silts [32–35]. The cyclic resistance of sands with less than 15% silts is expected to be sand-like and conservatively less than ‘‘limiting silts content’’ that is a matter of discussion among the researchers [36,37]. Nevertheless, the influence of increase in silts content even up to 15% on CRRtx  xR relationship is subsequently investigated in this paper. Table 1 presents properties of the soils involved in the database of isotropically consolidated cyclic triaxial tests. This collection is composed of 33 data of Huang et al. [13], 64 of Carraro et al. [12], 31 of Kanagalingam [38], 44 of Zhou and Chen [39], 36 of Silver et al. [40], and 9 of Amini and Qi [11] studies. Some data of Huang et al. [13] are shown in Fig. 2, for instance, in terms of CRRtx versus Nliq, where Nliq stands for the number of cycles required for liquefaction initiation. Although a particular target void ratio (or relative density) in triaxial tests cannot be exactly obtained [11], the values of relative density or void ratio cited in Table 1 are those achieved after the consolidation phase of tests procedure. The values of CRRtx at 15 uniform cycles of shear stress (referred to as CRRtx,15) were extracted from the CRRtx Nliq curves. The number of uniform cycles to cause liquefaction onset (Nliq) is simply related to earthquake moment magnitude. For example, Nliq ¼15 corresponds to moment magnitude of 7.5 according to the recommendation of Idriss and Boulanger [4]. The data of Vaid and Sivathayalan [24], which correspond to 10 uniform stress cycles to cause a single amplitude axial strain of 2.5%, were adjusted to 15 uniform cycles of loading using the magnitude scaling factor (MSF) proposed by Seed and Idriss [41]. A total number of 53 values of CRRtx,15 were obtained from the CRRtx  Nliq curves and the following equation was fitted using nonlinear regression to estimate CRRtx at 15 uniform cycles as a function of xR: CRRtx,15 ¼ 0:777ð0:2xR Þ4:47 þ0:139

Fig. 2. Cyclic resistance ratio versus the number of cycles required for liquefaction initiation for the specimens of Mai Liao sand at various percentages of non-plastic fines content (FC), relative densities, and void ratios, Huang et al. [13] cyclic triaxial tests data.

ð3Þ

The value of Q was considered to be 10 for all sands except for Ottawa sand that its measured values of Q were reported by Carraro et al. [12]. Fig. 3 shows the variations of the CRRtx,15 values versus their corresponding xR and the fitted curve (Eq. (3)). According to the figure, cyclic resistance ratio decreases with increase in xR while CRRtx,15 tends to be somewhat insensitive to the increment of xR. Most of the laboratory cases in Fig. 3 are in the range of  0.7 o xR o0.1 and the proposed relationship is only limited to the xR values lower than 0.2. Determination of Nliq depends on liquefaction onset criterion, which may be either excess pore pressure ratio (ru) of unity (e.g., [36]) or double amplitude axial strain (eDA) of a certain value (e.g., [12]). Fig. 4 illustrates that there is a clear separation between the data recorded by the first criterion (e.g., Polito’s [36] data) and those recorded by the second one. Accordingly, the data recorded with second criterion (double amplitude axial strain of 5%) were

Fig. 3. Cyclic resistance ratio in triaxial test condition as a function of relative state parameter index, cyclic triaxial tests data are from Carraro et al. [12], Huang et al. [13], Vaid and Sivathayalan [24], Kanagalingam [38], Zhou and Chen [39], Silver et al. [40], and Amini and Qi [11.]

Y. Jafarian et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 1061–1072

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Fig. 4. Segregation between the CRRtx data obtained from ru ¼1 and eDA ¼5% failure criteria.

Table 1 also presents the sample preparation method used for each dataset. The majority of the researchers mentioned in this table employed ‘‘moisture tamping’’ method for sample preparation, except for Zhou and Chen [39] and Vaid and Sivathayalan [24] who used ‘‘water sedimentation’’ method. Although the effect of sample preparation could be a function of soil type (e.g., [16]), Amini and Qi [11] and Amini and Sama [10] showed that there is no significant difference between the CRRtx values obtained by ‘‘moisture tamping’’ and ‘‘water sedimentation’’ methods in triaxial test condition. Their finding may be in accordance with the conclusion of Ishihara [42] indicating that sample preparation method affects liquefaction resistance, but the most significant difference exists between ‘‘moisture tamping’’ and ‘‘air pluviation’’ (or ‘‘dry pouring’’). Therefore, the influence of sample preparation for the specimens reconstituted by ‘‘moisture tamping’’ and ‘‘water sedimentation’’ methods is negligible and hence ignored in this study. The fitted curve shown in Fig. 3 approves this suggestion and provides sufficient robustness. 2.3. Effects of non-plastic fines content on CRRtx,15  xR relation The influence of silts content on CRRtx,15  xR relationship needs to be investigated because the data shown in Table 1 involve samples containing up to 15% silts. The presence of non-plastic fines may decrease the interlocking in between sand particles, their dilation propensity, and the resulting cyclic resistance. The majority of available laboratory results at a constant void ratio indicate that CRRtx,15 decreases with increase in silts content up to a specific amount known as ‘‘limiting fines content’’ and then increases. Although there is not a consensus among researchers regarding the exact value of limiting fines content [12,13,37,43,44,61], Koester [45] and Troncoso [46] concluded that limiting silts content is between 20% and 30%. In the current study, silts content is limited up to 15%, which is conservatively below the limiting fines content. Fig. 5(a) illustrates variations of the CRRtx,15 values obtained from Huang et al. [13] data at the constant void ratios and initial effective confining pressure (100 kPa) versus their corresponding xR values, while the percentages of silts fraction are shown adjacent to the spots. The figure confirms previous laboratory studies indicating that cyclic resistance decreases with increase in silts up to limiting fines content at the constant void ratio. On the other hand, Fig. 5(b) reveals that relative density decreases with increase in fines content at the constant void ratios. It means that

Fig. 5. Influences of increasing non-plastic fines content up to 15% on the: (a) variations of CRRtx versus xR at constant void ratios while the values indicated on the chart are silts fraction percentage; (b) variations of relative density against silts content at constant void ratios; and (c) variations of CRRtx versus xR at constant silts content.

the interactive effects of relative density (Dr), void ratio (e), and non-plastic fines content (FC) on the CRRtx,15 values have to be investigated concurrently. Note that the relative density values used in the current study are gross relative densities of mixtures, while skeletal relative density may lead to different results. Fig. 5(c) shows the variations of CRRtx,15 versus xR at the constant values of silts content of 0%, 5%, 10%, and 15% for the data of Huang et al. [13] and Carraro et al. [12] at the constant initial effective confining pressure of 100 kPa. It is clearly seen that there is no any apparent trend denoting the dependency of CRRtx,15  xR relationship to silts content. It means that the decreasing trend of CRRtx,15 against xR in Fig. 5(a) is mainly affected by the reduction

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of relative density that happens due to increase in silts content at the constant void ratios (Fig. 5b). Thus, the influence of increase in silts content up to 15% is inherently considered by xR due to its proportionality to relative density. This finding verifies the conclusion of Been and Jefferies [25] indicating that state parameter inherently takes the effects of fines content into account.

3. Evaluating the adjusted CRRtx,15  nR relation in field condition 3.1. Adjusting CRRtx,15  xR relation There are some uncertainties in determining field cyclic resistance ratio (CRRfield) from triaxial cyclic resistance ratio (CRRtx). The values of CRRtx for soils with high penetration resistance (e.g., NSPT ¼30–40) were shown to be conservative when they were compared with field observations [6]. However, the CRRtx,15  xR relationship is expected to be consistent with field data, if the required modifications are applied. The cyclic resistance ratio obtained from simple shear test (CRRss) has enough consistency with actual resistance in field condition. Field cyclic resistance ratio (CRRfield) is defined as the ratio of cyclic shear stress tcyc to initial effective overburden stress s0v0 . In addition, CRRtx has been defined as the ratio of maximum cyclic shear stress tmax (or half of the maximum cyclic deviatoric stress¼ sdev/2) to initial effective confining pressure (s03 ). The common relationship that correlates CRRtx, CRRss, and CRRfield for the same number of stress cycles is       tcyc tcyc sdev  0:9 0  0:9Cr ð4Þ 0 0 sv0 field sv0 ss 2s3 tx The reduction factor of 0.9 denotes on the findings of Seed et al. [19] who concluded that the pore water pressure generated under multi-directional loading in field conditions is greater than the one generated under unidirectional cyclic loading in simple shear test. Several researchers such as Finn et al. [20], Seed and Peacock [21], and Castro [22] proposed various constant Cr values per any given K0, as seen in Table 2. The proposed Cr coefficients convert CRRtx to the CRRss values of the same material at the same Dr and s0v0 . Vaid and Sivathayalan [24] found that Cr depends on initial relative density and initial effective confining pressure (Fig. 6) and indicated that constant Cr may cause over-conservative designs. The isotropic triaxial tests data presented by Vaid and Sivathayalan [24] for Frazer Delta Sand have been re-evaluated to obtain Cr values as a function of xR for Q¼10. This is an extension to the work of Vaid and Sivathayalan [24] since xR is an appropriate representative of Dr and s0v0 . Thus, the Cr values shown in Fig. 6 were fitted to their corresponding xR values and hereafter are called as Cr(VS): 2

Cr ðVSÞ ¼ 0:534xR þ 0:095xR þ 0:814,

0:6 r xR r 0

ð5Þ

Note that Eq. (5) was derived in order to examine and compare the results of Vaid and Sivathayalan’s [24] study with other proposed Cr values. Fig. 7 shows the data points reported in Vaid Table 2 Various recommended Cr values (after Seed [15]). Reference

Equations

Finn et al. [20] Seed and Peacock [21] Castro [22]

Cr ¼ (1+ K0)/2 – Cr ¼2(1 + 2K0)/(31.5)

Fig. 6. Effects of initial confining pressure and relative density on Cr coefficient (after Vaid and Sivathayalan [24]).

Fig. 7. Cr(VS) coefficient as a function of relative state parameter index, from Vaid and Sivathayalan’s [24] data.

and Sivathayalan [24] and the fitted curve while Cr(VS) increases with increasing xR. In order to obtain field cyclic resistance ratio from triaxial data, the CRRtx,15  xR relationship (Eq. (3)) can be multiplied by 0.9 and also any Cr coefficient presented in Table 2. The obtained field cyclic resistance ratio is hereafter referred to as CRRfield,7.5 because it stands for the liquefaction resistance at Mw ¼7.5, which is corresponding to 15 uniform cycles. Accordingly, the lower bound of the Cr range proposed by Seed and Peacock [21] for K0 ¼0.4, which is a reasonable value for normally consolidated deposits [15], leads to the following relationship for field cyclic resistance ratio: CRRfield,7:5 ¼ 0:385ð0:2xR Þ4:47 þ 0:069

ð6Þ

In addition, the following equation is obtained when the CRRtx,15  xR relationship (Eq. (3)) is multiplied by 0.9 and Cr(VS) (i.e. Eq. (5)): 2

CRRfield,7:5 ¼ 0:9½0:534xR þ 0:095xR þ 0:814½0:777ð0:2xR Þ4:47 þ 0:139 Cr for K0 ¼ 0.4

Cr for K0 ¼ 1

0.7 0.55–0.72 0.69

1.0 1.0 1.15

ð7Þ Although Eqs. (6) and (7) can adjust for the difference of initial in situ stress between field and triaxial tests, however, they might not be able to account for the inherent characteristic of soil in field conditions. This is mainly because CRRtx,15  xR relationship

Y. Jafarian et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 1061–1072

has been obtained from the results of tests on freshly reconstituted samples which do not include field associated factors. The field performance of these equations will be examined in the subsequent section. 3.2. Applying CRRfield,7.5  xR relationships to field liquefaction case histories Over the past decades, several researchers have tried to correlate field penetration resistance (e.g., SPT and CPT resistances) to the relative density of granular soils. The common form of the relationship between N1,60 and Dr is N1,60 ¼ Cd  Dr 2

ð8Þ

Skempton [48] recommended a Cd value of 44 for the relative density values between 30% and 90%. Cubrinovski and Ishihara [47] proposed a more comprehensive recommendation for Cd and demonstrated its dependence on the basic properties of soils. Idriss and Boulanger [4] indicated that considering Cd ¼46 for clean sands can be more realistic because it obtains a relative density of 81% for a corrected SPT blow counts of 30 (N1,60 ¼30). In the present study, Skempton’s [48] recommendation is used while it obtains the Dr value of 80% at N1,60 ¼ 30. Using Skempton’s [48] recommendation, Eq. (2) is rearranged in terms of N1,60: rffiffiffiffiffiffiffiffiffiffiffi 1 N1,60  xR ¼ ð9Þ Q lnð100p0 =Pa Þ 44 Boulanger [49] and Idriss and Boulanger [4] summarized the studies of Salgado et al. [50,51] on CPT and proposed the following equation to obtain Dr from the corrected values of CPT tip resistance (qC1N) for clean sands: Dr ¼ 0:478qC1N 0:264 1:063

ð10Þ

This relationship results in a relative density of about 80% at the limiting value of qC1N of 175 (i.e. qC1NðlimÞ ¼ 175). Eq. (10) was proposed for clean sands that can quickly dissipate the excess pore water pressure developed during sounding. However, Carraro et al.’s [12] study on calibration chambers shows that the sounding procedure of cone remains in drained condition even in the sand–silt mixtures having up to 15% silts content. Therefore, the application of Eq. (10) can be generalized to silty sands containing up to 15% silts. The following equation is obtained for field CPT data by substituting Eq. (10)

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into Eq. (2)

xR ¼

1 0:478qC1N 0:264 þ 1:063 Q lnð100p0 =Pa Þ

ð11Þ

Cetin et al. [5] and Moss et al. [7] presented two different liquefaction case history catalogs based on SPT and CPT liquefaction data, respectively. They performed a reasonable procedure to classify numerous case histories based on their qualities and compiled the final databases (201 SPT and 188 CPT data) with the most qualified data. In the current study, only the cases with fines content up to 15% (i.e., 202 data) have been selected among the total SPT and CPT based liquefaction data. A negligible error reveals since the fines content is assumed to be entirely non-plastic. Boulanger and Idriss [52] indicated that mixed soils with plasticity indices less than 7 (i.e. PIo7) exhibit sand-like behavior. Therefore, it is reasonable to expect that sand samples containing less than 15% fines exhibit sand-like behavior even though they contain a small fraction of clay. Since the liquefaction case histories have experienced earthquakes with different magnitudes, correction of equivalent uniform cyclic stress ratio (CSReq) for duration or the number of equivalent cycles to CSReq is necessary and defined by the following equation, according to Cetin et al. [5]: CSReq ¼ CSReq =DWFm

ð12Þ

where DWFm is the duration weighting factor (or magnitude scaling factor) that is required to account for the random nature of earthquake excitation and CSReq the equivalent uniform cyclic stress ratio, according to Cetin et al. [5]. The fact that cyclic resistance ratio reduces with increase in initial effective overburden pressure has been confirmed in laboratory and field conditions. This is a manifestation of critical state type of behavior. In fact, the suppression of dilatancy at increased effective stress decreases cyclic resistance ratio. Researchers has introduced an adjustment factor (Ks) to account for this effect of overburden pressure. However, this is not the case in the present study since xR parameter originates from critical state concept and inherently considers the suppression of dilatancy at increased effective stress. The values of relative state parameter were computed for the field SPT and CPT data using Eqs. (9) and (11) and the resulting values have been shown against CSReq in Fig. 8. This figure illustrates a reasonable classification between liquefied and

Fig. 8. Applying CRRtx–xR relation to field data and illustrating the influence of the previously proposed Cr coefficients.

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non-liquefied cases with a considerable overlapping (probabilistic) zone. The CRRtx,15  xR relationship that was derived from triaxial data (Eq. (3)) was superimposed on the liquefaction case histories data. It is observed that the CRRtx,15  xR curve denotes on the upper boundary of the probable zone with a reasonable accuracy. All the non-liquefied cases (except one case) have been clustered below this boundary. Reasonable agreement between laboratory studies and field observations is observed since Fig. 3 (i.e. laboratory data) and Fig. 8 (i.e. field data) concurrently indicate the absence of liquefied cases for the xR values lower than about 0.7. The CRRfield,7.5  xR curves adjusted by the Cr coefficients cited in Table 2 are also shown in Fig. 8. This figure demonstrates the excellent accuracy of the field curve (Eq. (6)) obtained by the lower bound of Seed and Peacock’s [21] recommendation for the xR values larger than 0.7 (i.e. its domain). Their recommendation converts the CRRtx,15  xR relationship in such a way that it gives the lower bound of the overlapping region for xR 4  0.7. Also, this boundary tends to be over-conservative for the more dilative cases corresponding to smaller xR. Therefore, the adjusted CRRtx,15  xR curves cannot be extrapolated towards the xR values lower than about 0.7. This over-conservatism, which is observed in CRRtx,15  xR and other adjusted curves, is in accordance with the suggestion of Robertson and Wride [6] indicating that triaxial test on sand with a high SPT  N value obtains conservative cyclic resistance ratio. Thus, the constant Cr coefficients that only correct for initial stress condition and ignore field associated factors such as aging, cementation, and overconsolidation may lead to questionable field cyclic resistance, especially for more dilative cases (i.e., lower values of xR), which are further affected by the mentioned factors. As discussed previously, majority of studies recommended constant coefficients for adjusting CRRtx. Vaid and Sivathayalan [24] may be the first researchers who showed the dependency of Cr on relative density and initial effective confining pressure. According to Fig. 8, the CRRfield,7.5  xR relationship obtained by Cr(VS) (Eq. (7)) provides reasonable classification for contractive cases and tends to be over-conservative with increase in dilation propensity. In fact, this figure confirms the success of the Vaid and Sivathayalan’s [24] Cr(VS) only at its applicable domain, i.e. 0.56o xR o0 (refer to Fig. 7) and demonstrates how its extrapolation leads to impractical and over-conservative field resistance. It is worthy of note that Vaid and Sivathayalan [24] plotted a boundary curve derived from their cyclic simple shear test data along with the field boundary curve proposed by Seed et al. [3] for clean sands. They observed that their laboratorybased boundary falls above the field boundary at low penetration resistances while it becomes more conservative than the field boundary for normalized SPT blow counts, N1,60, greater than about 10. The trend of the CRRfield,7.5  xR relationship obtained by Cr(VS) is in agreement with the mentioned difference between the field and simple shear resistances.

CRRtx values, which are obtained from triaxial tests on freshly reconstituted samples. Triaxial-to-field adjustment coefficient, Ct  f, is defined as the ratio of field to triaxial cyclic resistance ratios and can be compared with any given Cr multiplied by 0.9. Since Ct  f is directly obtained with the contribution of CRRfield,7.5 and triaxial cyclic resistance of freshly reconstituted samples (i.e. CRRtx,15), it can potentially correct CRRtx,15 with regard to stress condition, as well as other deficiencies of testing on reconstituted samples such as aging, fabric, and prior strain history. In order to obtain Ct–f, field cyclic resistance has to be defined as a function of xR because CRRtx,15 was obtained in terms of this parameter.

4.1. Evaluation of existing field limit state boundaries The existing SPT and CPT based limit state boundary curves can be mapped onto CRR–xR plane (Fig. 9) to be used as CRRfield and to calculate the corresponding Ct–f. As Idriss and Boulanger [53] indicated, it is anticipated that empirical limit state curves such as those proposed by Cetin et al. [5] for SPT data and Moss et al. [7] for CPT data provide reasonable agreement with the field data plotted on CSReq xR chart. The deterministic boundary curves of clean sands proposed by Cetin et al. [5] and Moss et al. [7] that are the most recent deterministic criteria for SPT and CPT data have been plotted on the CSReq xR chart (Fig. 9). Fig. 9 also shows a xR-based boundary curve of clean sands proposed by Idriss and Boulanger [4] who mapped two simplified SPT and CPT based criteria onto CRR  xR plane to derive this unique boundary. Another boundary curve illustrated in this figure is the one obtained from the CRRfield,7.5  xR relationship using Seed and Peacock’s [21] Cr value (Eq. (6)). These comparisons reveal that Moss et al.’s [7] deterministic boundary properly classifies CPT data, but it cannot be used as a reliable CRRfield,7.5 measure for the purpose of the current study. That is because several liquefied cases are located below this curve. In contrast, Cetin et al.’s [5] curve properly bounds liquefied data; however, it appears to be very conservative for xR values greater than 0.3. Moreover, there are several liquefied cases below the Idriss and Boulanger’s [4] boundary. Therefore, the need to obtain a new CRRfield,7.5 in terms of xR is sensible since the examined deterministic boundaries

4. Semi-empirical triaxial to field adjustment coefficient (Ct  f) The adjustment coefficients proposed by previous researchers (including constant Cr values and Cr(VS)) are unable to introduce a uniform level of liquefaction risk into the adjusted field liquefaction resistance. However with the lower bound value of Seed and Peacock’s [21] recommendation and Cr(VS), reasonable limit state curve has been obtained. Besides, all constant Cr and Cr(VS) coefficients provide field resistance curves that their extrapolations towards lower xR values lead to over-conservative field resistance measures. Therefore, a reliable adjustment coefficient with a specified level of liquefaction risk is required to correct

Fig. 9. Comparing various SPT and CPT based field liquefaction criteria and the proposed probabilistic xR-based criterion for 20% probability of initiation in xR CSReq chart.

Y. Jafarian et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 1061–1072

cannot guarantee a uniform level of liquefaction risk in CRRfield,7.5  xR plane. 4.2. Probabilistic xR-based field boundary using logistic regression Logistic regression can provide a suitable framework for the probabilistic evaluation of liquefaction potential in terms of relative state parameter index. Liao et al. [54] initially applied logistic regression to consider the uncertainties involved in deterministic criteria and to estimate the likelihood of liquefaction triggering. The scope of logistic regression is to establish an expression for conditional probability of liquefaction (PL) as a function of explanatory variables (X), which are factors that affect the occurrence of liquefaction. The components of explanatory variables vector (X) must be normally distributed and independent of each other. The PL function is derived from binary or dichotomous regression analyses because each case in liquefaction catalog is presented by a binary variable, which indicates whether or not liquefaction will occur. The probability function that should be fitted by employing field observation data can be defined as follows [54,55]: PL ¼

exp½b0 þ b1 x1 þ    þ bn xn  ¼ 1 þexp½b0 þ b1 x1 þ    þ bn xn 

"

1

1 þ exp  b0 þ

n P

1069

(containing less than 15% silt) in terms of xR parameter. This xRbased probabilistic criterion has been originated from critical state concept and considers SPT and CPT resistances together with effective overburden pressure. According to Fig. 10, the boundary curve representing 20% probability of liquefaction is sufficiently conservative and is suggested to be considered as a deterministic boundary curve that guarantees required safeties. Fig. 9 shows that the xR-based boundary curve at 20% liquefaction probability is close to Cetin et al.’s [5] deterministic boundary and CRRfield,7.5  xR relationship obtained by Seed and Peacock’s [21] Cr (i.e. Eq. (6)) at the low and high xR values, respectively. The conventional factor of safety (Fs) is considered as a reliability index (e.g., [58,59]) to obtain a relationship for estimating the probability of liquefaction occurrence at a given factor of safety. The factor of safety against liquefaction is obtained via dividing cyclic resistance ratio at 20% liquefaction probability CRRxR ,20% by CSReq Fig. 11 illustrates the probability of

!#

bi xi

i¼1

ð13Þ where PL is the likelihood of liquefaction occurrence and 0 rPL r1, n the total number of explanatory variables, X¼[x1, x2,y,xn] the vector of explanatory variables, b ¼[b0, b1,y,bn] the regression coefficients that are determined from logit analysis. Maximum likelihood of estimation is utilized to determine the vector of regression coefficients, b ¼[b0, b1,y,bn], by fitting the probability function to field observation data. The number of liquefied cases is more than non-liquefied cases in the field database and may affect the result by producing an undesirable bias in logistic regression. According to Mayfield [56], Moss et al. [7], and Cetin et al. [5], this bias is reduced by using a prior probability assigned to each liquefied or non-liquefied class such as the proportion of class’s population in the database. More details about the logistic regression can be found in Liao [57] and Liao et al. [54]. The experiences gathered during previous studies [5,7,54] reveal that lnðCSReq Þ is more naturally distributed than CSReq . Thus, in this study, earthquake magnitude (Mw), relative state parameter index (xR), and natural logarithm of CSReq , lnðCSReq Þ, are selected as explanatory variables and the following expression is obtained by fitting PL function to the 202 SPT and CPT data: PL ¼

Fig. 10. Probabilistic xR-based liquefaction criterion developed by logistic regression on SPT and CPT data, only for clean and silty sands containing up to 15% silts.

1 n o 3 1 þexp ½2:562 þ 1:835Mw þ 4:704lnðCSReq Þ þ 26:393xR  ð14Þ

Moreover, the xR-based cyclic resistance ratio for a given liquefaction probability is expressed as !   3 ln ð1=PL Þ1 þ1:835Mw þ 26:393xR 2:562 CRRxR ¼ exp ð15Þ 4:704 where CRRxR stands for cyclic resistance ratio at the various levels of liquefaction risk. Fig. 10 shows five family curves that are produced by Eq. (15) denoting on the contours of equivalent liquefaction probability for Mw ¼7.5. In contrast to the limit state curves obtained from deterministic approach, any probabilistic curve individually reflects a uniform level of risk. Eq. (15) can be used to evaluate probabilistic liquefaction potential of sands and silty sands

Fig. 11. Factor of safety versus xR-based liquefaction probability and comparison between the fitted curve and two previous FS  PL recommendations.

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liquefaction obtained from Eq. (14) versus the conventional factor of safety for all the SPT and CPT based cases. The following relationship is derived by nonlinear regression and Fig. 11 shows how it fits the field data: PL ¼

1 1þ 5:13Fs4:735

ð16Þ

Fig. 11 also compares Eq. (16) with the relationships proposed by Juang et al. [60] for SPT based data and Lai et al. [59] for CPT based data. Note that these researchers considered PL ¼50% as liquefaction triggering (i.e. Fs ¼1), as seen in Fig. 11.

4.3. The proposed probabilistic triaxial-to-field adjustment coefficient In order to obtain a probabilistic triaxial-to-field adjustment coefficient (Ct  f, PL), Eq. (15) is set for Mw ¼7.5 and is divided by the CRRtx,15  xR relationship to obtain the following equation:

Ctf ,PL ¼

    3 exp -0:213 ln ð1=PL Þ1 5:611xR 2:381 0:777ð0:2xR Þ4:47 þ0:139

ð17Þ

Eq. (17) can be employed to convert triaxial based cyclic resistance, CRRtx,15, to field resistance for a given level of liquefaction probability and soil xR. This relationship is best fitted for the earthquake magnitude of 7.5 because both field and laboratory based liquefaction resistances have been derived for Mw ¼7.5 and 15 uniform cycles, respectively. For the sites that are vulnerable to the strikes of other magnitudes, Eq. (17) can be directly used to convert the triaxial cyclic resistance ratio recorded at 15 uniform cycles (i.e., CRRtx,15) to an equivalent field resistance ratio (i.e., CRRfield,7.5). Since CRRfield,7.5 denotes the earthquake magnitude of 7.5, a magnitude scaling factor (e.g., [4,5,41]) can be applied in order to obtain cyclic resistance ratio for the given earthquake magnitude (i.e., CRRfield). Fig. 12 illustrates the probabilistic family curves of the Ct  f, PL (Eq. (17)) versus the corresponding xR values for various likelihoods of liquefaction initiation. According to Figs. 9 and 10, the field curve of 20% liquefaction probability has sufficient conservatism to be proposed as a deterministic xR-based field curve. Thus, PL ¼20% is substituted in Eq. (17) to obtain a conservative deterministic equation for Ct  f, which is equivalent

Fig. 13. Comparing the Cr coefficients recommended by various researchers multiplied by 0.9 and the proposed semi-empirical Ct  f coefficient obtained from composing the CRRtx  xR relation and the probabilistic CRRfield  xR relation at 20% probability of liquefaction.

to Ct  f,

PL

at 20% liquefaction probability: 3

Ctf ¼

expð2:6765:61xR Þ 0:777ð0:2xR Þ

4:47

þ0:139

ð18Þ

Fig. 13 demonstrates how Ct  f varies against relative state parameter index. Also, the previous recommendations of constant Cr values together with the Cr(VS) relationship obtained from Vaid and Sivathayalan’s [24] data (i.e., Eq. (5)) were multiplied by 0.9 and plotted in this figure. Note that the applicable domain of Ct  f equation is  0.7 o xR o0.1 because CRRtx,15 was defined in this range (Fig. 3). The figure shows that Ct  f varies between 0.45 and 0.76 versus xR variations. The proposed Ct–f locates within the range proposed by Seed and Peacock [21] only at 0.66o xR o  0.55 while it corresponds with the lower bound of this range for xR 4 0.02. It is reasonable to conclude that the Cr values proposed by previous researchers is over-conservative at xR o  0.66 and lead to unsafe results at the wide range of  0.55o xR o  0.02. The incremental trend of Ct–f for xR o  0.4 is in agreement with the conclusion of Robertson and Wride [6] who indicated that cyclic triaxial testing on the soil samples with high penetration resistance or relative density obtains conservative results.

5. Summary and conclusions

Fig. 12. Variations of the triaxial-to-field adjustment coefficient versus xR for any given likelihood of liquefaction triggering.

Relative state parameter index (xR) has been found to be a useful parameter to link triaxial-based and field-based cyclic resistance ratios of sands. Numerous results of the cyclic triaxial tests conducted on freshly reconstituted samples have been collected from the literature and a relationship has been established between triaxial cyclic resistance ratio at 15 uniform stress cycles (CRRtx,15) and xR. The CRRtx,15  xR relationship has been modified by the Cr coefficients proposed by some researchers to correct for the difference between the stress conditions of triaxial test and field. The modified CRRtx,15  xR relationship has been applied to field liquefaction case histories consisting of both SPT and CPT based data. It has been observed that the CRRtx,15  xR relationships modified by the lower bound of the Cr range

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proposed by Seed and Peacock [21] (for K0 ¼0.4) and the adjustment coefficient extracted from Vaid and Sivathayalan’s [24] study reasonably classify liquefied and non-liquefied case histories for contractive soils; however, they tend to be overconservative for dilative soils. This finding shows that the adjustment coefficients that only account for stress conditions cannot consider all differences between field cyclic resistance and the cyclic resistance obtained from triaxial tests on freshly reconstituted samples. This difference is significant for dilative soils that are more affected by the influences of the inherent characteristics of soil in field conditions such as aging, fabric, cementation, and prior strain history. Logistic regression approach has been employed to obtain a probabilistic CRRfield  xR relationship that is consistent with both SPT and CPT data. The probabilistic xR-based CRRfield  xR relationship yields boundary curves corresponding to the given probabilities of liquefaction initiation. It has been observed that the boundary curve corresponding to 20% liquefaction probability involves sufficient conservatism to be considered as a deterministic boundary curve in terms of xR. Considering factor of safety against liquefaction as a reliability index, a robust correlation has been proposed to relate factor of safety and liquefaction probability. This could be useful for evaluating the level of risk for deterministic methods that only consider factor of safety as failure criterion. Finally, the xR-based relationships of field and triaxial cyclic resistances have been composed to obtain a probabilistic triaxial-to-field adjustment coefficient. The boundary curve corresponding to 20% liquefaction probability has been utilized to propose the most conservative semi-empirical adjustment coefficient that is useful to convert the triaxial cyclic resistance ratio of freshly reconstituted sand samples to field resistance. Variations of the proposed Ct  f coefficient versus xR have been compared with previous recommendations. It has been concluded that the previous recommendations of Cr obtain either unsafe or overconservative field resistance depends on the dilation propensity of soil and also the agreed level of liquefaction risk. The most important advantage of the proposed triaxial-to-field adjustment coefficient over the previous Cr is its capability in considering relative density and initial effective confining pressure of soil as well as the likelihood of liquefaction initiation. Furthermore, it can consider field associated features and reduce the errors of sample disturbance in triaxial testing on reconstituted specimens because it has been obtained using direct correlation between triaxial and field cyclic resistances. The results of this study are limited to the silty sands having less than 15% silts content and stress-density state that falls within the range of 0.7 o xR o0.1.

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